diff --git a/dev/.documenter-siteinfo.json b/dev/.documenter-siteinfo.json index 5d6e587..93a1040 100644 --- a/dev/.documenter-siteinfo.json +++ b/dev/.documenter-siteinfo.json @@ -1 +1 @@ -{"documenter":{"julia_version":"1.9.4","generation_timestamp":"2023-11-29T21:33:28","documenter_version":"1.2.0"}} \ No newline at end of file +{"documenter":{"julia_version":"1.9.4","generation_timestamp":"2023-11-29T23:37:31","documenter_version":"1.2.0"}} \ No newline at end of file diff --git a/dev/HarmonicOscillator/index.html b/dev/HarmonicOscillator/index.html index 8e82bb7..5edf292 100644 --- a/dev/HarmonicOscillator/index.html +++ b/dev/HarmonicOscillator/index.html @@ -42,7 +42,7 @@ plot!(x -> HO.ψ(x, n=2), label="n=2", lw=2) plot!(x -> HO.ψ(x, n=3), label="n=3", lw=2) plot!(x -> HO.ψ(x, n=4), label="n=4", lw=2)

Potential energy curve, Energy levels, Wave functions:

using Plots
-plot(xlim=(-5.5,5.5), ylim=(-0.2,5.4), xlabel="\$x\$", ylabel="\$V(x),~E_n,~\\psi_n(x)\\times0.5+E_n\$", size=(480,400), dpi=400)
+plot(xlim=(-5.5,5.5), ylim=(-0.2,5.4), xlabel="\$x\$", ylabel="\$V(x),~E_n,~\\psi_n(x)\\times0.5+E_n\$", size=(480,400), dpi=300)
 for n in 0:4
   # energy
   hline!([HO.E(n=n)], lc=:black, ls=:dash, label="")
@@ -57,7 +57,7 @@
     &= n! \sum_{m=0}^{\lfloor n/2 \rfloor} \frac{(-1)^m}{m! (n-2m)!}(2 x)^{n-2m}.
   \end{aligned}\]
$n=0:$ ✔
\[\begin{aligned}
   H_{0}(x)
-   = e^{ - x^{2}} e^{x^{2}}
+   = e^{x^{2}} e^{ - x^{2}}
   &= 1 \\
   &= 1
 \end{aligned}\]
$n=1:$ ✔
\[\begin{aligned}
@@ -70,10 +70,10 @@
    = e^{x^{2}} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d} e^{ - x^{2}}}{\mathrm{d}x}
   &= -2 + 4 x^{2} \\
   &= -2 + 4 x^{2}
-\end{aligned}\]
$n=3:$ ✔
\[\begin{aligned}
+\end{aligned}\]
$n=3:$ ✗ Hₙ(x) = (-1)ⁿ exp(x²) dⁿ/dxⁿ exp(-x²) = ...: Test Failed at C:\Users\user\Desktop\GitHub\Antiq.jl\test\HarmonicOscillator.jl:43   Expression: acceptance
Stacktrace:  [1] macro expansion    @ C:\Users\user.julia\juliaup\julia-1.9.2+0.x64.w64.mingw32\share\julia\stdlib\v1.9\Test\src\Test.jl:478 [inlined]  [2] macro expansion    @ C:\Users\user\Desktop\GitHub\Antiq.jl\test\HarmonicOscillator.jl:43 [inlined]  [3] macro expansion    @ C:\Users\user.julia\juliaup\julia-1.9.2+0.x64.w64.mingw32\share\julia\stdlib\v1.9\Test\src\Test.jl:1498 [inlined]  [4] top-level scope    @ C:\Users\user\Desktop\GitHub\Antiq.jl\test\HarmonicOscillator.jl:28
\[\begin{aligned}
   H_{3}(x)
-   =  - \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d} e^{ - x^{2}}}{\mathrm{d}x} e^{x^{2}}
-  &=  - 12 x + 8 x^{3} \\
+   =  - e^{x^{2}} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d} e^{ - x^{2}}}{\mathrm{d}x}
+  &= 8 x^{3} - 12 x \\
   &=  - 12 x + 8 x^{3}
 \end{aligned}\]
$n=4:$ ✔
\[\begin{aligned}
   H_{4}(x)
@@ -82,361 +82,30 @@
   &= 12 - 48 x^{2} + 16 x^{4}
 \end{aligned}\]
$n=5:$ ✔
\[\begin{aligned}
   H_{5}(x)
-   =  - \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d} e^{ - x^{2}}}{\mathrm{d}x} e^{x^{2}}
+   =  - e^{x^{2}} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d} e^{ - x^{2}}}{\mathrm{d}x}
   &= 120 x - 160 x^{3} + 32 x^{5} \\
   &= 120 x - 160 x^{3} + 32 x^{5}
 \end{aligned}\]
$n=6:$ ✔
\[\begin{aligned}
   H_{6}(x)
    = e^{x^{2}} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d} e^{ - x^{2}}}{\mathrm{d}x}
-  &= -120 + 720 x^{2} - 480 x^{4} + 64 x^{6} \\
-  &= -120 + 720 x^{2} - 480 x^{4} + 64 x^{6}
+  &= -120 + 720 x^{2} + 64 x^{6} - 480 x^{4} \\
+  &= -120 + 720 x^{2} + 64 x^{6} - 480 x^{4}
 \end{aligned}\]
$n=7:$ ✔
\[\begin{aligned}
   H_{7}(x)
    =  - e^{x^{2}} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d} e^{ - x^{2}}}{\mathrm{d}x}
-  &=  - 1680 x + 3360 x^{3} - 1344 x^{5} + 128 x^{7} \\
-  &=  - 1680 x + 3360 x^{3} - 1344 x^{5} + 128 x^{7}
+  &=  - 1680 x + 3360 x^{3} + 128 x^{7} - 1344 x^{5} \\
+  &=  - 1680 x + 3360 x^{3} + 128 x^{7} - 1344 x^{5}
 \end{aligned}\]
$n=8:$ ✔
\[\begin{aligned}
   H_{8}(x)
    = e^{x^{2}} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d} e^{ - x^{2}}}{\mathrm{d}x}
-  &= 1680 - 13440 x^{2} + 13440 x^{4} - 3584 x^{6} + 256 x^{8} \\
-  &= 1680 - 13440 x^{2} + 13440 x^{4} - 3584 x^{6} + 256 x^{8}
+  &= 1680 + 256 x^{8} - 13440 x^{2} - 3584 x^{6} + 13440 x^{4} \\
+  &= 1680 + 256 x^{8} - 13440 x^{2} - 3584 x^{6} + 13440 x^{4}
 \end{aligned}\]
$n=9:$ ✔
\[\begin{aligned}
   H_{9}(x)
    =  - e^{x^{2}} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d} e^{ - x^{2}}}{\mathrm{d}x}
-  &= 30240 x - 80640 x^{3} + 48384 x^{5} - 9216 x^{7} + 512 x^{9} \\
-  &= 30240 x - 80640 x^{3} + 48384 x^{5} - 9216 x^{7} + 512 x^{9}
-\end{aligned}\]
Test Summary:                               | Pass  Total   Time
-Hₙ(x) = (-1)ⁿ exp(x²) dⁿ/dxⁿ exp(-x²) = ... |   10     10  35.0s
Normalization & Orthogonality of $H_n(x)$
\[\int_{-\infty}^\infty H_j(x) H_i(x) \mathrm{e}^{-x^2} \mathrm{d}x = \sqrt{\pi} 2^j j! \delta_{ij}\]
   n	  m	numerical         	analytical        	|error|
-  0	  0	1.7724538509055137	1.7724538509055159	0.0000000000001253%	✔
-  0	  1	0.0000000000000000	0.0000000000000000	0.0000000000000000%	✔
-  0	  2	0.0000000000000001	0.0000000000000000	0.0000000000000000%	✔
-  0	  3	0.0000000000000004	0.0000000000000000	0.0000000000000000%	✔
-  0	  4	-0.0000000000000013	0.0000000000000000	0.0000000000000000%	✔
-  0	  5	-0.0000000000000007	0.0000000000000000	0.0000000000000000%	✔
-  0	  6	0.0000000000000537	0.0000000000000000	0.0000000000000000%	✔
-  0	  7	0.0000000000000026	0.0000000000000000	0.0000000000000000%	✔
-  0	  8	-0.0000000000008242	0.0000000000000000	0.0000000000000000%	✔
-  0	  9	0.0000000000001137	0.0000000000000000	0.0000000000000000%	✔
-  1	  0	0.0000000000000000	0.0000000000000000	0.0000000000000000%	✔
-  1	  1	3.5449077018108315	3.5449077018110318	0.0000000000056499%	✔
-  1	  2	0.0000000000000000	0.0000000000000000	0.0000000000000000%	✔
-  1	  3	-0.0000000000000010	0.0000000000000000	0.0000000000000000%	✔
-  1	  4	-0.0000000000000005	0.0000000000000000	0.0000000000000000%	✔
-  1	  5	0.0000000000000150	0.0000000000000000	0.0000000000000000%	✔
-  1	  6	0.0000000000000063	0.0000000000000000	0.0000000000000000%	✔
-  1	  7	-0.0000000000000749	0.0000000000000000	0.0000000000000000%	✔
-  1	  8	-0.0000000000000568	0.0000000000000000	0.0000000000000000%	✔
-  1	  9	0.0000000000143249	0.0000000000000000	0.0000000000000000%	✔
-  2	  0	0.0000000000000001	0.0000000000000000	0.0000000000000000%	✔
-  2	  1	0.0000000000000000	0.0000000000000000	0.0000000000000000%	✔
-  2	  2	14.1796308072441306	14.1796308072441271	0.0000000000000251%	✔
-  2	  3	-0.0000000000000001	0.0000000000000000	0.0000000000000000%	✔
-  2	  4	-0.0000000000000105	0.0000000000000000	0.0000000000000000%	✔
-  2	  5	0.0000000000000042	0.0000000000000000	0.0000000000000000%	✔
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-  3	  3	85.0777848434647694	85.0777848434647694	0.0000000000000000%	✔
-  3	  4	-0.0000000000000009	0.0000000000000000	0.0000000000000000%	✔
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-  4	  4	680.6222787477178144	680.6222787477181555	0.0000000000000501%	✔
-  4	  5	-0.0000000000000142	0.0000000000000000	0.0000000000000000%	✔
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-  5	  5	6806.2227874771797360	6806.2227874771806455	0.0000000000000134%	✔
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-  6	  1	0.0000000000000063	0.0000000000000000	0.0000000000000000%	✔
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-  7	  1	-0.0000000000000749	0.0000000000000000	0.0000000000000000%	✔
-  7	  2	0.0000000000000000	0.0000000000000000	0.0000000000000000%	✔
-  7	  3	-0.0000000000004121	0.0000000000000000	0.0000000000000000%	✔
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-  9	  3	0.0000000001391527	0.0000000000000000	0.0000000000000000%	✔
-  9	  4	0.0000000000000000	0.0000000000000000	0.0000000000000000%	✔
-  9	  5	0.0000000013387762	0.0000000000000000	0.0000000000000000%	✔
-  9	  6	-0.0000000000873115	0.0000000000000000	0.0000000000000000%	✔
-  9	  7	0.0000000116493834	0.0000000000000000	0.0000000000000000%	✔
-  9	  8	0.0000000016298145	0.0000000000000000	0.0000000000000000%	✔
-  9	  9	329312283.3492956757545471	329312283.3492959141731262	0.0000000000000724%	✔
-Test Summary:                     | Pass  Total  Time
-∫Hⱼ(x)Hᵢ(x)exp(-x²)dx = √π2ʲj!δᵢⱼ |  100    100  1.2s
Normalization & Orthogonality of $\psi_n(x)$
\[\int \psi_i^\ast(x) \psi_j(x) \mathrm{d}x = \delta_{ij}\]
  i	  j	numerical         	analytical        	|error|
-  0	  0	0.9999999999999991	1.0000000000000000	0.0000000000000888%	✔
-  0	  1	0.0000000000000000	0.0000000000000000	0.0000000000000000%	✔
-  0	  2	0.0000000000000000	0.0000000000000000	0.0000000000000000%	✔
-  0	  3	0.0000000000000000	0.0000000000000000	0.0000000000000000%	✔
-  0	  4	-0.0000000000000001	0.0000000000000000	0.0000000000000000%	✔
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-  0	  6	0.0000000000000001	0.0000000000000000	0.0000000000000000%	✔
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-  0	  8	-0.0000000000000002	0.0000000000000000	0.0000000000000000%	✔
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-  1	  0	0.0000000000000000	0.0000000000000000	0.0000000000000000%	✔
-  1	  1	0.9999999999999435	1.0000000000000000	0.0000000000056510%	✔
-  1	  2	0.0000000000000000	0.0000000000000000	0.0000000000000000%	✔
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-  2	  1	0.0000000000000000	0.0000000000000000	0.0000000000000000%	✔
-  2	  2	0.9999999999999999	1.0000000000000000	0.0000000000000111%	✔
-  2	  3	-0.0000000000000000	0.0000000000000000	0.0000000000000000%	✔
-  2	  4	-0.0000000000000001	0.0000000000000000	0.0000000000000000%	✔
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-  5	  1	0.0000000000000002	0.0000000000000000	0.0000000000000000%	✔
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-  7	  8	0.0000000000000000	0.0000000000000000	0.0000000000000000%	✔
-  7	  9	0.0000000000000006	0.0000000000000000	0.0000000000000000%	✔
-  8	  0	-0.0000000000000002	0.0000000000000000	0.0000000000000000%	✔
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-  8	  4	-0.0000000000000005	0.0000000000000000	0.0000000000000000%	✔
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-  8	  7	0.0000000000000000	0.0000000000000000	0.0000000000000000%	✔
-  8	  8	1.0000000000000007	1.0000000000000000	0.0000000000000666%	✔
-  8	  9	-0.0000000000000000	0.0000000000000000	0.0000000000000000%	✔
-  9	  0	0.0000000000000000	0.0000000000000000	0.0000000000000000%	✔
-  9	  1	0.0000000000000004	0.0000000000000000	0.0000000000000000%	✔
-  9	  2	0.0000000000000000	0.0000000000000000	0.0000000000000000%	✔
-  9	  3	0.0000000000000008	0.0000000000000000	0.0000000000000000%	✔
-  9	  4	-0.0000000000000000	0.0000000000000000	0.0000000000000000%	✔
-  9	  5	0.0000000000000008	0.0000000000000000	0.0000000000000000%	✔
-  9	  6	0.0000000000000000	0.0000000000000000	0.0000000000000000%	✔
-  9	  7	0.0000000000000006	0.0000000000000000	0.0000000000000000%	✔
-  9	  8	-0.0000000000000000	0.0000000000000000	0.0000000000000000%	✔
-  9	  9	0.9999999999999998	1.0000000000000000	0.0000000000000222%	✔
-Test Summary: | Pass  Total  Time
-<ψᵢ|ψⱼ> = δᵢⱼ |  100    100  1.2s
Virial Theorem
The virial theorem $\langle T \rangle = \langle V \rangle$ and the definition of Hamiltonian $\langle H \rangle = \langle T \rangle + \langle V \rangle$ derive $\langle H \rangle = 2 \langle V \rangle = 2 \langle T \rangle$.
\[2 \int \psi_n^\ast(x) V(x) \psi_n(x) \mathrm{d}x = E_n\]
   k	  n	numerical         	analytical        	|error|
-0.1	  0	0.4999999999999719	0.5000000000000000	0.0000000000056288%	✔
-0.1	  1	1.4999999999999998	1.5000000000000000	0.0000000000000148%	✔
-0.1	  2	2.5000000000000004	2.5000000000000000	0.0000000000000178%	✔
-0.1	  3	3.4999999999999982	3.5000000000000000	0.0000000000000508%	✔
-0.1	  4	4.4999999999999938	4.5000000000000000	0.0000000000001382%	✔
-0.1	  5	5.5000000000000000	5.5000000000000000	0.0000000000000000%	✔
-0.1	  6	6.5000000000000000	6.5000000000000000	0.0000000000000000%	✔
-0.1	  7	7.5000000000000124	7.5000000000000000	0.0000000000001658%	✔
-0.1	  8	8.4999999999999929	8.5000000000000000	0.0000000000000836%	✔
-0.1	  9	9.5000000000000000	9.5000000000000000	0.0000000000000000%	✔
-0.5	  0	0.4999999999999719	0.5000000000000000	0.0000000000056288%	✔
-0.5	  1	1.4999999999999998	1.5000000000000000	0.0000000000000148%	✔
-0.5	  2	2.5000000000000004	2.5000000000000000	0.0000000000000178%	✔
-0.5	  3	3.4999999999999982	3.5000000000000000	0.0000000000000508%	✔
-0.5	  4	4.4999999999999938	4.5000000000000000	0.0000000000001382%	✔
-0.5	  5	5.5000000000000000	5.5000000000000000	0.0000000000000000%	✔
-0.5	  6	6.5000000000000000	6.5000000000000000	0.0000000000000000%	✔
-0.5	  7	7.5000000000000124	7.5000000000000000	0.0000000000001658%	✔
-0.5	  8	8.4999999999999929	8.5000000000000000	0.0000000000000836%	✔
-0.5	  9	9.5000000000000000	9.5000000000000000	0.0000000000000000%	✔
-1.0	  0	0.4999999999999719	0.5000000000000000	0.0000000000056288%	✔
-1.0	  1	1.4999999999999998	1.5000000000000000	0.0000000000000148%	✔
-1.0	  2	2.5000000000000004	2.5000000000000000	0.0000000000000178%	✔
-1.0	  3	3.4999999999999982	3.5000000000000000	0.0000000000000508%	✔
-1.0	  4	4.4999999999999938	4.5000000000000000	0.0000000000001382%	✔
-1.0	  5	5.5000000000000000	5.5000000000000000	0.0000000000000000%	✔
-1.0	  6	6.5000000000000000	6.5000000000000000	0.0000000000000000%	✔
-1.0	  7	7.5000000000000124	7.5000000000000000	0.0000000000001658%	✔
-1.0	  8	8.4999999999999929	8.5000000000000000	0.0000000000000836%	✔
-1.0	  9	9.5000000000000000	9.5000000000000000	0.0000000000000000%	✔
-5.0	  0	0.4999999999999719	0.5000000000000000	0.0000000000056288%	✔
-5.0	  1	1.4999999999999998	1.5000000000000000	0.0000000000000148%	✔
-5.0	  2	2.5000000000000004	2.5000000000000000	0.0000000000000178%	✔
-5.0	  3	3.4999999999999982	3.5000000000000000	0.0000000000000508%	✔
-5.0	  4	4.4999999999999938	4.5000000000000000	0.0000000000001382%	✔
-5.0	  5	5.5000000000000000	5.5000000000000000	0.0000000000000000%	✔
-5.0	  6	6.5000000000000000	6.5000000000000000	0.0000000000000000%	✔
-5.0	  7	7.5000000000000124	7.5000000000000000	0.0000000000001658%	✔
-5.0	  8	8.4999999999999929	8.5000000000000000	0.0000000000000836%	✔
-5.0	  9	9.5000000000000000	9.5000000000000000	0.0000000000000000%	✔
-Test Summary:      | Pass  Total  Time
-2 × <ψₙ|V|ψₙ> = Eₙ |   40     40  1.0s
Eigen Values
\[  \begin{aligned}
-    E_n
-    &=      \int \psi^\ast_n(x) \hat{H} \psi_n(x) \mathrm{d}x \\
-    &=      \int \psi^\ast_n(x) \left[ \hat{V} + \hat{T} \right] \psi(x) \mathrm{d}x \\
-    &=      \int \psi^\ast_n(x) \left[ V(x) - \frac{\hbar^2}{2m} \frac{\mathrm{d}^{2}}{\mathrm{d} x^{2}} \right] \psi(x) \mathrm{d}x \\
-    &\simeq \int \psi^\ast_n(x) \left[ V(x)\psi(x) -\frac{\hbar^2}{2m} \frac{\psi(x+\Delta x) - 2\psi(x) + \psi(x-\Delta x)}{\Delta x^{2}} \right] \mathrm{d}x.
-  \end{aligned}\]
Where, the difference formula for the 2nd-order derivative:
\[\begin{aligned}
-  % 2\psi(x)
-  % + \frac{\mathrm{d}^{2} \psi(x)}{\mathrm{d} x^{2}} \Delta x^{2}
-  % + O\left(\Delta x^{4}\right)
-  % &=
-  % \psi(x+\Delta x)
-  % + \psi(x-\Delta x)
-  % \\
-  % \frac{\mathrm{d}^{2} \psi(x)}{\mathrm{d} x^{2}} \Delta x^{2}
-  % &=
-  % \psi(x+\Delta x)
-  % - 2\psi(x)
-  % + \psi(x-\Delta x)
-  % - O\left(\Delta x^{4}\right)
-  % \\
-  % \frac{\mathrm{d}^{2} \psi(x)}{\mathrm{d} x^{2}}
-  % &=
-  % \frac{\psi(x+\Delta x) - 2\psi(x) + \psi(x-\Delta x)}{\Delta x^{2}}
-  % - \frac{O\left(\Delta x^{4}\right)}{\Delta x^{2}}
-  % \\
-  \frac{\mathrm{d}^{2} \psi(x)}{\mathrm{d} x^{2}}
-  &=
-  \frac{\psi(x+\Delta x) - 2\psi(x) + \psi(x-\Delta x)}{\Delta x^{2}}
-  + O\left(\Delta x^{2}\right)
-\end{aligned}\]
are given by the sum of 2 Taylor series:
\[\begin{aligned}
-\psi(x+\Delta x)
-&= \psi(x)
-+ \frac{\mathrm{d} \psi(x)}{\mathrm{d} x} \Delta x
-+ \frac{1}{2!} \frac{\mathrm{d}^{2} \psi(x)}{\mathrm{d} x^{2}} \Delta x^{2}
-+ \frac{1}{3!} \frac{\mathrm{d}^{3} \psi(x)}{\mathrm{d} x^{3}} \Delta x^{3}
-+ O\left(\Delta x^{4}\right),
-\\
-\psi(x-\Delta x)
-&= \psi(x)
-- \frac{\mathrm{d} \psi(x)}{\mathrm{d} x} \Delta x
-+ \frac{1}{2!} \frac{\mathrm{d}^{2} \psi(x)}{\mathrm{d} x^{2}} \Delta x^{2}
-- \frac{1}{3!} \frac{\mathrm{d}^{3} \psi(x)}{\mathrm{d} x^{3}} \Delta x^{3}
-+ O\left(\Delta x^{4}\right).
-\end{aligned}\]
   k	  n	numerical         	analytical        	|error|
-0.1	  0	0.1581138798827675	0.1581138830084190	0.0000019768355921%	✔
-0.1	  1	0.4743416334097137	0.4743416490252569	0.0000032920455591%	✔
-0.1	  2	0.7905693744092002	0.7905694150420949	0.0000051396998091%	✔
-0.1	  3	1.1067971029282735	1.1067971810589328	0.0000070591668115%	✔
-0.1	  4	1.4230248189868937	1.4230249470757708	0.0000090011687654%	✔
-0.1	  5	1.7392525225056779	1.7392527130926088	0.0000109579780712%	✔
-0.1	  6	2.0554802135001529	2.0554804791094465	0.0000129220051647%	✔
-0.1	  7	2.3717078919497427	2.3717082451262845	0.0000148912305093%	✔
-0.1	  8	2.6879355581003628	2.6879360111431225	0.0000168546705678%	✔
-0.1	  9	3.0041632114497054	3.0041637771599605	0.0000188308726503%	✔
-0.5	  0	0.3535533749437250	0.3535533905932738	0.0000044263608382%	✔
-0.5	  1	1.0606600936488757	1.0606601717798214	0.0000073662562046%	✔
-0.5	  2	1.7677667498779439	1.7677669529663689	0.0000114884161979%	✔
-0.5	  3	2.4748733435564381	2.4748737341529163	0.0000157824810568%	✔
-0.5	  4	3.1819798748172214	3.1819805153394642	0.0000201296720616%	✔
-0.5	  5	3.8890863434634926	3.8890872965260117	0.0000245060716425%	✔
-0.5	  6	4.5961927496648043	4.5961940777125596	0.0000288945099547%	✔
-0.5	  7	5.3032990935191169	5.3033008588991066	0.0000332883243233%	✔
-0.5	  8	6.0104053741967718	6.0104076400856545	0.0000376994210448%	✔
-0.5	  9	6.7175115932661038	6.7175144212722016	0.0000420989955569%	✔
-1.0	  0	0.4999999687733652	0.5000000000000000	0.0000062453269556%	✔
-1.0	  1	1.4999998437740314	1.5000000000000000	0.0000104150645737%	✔
-1.0	  2	2.4999995937642381	2.5000000000000000	0.0000162494304767%	✔
-1.0	  3	3.4999992187323192	3.5000000000000000	0.0000223219337363%	✔
-1.0	  4	4.4999987187474328	4.5000000000000000	0.0000284722792701%	✔
-1.0	  5	5.4999980937550053	5.5000000000000000	0.0000346589999044%	✔
-1.0	  6	6.4999973436023444	6.5000000000000000	0.0000408676562403%	✔
-1.0	  7	7.4999964688869518	7.5000000000000000	0.0000470815073091%	✔
-1.0	  8	8.4999954688433572	8.5000000000000000	0.0000533077252092%	✔
-1.0	  9	9.4999943434450671	9.5000000000000000	0.0000595426835042%	✔
-5.0	  0	1.1180338325225039	1.1180339887498949	0.0000139734026511%	✔
-5.0	  1	3.3541011849691116	3.3541019662496847	0.0000232932862801%	✔
-5.0	  2	5.5901679125242794	5.5901699437494745	0.0000363356609118%	✔
-5.0	  3	7.8262340149842551	7.8262379212492643	0.0000499124234216%	✔
-5.0	  4	10.0622994924938265	10.0623058987490541	0.0000636658763120%	✔
-5.0	  5	12.2983643449971325	12.2983738762488439	0.0000775000972270%	✔
-5.0	  6	14.5344285723091655	14.5344418537486337	0.0000913790815076%	✔
-5.0	  7	16.7704921752222305	16.7705098312484253	0.0001052801994246%	✔
-5.0	  8	19.0065551524155296	19.0065778087482116	0.0001192025882298%	✔
-5.0	  9	21.2426175047498660	21.2426457862480049	0.0001331354786194%	✔
-Test Summary:              | Pass  Total  Time
-∫ψₙ*Hψₙdx = <ψₙ|H|ψₙ> = Eₙ |   40     40  1.4s
-

+  &= 30240 x - 80640 x^{3} + 512 x^{9} - 9216 x^{7} + 48384 x^{5} \\
+  &= 30240 x - 80640 x^{3} + 512 x^{9} - 9216 x^{7} + 48384 x^{5}
+\end{aligned}\]
Test Summary:                               | Pass  Fail  Total   Time
+Hₙ(x) = (-1)ⁿ exp(x²) dⁿ/dxⁿ exp(-x²) = ... |    9     1     10  29.5s
+Error: LoadError: Some tests did not pass: 9 passed, 1 failed, 0 errored, 0 broken.
+in expression starting at C:\Users\user\Desktop\GitHub\Antiq.jl\test\HarmonicOscillator.jl:27

diff --git a/dev/HydrogenAtom/index.html b/dev/HydrogenAtom/index.html
index ff7a3c5..146deea 100644
--- a/dev/HydrogenAtom/index.html
+++ b/dev/HydrogenAtom/index.html
@@ -95,14 +95,14 @@
 ΔE = 9.427622831641132e-25 J
 ν = ΔE/h = 1422.8075794882932 MHz
 λ = hc/ΔE = 21.070485027063118 cm

Potential energy curve:

using Plots
-plot(xlims=(0.0,15.0), ylims=(-0.6,0.05), xlabel="\$r~/~a_0\$", ylabel="\$V(r)/E_\\mathrm{h},~E_n/E_\\mathrm{h}\$", legend=:bottomright, size=(480,400), dpi=400)
+plot(xlims=(0.0,15.0), ylims=(-0.6,0.05), xlabel="\$r~/~a_0\$", ylabel="\$V(r)/E_\\mathrm{h},~E_n/E_\\mathrm{h}\$", legend=:bottomright, size=(480,400))
 plot!(0.1:0.01:15, r -> H.V(r), lc=:black, lw=2, label="") # potential

Potential energy curve, Energy levels:

using Plots
-plot(xlims=(0.0,15.0), ylims=(-0.6,0.05), xlabel="\$r~/~a_0\$", ylabel="\$V(r)/E_\\mathrm{h}\$", legend=:bottomright, size=(480,400), dpi=400)
+plot(xlims=(0.0,15.0), ylims=(-0.6,0.05), xlabel="\$r~/~a_0\$", ylabel="\$V(r)/E_\\mathrm{h}\$", legend=:bottomright, size=(480,400))
 for n in 0:10
   plot!(0.0:0.01:15, r -> H.E(n=n) > H.V(r) ? H.E(n=n) : NaN, lc=n, lw=1, label="") # energy level
 end
 plot!(0.1:0.01:15, r -> H.V(r), lc=:black, lw=2, label="") # potential

Radial functions:

using Plots
-plot(xlabel="\$r~/~a_0\$", ylabel="\$r^2|R_{nl}(r)|^2~/~a_0^{-1}\$", ylims=(-0.01,0.55), xticks=0:1:20, size=(480,400), dpi=400)
+plot(xlabel="\$r~/~a_0\$", ylabel="\$r^2|R_{nl}(r)|^2~/~a_0^{-1}\$", ylims=(-0.01,0.55), xticks=0:1:20, size=(480,400), dpi=300)
 for n in 1:3
   for l in 0:n-1
     plot!(0:0.01:20, r->r^2*H.R(r,n=n,l=l)^2, lc=n, lw=2, ls=[:solid,:dash,:dot,:dashdot,:dashdotdot][l+1], label="\$n = $n, l=$l\$")
@@ -151,8 +151,8 @@
 \end{aligned}\]
$n=3, m=1:$ ✔
\[\begin{aligned}
   P_{3}^{1}(x)
     = \left( 1 - x^{2} \right)^{\frac{1}{2}} \frac{\mathrm{d}}{\mathrm{d}x} \frac{1}{48} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \left( -1 + x^{2} \right)^{3}
-  &=  - \frac{3}{2} \left( 1 - x^{2} \right)^{\frac{1}{2}} + \frac{15}{2} x^{2} \left( 1 - x^{2} \right)^{\frac{1}{2}} \\
-  &=  - \frac{3}{2} \left( 1 - x^{2} \right)^{\frac{1}{2}} + \frac{15}{2} x^{2} \left( 1 - x^{2} \right)^{\frac{1}{2}}
+  &=  - \frac{3}{2} \left( 1 - x^{2} \right)^{\frac{1}{2}} + \frac{15}{2} \left( 1 - x^{2} \right)^{\frac{1}{2}} x^{2} \\
+  &=  - \frac{3}{2} \left( 1 - x^{2} \right)^{\frac{1}{2}} + \frac{15}{2} \left( 1 - x^{2} \right)^{\frac{1}{2}} x^{2}
 \end{aligned}\]
$n=3, m=2:$ ✔
\[\begin{aligned}
   P_{3}^{2}(x)
     = \left( 1 - x^{2} \right) \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{1}{48} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \left( -1 + x^{2} \right)^{3}
@@ -171,8 +171,8 @@
 \end{aligned}\]
$n=4, m=1:$ ✔
\[\begin{aligned}
   P_{4}^{1}(x)
     = \left( 1 - x^{2} \right)^{\frac{1}{2}} \frac{\mathrm{d}}{\mathrm{d}x} \frac{1}{384} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \left( -1 + x^{2} \right)^{4}
-  &=  - \frac{15}{2} \left( 1 - x^{2} \right)^{\frac{1}{2}} x + \frac{35}{2} x^{3} \left( 1 - x^{2} \right)^{\frac{1}{2}} \\
-  &=  - \frac{15}{2} \left( 1 - x^{2} \right)^{\frac{1}{2}} x + \frac{35}{2} x^{3} \left( 1 - x^{2} \right)^{\frac{1}{2}}
+  &=  - \frac{15}{2} \left( 1 - x^{2} \right)^{\frac{1}{2}} x + \frac{35}{2} \left( 1 - x^{2} \right)^{\frac{1}{2}} x^{3} \\
+  &=  - \frac{15}{2} \left( 1 - x^{2} \right)^{\frac{1}{2}} x + \frac{35}{2} \left( 1 - x^{2} \right)^{\frac{1}{2}} x^{3}
 \end{aligned}\]
$n=4, m=2:$ ✔
\[\begin{aligned}
   P_{4}^{2}(x)
     = \left( 1 - x^{2} \right) \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{1}{384} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \left( -1 + x^{2} \right)^{4}
@@ -189,7 +189,7 @@
   &= 105 \left( 1 - x^{2} \right)^{2} \\
   &= 105 \left( 1 - x^{2} \right)^{2}
 \end{aligned}\]
Test Summary:                                                   | Pass  Total  Time
-Pₙᵐ(x) = √(1-x²)ᵐ dᵐ/dxᵐ Pₙ(x); Pₙ(x) = 1/(2ⁿn!) dⁿ/dxⁿ (x²-1)ⁿ |   15     15  1.9s
Normalization & Orthogonality of $P_n^m(x)$
\[\int_{-1}^{1} P_i^m(x) P_j^m(x) \mathrm{d}x = \frac{2(j+m)!}{(2j+1)(j-m)!} \delta_{ij}\]
  m	  i	  j	numerical         	analytical        	|error|
+Pₙᵐ(x) = √(1-x²)ᵐ dᵐ/dxᵐ Pₙ(x); Pₙ(x) = 1/(2ⁿn!) dⁿ/dxⁿ (x²-1)ⁿ |   15     15  1.7s
Normalization & Orthogonality of $P_n^m(x)$
\[\int_{-1}^{1} P_i^m(x) P_j^m(x) \mathrm{d}x = \frac{2(j+m)!}{(2j+1)(j-m)!} \delta_{ij}\]
  m	  i	  j	numerical         	analytical        	|error|
   0	  0	  0	2.0000000000000000	2.0000000000000000	0.0000000000000000%	✔
   0	  0	  1	0.0000000000000000	0.0000000000000000	0.0000000000000000%	✔
   0	  0	  2	0.0000000000000000	0.0000000000000000	0.0000000000000000%	✔
@@ -546,7 +546,7 @@
   5	  9	  8	-0.0000000013969839	0.0000000000000000	0.0000000000000000%	✔
   5	  9	  9	382360926.3157894611358643	382360926.3157894611358643	0.0000000000000000%	✔
 Test Summary:                              | Pass  Total  Time
-∫Pᵢᵐ(x)Pⱼᵐ(x)dx = 2(j+m)!/(2j+1)(j-m)! δᵢⱼ |  355    355  1.5s
Normalization & Orthogonality of $Y_{lm}(\theta,\varphi)$
\[\int_0^{2\pi}
+∫Pᵢᵐ(x)Pⱼᵐ(x)dx = 2(j+m)!/(2j+1)(j-m)! δᵢⱼ |  355    355  1.2s

Normalization & Orthogonality of $Y_{lm}(\theta,\varphi)$

\[\int_0^{2\pi} \int_0^\pi Y_{lm}(\theta,\varphi)^* Y_{l'm'}(\theta,\varphi) \sin(\theta) ~\mathrm{d}\theta \mathrm{d}\varphi @@ -633,7 +633,7 @@ 2 2 2 1 0.0000000000000000 0.0000000000000000 0.0000000000000000% ✔ 2 2 2 2 1.0000000000000002 1.0000000000000000 0.0000000000000222% ✔ Test Summary: | Pass Total Time -∫Yₗ₁ₘ₁(θ,φ)Yₗ₂ₘ₂(θ,φ)sinθdθdφ = δₗ₁ₗ₂δₘ₁ₘ₂ | 81 81 3.1s

Associated Laguerre Polynomials $L_n^{k}(x)$

\[ \begin{aligned} +∫Yₗ₁ₘ₁(θ,φ)Yₗ₂ₘ₂(θ,φ)sinθdθdφ = δₗ₁ₗ₂δₘ₁ₘ₂ | 81 81 2.7s

Associated Laguerre Polynomials $L_n^{k}(x)$

\[ \begin{aligned} L_n^{k}(x) &= \frac{\mathrm{d}^k}{\mathrm{d}x^k} L_n(x) \\ &= \frac{\mathrm{d}^k}{\mathrm{d}x^k} \frac{1}{n!} \mathrm{e}^x \frac{\mathrm{d}^n}{\mathrm{d}x ^n} \left( \mathrm{e}^{-x} x^n \right) \\ @@ -641,91 +641,91 @@ &= (-1)^k L_{n-k}^{(k)}(x) \end{aligned}\]

$n=0, k=0:$ ✔

\[\begin{aligned} L_{0}^{0}(x) - = e^{ - x} e^{x} + = e^{x} e^{ - x} &= 1 \\ &= 1 \\ &= 1 \end{aligned}\]

$n=1, k=0:$ ✔

\[\begin{aligned} L_{1}^{0}(x) - = \frac{\mathrm{d}}{\mathrm{d}x} x e^{ - x} e^{x} + = e^{x} \frac{\mathrm{d}}{\mathrm{d}x} x e^{ - x} &= 1 - x \\ &= 1 - x \\ &= 1 - x \end{aligned}\]

$n=1, k=1:$ ✔

\[\begin{aligned} L_{1}^{1}(x) - = \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} x e^{ - x} e^{x} + = \frac{\mathrm{d}}{\mathrm{d}x} e^{x} \frac{\mathrm{d}}{\mathrm{d}x} x e^{ - x} &= -1 \\ &= -1 \\ &= -1 \end{aligned}\]

$n=2, k=0:$ ✔

\[\begin{aligned} L_{2}^{0}(x) - = \frac{1}{2} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} x^{2} e^{ - x} e^{x} + = \frac{1}{2} e^{x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} x^{2} e^{ - x} &= 1 - 2 x + \frac{1}{2} x^{2} \\ &= 1 - 2 x + \frac{1}{2} x^{2} \\ &= 1 - 2 x + \frac{1}{2} x^{2} \end{aligned}\]

$n=2, k=1:$ ✔

\[\begin{aligned} L_{2}^{1}(x) - = \frac{\mathrm{d}}{\mathrm{d}x} \frac{1}{2} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} x^{2} e^{ - x} e^{x} + = \frac{\mathrm{d}}{\mathrm{d}x} \frac{1}{2} e^{x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} x^{2} e^{ - x} &= -2 + x \\ &= -2 + x \\ &= -2 + x \end{aligned}\]

$n=2, k=2:$ ✔

\[\begin{aligned} L_{2}^{2}(x) - = \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{1}{2} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} x^{2} e^{ - x} e^{x} + = \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{1}{2} e^{x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} x^{2} e^{ - x} &= 1 \\ &= 1 \\ &= 1 \end{aligned}\]

$n=3, k=0:$ ✔

\[\begin{aligned} L_{3}^{0}(x) - = \frac{1}{6} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} x^{3} e^{ - x} e^{x} - &= 1 - 3 x + \frac{3}{2} x^{2} - \frac{1}{6} x^{3} \\ - &= 1 - 3 x + \frac{3}{2} x^{2} - \frac{1}{6} x^{3} \\ - &= 1 - 3 x + \frac{3}{2} x^{2} - \frac{1}{6} x^{3} + = \frac{1}{6} e^{x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} x^{3} e^{ - x} + &= 1 - \frac{1}{6} x^{3} - 3 x + \frac{3}{2} x^{2} \\ + &= 1 - \frac{1}{6} x^{3} - 3 x + \frac{3}{2} x^{2} \\ + &= 1 - \frac{1}{6} x^{3} - 3 x + \frac{3}{2} x^{2} \end{aligned}\]

$n=3, k=1:$ ✔

\[\begin{aligned} L_{3}^{1}(x) - = \frac{\mathrm{d}}{\mathrm{d}x} \frac{1}{6} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} x^{3} e^{ - x} e^{x} + = \frac{\mathrm{d}}{\mathrm{d}x} \frac{1}{6} e^{x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} x^{3} e^{ - x} &= -3 + 3 x - \frac{1}{2} x^{2} \\ &= -3 + 3 x - \frac{1}{2} x^{2} \\ &= -3 + 3 x - \frac{1}{2} x^{2} \end{aligned}\]

$n=3, k=2:$ ✔

\[\begin{aligned} L_{3}^{2}(x) - = \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{1}{6} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} x^{3} e^{ - x} e^{x} + = \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{1}{6} e^{x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} x^{3} e^{ - x} &= 3 - x \\ &= 3 - x \\ &= 3 - x \end{aligned}\]

$n=3, k=3:$ ✔

\[\begin{aligned} L_{3}^{3}(x) - = \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{1}{6} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} x^{3} e^{ - x} e^{x} + = \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{1}{6} e^{x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} x^{3} e^{ - x} &= -1 \\ &= -1 \\ &= -1 \end{aligned}\]

$n=4, k=0:$ ✔

\[\begin{aligned} L_{4}^{0}(x) - = \frac{1}{24} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} x^{4} e^{ - x} e^{x} - &= 1 - 4 x + 3 x^{2} - \frac{2}{3} x^{3} + \frac{1}{24} x^{4} \\ - &= 1 - 4 x + 3 x^{2} - \frac{2}{3} x^{3} + \frac{1}{24} x^{4} \\ - &= 1 - 4 x + 3 x^{2} - \frac{2}{3} x^{3} + \frac{1}{24} x^{4} + = \frac{1}{24} e^{x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} x^{4} e^{ - x} + &= 1 - \frac{2}{3} x^{3} - 4 x + 3 x^{2} + \frac{1}{24} x^{4} \\ + &= 1 - \frac{2}{3} x^{3} - 4 x + 3 x^{2} + \frac{1}{24} x^{4} \\ + &= 1 - \frac{2}{3} x^{3} - 4 x + 3 x^{2} + \frac{1}{24} x^{4} \end{aligned}\]

$n=4, k=1:$ ✔

\[\begin{aligned} L_{4}^{1}(x) - = \frac{\mathrm{d}}{\mathrm{d}x} \frac{1}{24} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} x^{4} e^{ - x} e^{x} - &= -4 + 6 x - 2 x^{2} + \frac{1}{6} x^{3} \\ - &= -4 + 6 x - 2 x^{2} + \frac{1}{6} x^{3} \\ - &= -4 + 6 x - 2 x^{2} + \frac{1}{6} x^{3} + = \frac{\mathrm{d}}{\mathrm{d}x} \frac{1}{24} e^{x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} x^{4} e^{ - x} + &= -4 + \frac{1}{6} x^{3} + 6 x - 2 x^{2} \\ + &= -4 + \frac{1}{6} x^{3} + 6 x - 2 x^{2} \\ + &= -4 + \frac{1}{6} x^{3} + 6 x - 2 x^{2} \end{aligned}\]

$n=4, k=2:$ ✔

\[\begin{aligned} L_{4}^{2}(x) - = \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{1}{24} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} x^{4} e^{ - x} e^{x} + = \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{1}{24} e^{x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} x^{4} e^{ - x} &= 6 - 4 x + \frac{1}{2} x^{2} \\ &= 6 - 4 x + \frac{1}{2} x^{2} \\ &= 6 - 4 x + \frac{1}{2} x^{2} \end{aligned}\]

$n=4, k=3:$ ✔

\[\begin{aligned} L_{4}^{3}(x) - = \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{1}{24} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} x^{4} e^{ - x} e^{x} + = \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{1}{24} e^{x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} x^{4} e^{ - x} &= -4 + x \\ &= -4 + x \\ &= -4 + x \end{aligned}\]

$n=4, k=4:$ ✔

\[\begin{aligned} L_{4}^{4}(x) - = \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{1}{24} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} x^{4} e^{ - x} e^{x} + = \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{1}{24} e^{x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} x^{4} e^{ - x} &= 1 \\ &= 1 \\ &= 1 @@ -936,7 +936,7 @@ 7 7 6 5039.9999999999854481 5040.0000000000000000 0.0000000000002887% ✔ 7 7 7 5040.0000000000000000 5040.0000000000000000 0.0000000000000000% ✔ Test Summary: | Pass Total Time -∫exp(-x)xᵏLᵢᵏ(x)Lⱼᵏ(x)dx = (2i+k)!/(i+k)! δᵢⱼ | 204 204 1.4s

Normalization of $R_{nl}(r)$

\[\int |R_{nl}(r)|^2 r^2 \mathrm{d}r = 1\]

  n	  l	numerical         	analytical        	|error|
+∫exp(-x)xᵏLᵢᵏ(x)Lⱼᵏ(x)dx = (2i+k)!/(i+k)! δᵢⱼ |  204    204  1.0s

Normalization of $R_{nl}(r)$

\[\int |R_{nl}(r)|^2 r^2 \mathrm{d}r = 1\]

  n	  l	numerical         	analytical        	|error|
   1	  0	1.0000000000000286	1.0000000000000000	0.0000000000028644%	✔
   2	  0	1.0000000000000002	1.0000000000000000	0.0000000000000222%	✔
   2	  1	1.0000000000000000	1.0000000000000000	0.0000000000000000%	✔
@@ -983,7 +983,7 @@
   9	  7	1.0000000000000007	1.0000000000000000	0.0000000000000666%	✔
   9	  8	0.9999999999999994	1.0000000000000000	0.0000000000000555%	✔
 Test Summary:               | Pass  Total  Time
-∫|Rₙₗ(r)|²r²dr = δₙ₁ₙ₂δₗ₁ₗ₂ |   45     45  1.1s

Expected Value of $r$

\[\langle r \rangle +∫|Rₙₗ(r)|²r²dr = δₙ₁ₙ₂δₗ₁ₗ₂ | 45 45 0.9s

Expected Value of $r$

\[\langle r \rangle = \int r |R_{n_1 l_1}(r)|^2 r^2 \mathrm{d}r = \frac{a_\mu}{2Z} \left[ 3n^2 - l(l+1) \right] \\ a_\mu = a_0 \frac{m_\mathrm{e}}{\mu} \\ @@ -1034,7 +1034,7 @@ 9 7 93.5000000000000000 93.5000000000000000 0.0000000000000000% ✔ 9 8 85.4999999999999716 85.5000000000000000 0.0000000000000332% ✔ Test Summary: | Pass Total Time -∫r|Rₙₗ(r)|²r²dr = (a₀×mₑ/μ)/2Z × [3n²-l(l+1)]; 1/μ = 1/mₑ + 1/mₚ | 45 45 0.9s

Expected Value of $r^2$

\[\langle r^2 \rangle +∫r|Rₙₗ(r)|²r²dr = (a₀×mₑ/μ)/2Z × [3n²-l(l+1)]; 1/μ = 1/mₑ + 1/mₚ | 45 45 0.8s

Expected Value of $r^2$

\[\langle r^2 \rangle = \int r^2 |R_{n_1 l_1}(r)|^2 r^2 \mathrm{d}r = \frac{a_\mu^2}{2Z^2} n^2 \left[ 5n^2 + 1 - 3l(l+1) \right] \\ a_\mu = a_0 \frac{m_\mathrm{e}}{\mu} \\ @@ -1085,7 +1085,7 @@ 9 7 9638.9999999999909051 9639.0000000000000000 0.0000000000000944% ✔ 9 8 7694.9999999999981810 7695.0000000000000000 0.0000000000000236% ✔ Test Summary: | Pass Total Time -∫r²|Rₙₗ(r)|²r²dr = (a₀×mₑ/μ)²/2Z² × n²[5n²+1-3l(l+1)]; 1/μ = 1/mₑ + 1/mₚ | 45 45 1.0s

Virial Theorem

The virial theorem $2\langle T \rangle + \langle V \rangle = 0$ and the definition of Hamiltonian $\langle H \rangle = \langle T \rangle + \langle V \rangle$ derive $\langle H \rangle = \frac{1}{2} \langle V \rangle$ and $\langle H \rangle = -\langle T \rangle$.

\[\frac{1}{2} \int \psi_n^\ast(x) V(x) \psi_n(x) \mathrm{d}x = E_n\]

  n	numerical         	analytical        	|error|
+∫r²|Rₙₗ(r)|²r²dr = (a₀×mₑ/μ)²/2Z² × n²[5n²+1-3l(l+1)]; 1/μ = 1/mₑ + 1/mₚ |   45     45  0.8s

Virial Theorem

The virial theorem $2\langle T \rangle + \langle V \rangle = 0$ and the definition of Hamiltonian $\langle H \rangle = \langle T \rangle + \langle V \rangle$ derive $\langle H \rangle = \frac{1}{2} \langle V \rangle$ and $\langle H \rangle = -\langle T \rangle$.

\[\frac{1}{2} \int \psi_n^\ast(x) V(x) \psi_n(x) \mathrm{d}x = E_n\]

  n	numerical         	analytical        	|error|
   1	-0.9999999999999999	-1.0000000000000000	0.0000000000000111%	✔
   2	-0.2500000000000001	-0.2500000000000000	0.0000000000000444%	✔
   3	-0.1111111111111111	-0.1111111111111111	0.0000000000000125%	✔
@@ -1097,7 +1097,7 @@
   9	-0.0123456790123456	-0.0123456790123457	0.0000000000002389%	✔
  10	-0.0100000000000004	-0.0100000000000000	0.0000000000036256%	✔
 Test Summary:      | Pass  Total  Time
-<ψₙ|V|ψₙ> / 2 = Eₙ |   10     10  0.9s

Normalization & Orthogonality of $\psi_n(r,\theta,\varphi)$

\[\int \psi_i^\ast(r,\theta,\varphi) \psi_j(r,\theta,\varphi) r^2 \mathrm{d}r \mathrm{d}\theta \mathrm{d}\varphi = \delta_{ij}\]

 n1	 n2	 l1	 l2	 m1	 m2	numerical         	analytical        	|error|
+<ψₙ|V|ψₙ> / 2 = Eₙ |   10     10  0.8s

Normalization & Orthogonality of $\psi_n(r,\theta,\varphi)$

\[\int \psi_i^\ast(r,\theta,\varphi) \psi_j(r,\theta,\varphi) r^2 \mathrm{d}r \mathrm{d}\theta \mathrm{d}\varphi = \delta_{ij}\]

 n1	 n2	 l1	 l2	 m1	 m2	numerical         	analytical        	|error|
   1	  1	  0	  0	  0	  0	1.0000000002519327	1.0000000000000000	0.0000000251932697%	✔
   1	  2	  0	  0	  0	  0	-0.0000000112226007	0.0000000000000000	0.0000000000000000%	✔
   1	  2	  0	  1	  0	 -1	-0.0000000000000000	0.0000000000000000	0.0000000000000000%	✔
@@ -1294,6 +1294,6 @@
   3	  3	  2	  2	  2	  0	-0.0000000000000175	0.0000000000000000	0.0000000000000000%	✔
   3	  3	  2	  2	  2	  1	0.0000000000000000	0.0000000000000000	0.0000000000000000%	✔
   3	  3	  2	  2	  2	  2	1.0003006285656155	1.0000000000000000	0.0300628565615524%	✔
-Test Summary:                       | Pass  Total   Time
-<ψₙ₁ₗ₁ₘ₁|ψₙ₂ₗ₂ₘ₂> = δₙ₁ₙ₂δₗ₁ₗ₂δₘ₁ₘ₂ |  196    196  26.8s
-
+Test Summary: | Pass Total Time +<ψₙ₁ₗ₁ₘ₁|ψₙ₂ₗ₂ₘ₂> = δₙ₁ₙ₂δₗ₁ₗ₂δₘ₁ₘ₂ | 196 196 8.0s + diff --git a/dev/InfinitePotentialWell/index.html b/dev/InfinitePotentialWell/index.html index 0f2246c..8ceb585 100644 --- a/dev/InfinitePotentialWell/index.html +++ b/dev/InfinitePotentialWell/index.html @@ -25,7 +25,7 @@ plot!(x -> IPW.ψ(x, n=4), label="n=4", lw=2) plot!(x -> IPW.ψ(x, n=5), label="n=5", lw=2)

Potential energy curve, Energy levels, Wave functions:

L = 1
 using Plots
-plot(xlim=(-0.5,1.5), ylim=(-5,140), xlabel="\$x\$", ylabel="\$V(x),~E_n,~\\psi_n(x)\\times5+E_n\$", size=(480,400), dpi=400)
+plot(xlim=(-0.5,1.5), ylim=(-5,140), xlabel="\$x\$", ylabel="\$V(x),~E_n,~\\psi_n(x)\\times5+E_n\$", size=(480,400), dpi=300)
 for n in 1:5
   # energy
   plot!([0,L], fill(IPW.E(n=n,L=L),2), lc=:black, lw=2, label="")
@@ -135,7 +135,7 @@
  10	  9	0.0000000000000001	0.0000000000000000	0.0000000000000000%	✔
  10	 10	1.0000000000000002	1.0000000000000000	0.0000000000000222%	✔
 Test Summary:            | Pass  Total  Time
-<ψᵢ|ψⱼ> = ∫ψₙ*ψₙdx = δᵢⱼ |  100    100  1.5s

Eigen Values

\[ \begin{aligned} +<ψᵢ|ψⱼ> = ∫ψₙ*ψₙdx = δᵢⱼ | 100 100 2.3s

Eigen Values

\[ \begin{aligned} E_n &= \int_0^L \psi^\ast_n(x) \hat{H} \psi_n(x) ~\mathrm{d}x \\ &= \int_0^L \psi^\ast_n(x) \left[ \hat{V} + \hat{T} \right] \psi(x) ~\mathrm{d}x \\ @@ -261,7 +261,7 @@ 1.0 1.0 1.0 9 399.718711951912 399.718978244119 0.000066619856% ✔ 1.0 1.0 1.0 10 493.479814178266 493.480220054468 0.000082247714% ✔ Test Summary: | Pass Total Time -<ψₙ|H|ψₙ> = ∫ψₙ*Tψₙdx = Eₙ | 80 80 1.2s

Expected Value of $x$

\[\langle x \rangle_{n=1} +<ψₙ|H|ψₙ> = ∫ψₙ*Tψₙdx = Eₙ | 80 80 0.9s

Expected Value of $x$

\[\langle x \rangle_{n=1} = \int_{0}^{L} \psi_1^\ast(x) \hat{x} \psi_1(x) ~\mathrm{d}x = \frac{2(2a)^2}{\pi^3} \left( \frac{\pi^3}{6} - \frac{\pi}{4} \right)\]

for only $n=1$.

Reference:

   L	  n	numerical         	analytical        	|error|
 0.1	  1	0.0500000000000000	0.0500000000000000	0.0000000000000278%	✔
@@ -269,7 +269,7 @@
 1.0	  1	0.5000000000000002	0.5000000000000000	0.0000000000000444%	✔
 7.0	  1	3.5000000000000000	3.5000000000000000	0.0000000000000000%	✔
 Test Summary:    | Pass  Total  Time
-<ψₙ|x|ψₙ>  = L/2 |    4      4  0.7s

Expected Value of $x^2$

\[\langle x^2 \rangle_{n=1} +<ψₙ|x|ψₙ> = L/2 | 4 4 0.5s

Expected Value of $x^2$

\[\langle x^2 \rangle_{n=1} = \int_{0}^{L} \psi_1^\ast(x) \hat{x}^2 \psi_1(x) ~\mathrm{d}x = \frac{2(2a)^2}{\pi^3} \left( \frac{\pi^3}{6} - \frac{\pi}{4} \right)\]

Reference:

   L	  n	numerical         	analytical        	|error|
 0.1	  1	0.0028267274151216	0.0028267274151216	0.0000000000000460%	✔
@@ -326,7 +326,7 @@
 1.0	  1	0.0000000000001066	0.0000000000000000	0.0000000000000000%	✔
 7.0	  1	0.0000000000000252	0.0000000000000000	0.0000000000000000%	✔
 Test Summary:                      | Pass  Total  Time
-<ψₙ|p|ψₙ>  = ∫ψₙ*(-iℏd/dx)ψₙdx = 0 |    4      4  0.9s

Expected Value of $p^2$

\[\langle p^2 \rangle +<ψₙ|p|ψₙ> = ∫ψₙ*(-iℏd/dx)ψₙdx = 0 | 4 4 0.7s

Expected Value of $p^2$

\[\langle p^2 \rangle = \int_{0}^{L} \psi_1^\ast(x) \hat{p}^2 \psi_1(x) ~\mathrm{d}x = \frac{\pi^2\hbar^2}{L^2}\]

Reference:

\[ \begin{aligned} \langle p^2 \rangle @@ -377,5 +377,5 @@ 1.0 1 9.8696043189632228 9.8696044010893580 0.0000008321117229% ✔ 7.0 1 0.2014204963826796 0.2014204979814155 0.0000007937304979% ✔ Test Summary: | Pass Total Time -<ψₙ|p²|ψₙ> = ∫ψₙ*(-ℏ²d²/dx²)ψₙdx = π²ℏ²/L² | 4 4 0.4s -

+<ψₙ|p²|ψₙ> = ∫ψₙ*(-ℏ²d²/dx²)ψₙdx = π²ℏ²/L² | 4 4 0.5s + diff --git a/dev/MorsePotential/index.html b/dev/MorsePotential/index.html index 46ba758..65a9826 100644 --- a/dev/MorsePotential/index.html +++ b/dev/MorsePotential/index.html @@ -50,7 +50,7 @@ plot!(x -> MP.ψ(x, n=5), label="n=5", lw=2)

Potential energy curve, Energy levels, Comparison with harmonic oscillator:

MP = antiq(:MorsePotential)
 HO = antiq(:HarmonicOscillator, k=MP.k, m=MP.μ)
 using Plots
-plot(xlims=(0.1,9.1), ylims=(-0.11,0.01), xlabel="\$r\$", ylabel="\$V(r), E_n\$", legend=:bottomright, size=(480,400), dpi=400)
+plot(xlims=(0.1,9.1), ylims=(-0.11,0.01), xlabel="\$r\$", ylabel="\$V(r), E_n\$", legend=:bottomright, size=(480,400), dpi=300)
 for n in 0:MP.nₘₐₓ()
   # energy
   EM = MP.E(n=n)
@@ -66,12 +66,12 @@
     &= \sum_{k=0}^n(-1)^k \frac{\Gamma(\alpha+n+1)}{\Gamma(\alpha+k+1)\Gamma(n-k+1)} \frac{x^k}{k !}.
   \end{aligned}\]
$n=0, α=0:$ ✔
\[\begin{aligned}
   L_{0}^{(0)}(x)
-   = e^{ - x} e^{x}
+   = e^{x} e^{ - x}
   &= 1 \\
   &= 1
 \end{aligned}\]
$n=1, α=0:$ ✔
\[\begin{aligned}
   L_{1}^{(0)}(x)
-   = \frac{\mathrm{d}}{\mathrm{d}x} x e^{ - x} e^{x}
+   = e^{x} \frac{\mathrm{d}}{\mathrm{d}x} x e^{ - x}
   &= 1 - x \\
   &= 1 - x
 \end{aligned}\]
$n=1, α=1:$ ✔
\[\begin{aligned}
@@ -81,654 +81,65 @@
   &= 2 - x
 \end{aligned}\]
$n=2, α=0:$ ✔
\[\begin{aligned}
   L_{2}^{(0)}(x)
-   = \frac{1}{2} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} x^{2} e^{ - x} e^{x}
+   = \frac{1}{2} e^{x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} x^{2} e^{ - x}
   &= 1 - 2 x + \frac{1}{2} x^{2} \\
   &= 1 - 2 x + \frac{1}{2} x^{2}
 \end{aligned}\]
$n=2, α=1:$ ✔
\[\begin{aligned}
   L_{2}^{(1)}(x)
-   = \frac{\frac{1}{2} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} x^{3} e^{ - x} e^{x}}{x}
+   = \frac{\frac{1}{2} e^{x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} x^{3} e^{ - x}}{x}
   &= 3 - 3 x + \frac{1}{2} x^{2} \\
   &= 3 - 3 x + \frac{1}{2} x^{2}
 \end{aligned}\]
$n=2, α=2:$ ✔
\[\begin{aligned}
   L_{2}^{(2)}(x)
-   = \frac{\frac{1}{2} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} x^{4} e^{ - x} e^{x}}{x^{2}}
+   = \frac{\frac{1}{2} e^{x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} x^{4} e^{ - x}}{x^{2}}
   &= 6 - 4 x + \frac{1}{2} x^{2} \\
   &= 6 - 4 x + \frac{1}{2} x^{2}
 \end{aligned}\]
$n=3, α=0:$ ✔
\[\begin{aligned}
   L_{3}^{(0)}(x)
-   = \frac{1}{6} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} x^{3} e^{ - x} e^{x}
-  &= 1 - 3 x + \frac{3}{2} x^{2} - \frac{1}{6} x^{3} \\
-  &= 1 - 3 x + \frac{3}{2} x^{2} - \frac{1}{6} x^{3}
-\end{aligned}\]
$n=3, α=1:$ ✔
\[\begin{aligned}
+   = \frac{1}{6} e^{x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} x^{3} e^{ - x}
+  &= 1 - \frac{1}{6} x^{3} - 3 x + \frac{3}{2} x^{2} \\
+  &= 1 - \frac{1}{6} x^{3} - 3 x + \frac{3}{2} x^{2}
+\end{aligned}\]
$n=3, α=1:$ ✗ Lₙ⁽ᵅ⁾(x) = x⁻ᵅeˣ/n! dⁿ/dxⁿ xⁿ⁺ᵅe⁻ˣ: Test Failed at C:\Users\user\Desktop\GitHub\Antiq.jl\test\MorsePotential.jl:44   Expression: acceptance
Stacktrace:  [1] macro expansion    @ C:\Users\user.julia\juliaup\julia-1.9.2+0.x64.w64.mingw32\share\julia\stdlib\v1.9\Test\src\Test.jl:478 [inlined]  [2] macro expansion    @ C:\Users\user\Desktop\GitHub\Antiq.jl\test\MorsePotential.jl:44 [inlined]  [3] macro expansion    @ C:\Users\user.julia\juliaup\julia-1.9.2+0.x64.w64.mingw32\share\julia\stdlib\v1.9\Test\src\Test.jl:1498 [inlined]  [4] top-level scope    @ C:\Users\user\Desktop\GitHub\Antiq.jl\test\MorsePotential.jl:28
\[\begin{aligned}
   L_{3}^{(1)}(x)
    = \frac{\frac{1}{6} e^{x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} x^{4} e^{ - x}}{x}
-  &= 4 - 6 x + 2 x^{2} - \frac{1}{6} x^{3} \\
-  &= 4 - 6 x + 2 x^{2} - \frac{1}{6} x^{3}
-\end{aligned}\]
$n=3, α=2:$ ✔
\[\begin{aligned}
+  &= 4 - 6 x - \frac{1}{6} x^{3} + 2 x^{2} \\
+  &= 4 - \frac{1}{6} x^{3} - 6 x + 2 x^{2}
+\end{aligned}\]
$n=3, α=2:$ ✗ Lₙ⁽ᵅ⁾(x) = x⁻ᵅeˣ/n! dⁿ/dxⁿ xⁿ⁺ᵅe⁻ˣ: Test Failed at C:\Users\user\Desktop\GitHub\Antiq.jl\test\MorsePotential.jl:44   Expression: acceptance
Stacktrace:  [1] macro expansion    @ C:\Users\user.julia\juliaup\julia-1.9.2+0.x64.w64.mingw32\share\julia\stdlib\v1.9\Test\src\Test.jl:478 [inlined]  [2] macro expansion    @ C:\Users\user\Desktop\GitHub\Antiq.jl\test\MorsePotential.jl:44 [inlined]  [3] macro expansion    @ C:\Users\user.julia\juliaup\julia-1.9.2+0.x64.w64.mingw32\share\julia\stdlib\v1.9\Test\src\Test.jl:1498 [inlined]  [4] top-level scope    @ C:\Users\user\Desktop\GitHub\Antiq.jl\test\MorsePotential.jl:28
\[\begin{aligned}
   L_{3}^{(2)}(x)
-   = \frac{\frac{1}{6} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} x^{5} e^{ - x} e^{x}}{x^{2}}
-  &= 10 - 10 x + \frac{5}{2} x^{2} - \frac{1}{6} x^{3} \\
-  &= 10 - 10 x + \frac{5}{2} x^{2} - \frac{1}{6} x^{3}
-\end{aligned}\]
$n=3, α=3:$ ✔
\[\begin{aligned}
+   = \frac{\frac{1}{6} e^{x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} x^{5} e^{ - x}}{x^{2}}
+  &= 10 - 10 x - \frac{1}{6} x^{3} + \frac{5}{2} x^{2} \\
+  &= 10 - \frac{1}{6} x^{3} - 10 x + \frac{5}{2} x^{2}
+\end{aligned}\]
$n=3, α=3:$ ✗ Lₙ⁽ᵅ⁾(x) = x⁻ᵅeˣ/n! dⁿ/dxⁿ xⁿ⁺ᵅe⁻ˣ: Test Failed at C:\Users\user\Desktop\GitHub\Antiq.jl\test\MorsePotential.jl:44   Expression: acceptance
Stacktrace:  [1] macro expansion    @ C:\Users\user.julia\juliaup\julia-1.9.2+0.x64.w64.mingw32\share\julia\stdlib\v1.9\Test\src\Test.jl:478 [inlined]  [2] macro expansion    @ C:\Users\user\Desktop\GitHub\Antiq.jl\test\MorsePotential.jl:44 [inlined]  [3] macro expansion    @ C:\Users\user.julia\juliaup\julia-1.9.2+0.x64.w64.mingw32\share\julia\stdlib\v1.9\Test\src\Test.jl:1498 [inlined]  [4] top-level scope    @ C:\Users\user\Desktop\GitHub\Antiq.jl\test\MorsePotential.jl:28
\[\begin{aligned}
   L_{3}^{(3)}(x)
-   = \frac{\frac{1}{6} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} x^{6} e^{ - x} e^{x}}{x^{3}}
-  &= 20 - 15 x + 3 x^{2} - \frac{1}{6} x^{3} \\
-  &= 20 - 15 x + 3 x^{2} - \frac{1}{6} x^{3}
+   = \frac{\frac{1}{6} e^{x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} x^{6} e^{ - x}}{x^{3}}
+  &= 20 - 15 x - \frac{1}{6} x^{3} + 3 x^{2} \\
+  &= 20 - \frac{1}{6} x^{3} - 15 x + 3 x^{2}
 \end{aligned}\]
$n=4, α=0:$ ✔
\[\begin{aligned}
   L_{4}^{(0)}(x)
-   = \frac{1}{24} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} x^{4} e^{ - x} e^{x}
-  &= 1 - 4 x + 3 x^{2} - \frac{2}{3} x^{3} + \frac{1}{24} x^{4} \\
-  &= 1 - 4 x + 3 x^{2} - \frac{2}{3} x^{3} + \frac{1}{24} x^{4}
-\end{aligned}\]
$n=4, α=1:$ ✔
\[\begin{aligned}
+   = \frac{1}{24} e^{x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} x^{4} e^{ - x}
+  &= 1 - \frac{2}{3} x^{3} - 4 x + 3 x^{2} + \frac{1}{24} x^{4} \\
+  &= 1 - \frac{2}{3} x^{3} - 4 x + 3 x^{2} + \frac{1}{24} x^{4}
+\end{aligned}\]
$n=4, α=1:$ ✗ Lₙ⁽ᵅ⁾(x) = x⁻ᵅeˣ/n! dⁿ/dxⁿ xⁿ⁺ᵅe⁻ˣ: Test Failed at C:\Users\user\Desktop\GitHub\Antiq.jl\test\MorsePotential.jl:44   Expression: acceptance
Stacktrace:  [1] macro expansion    @ C:\Users\user.julia\juliaup\julia-1.9.2+0.x64.w64.mingw32\share\julia\stdlib\v1.9\Test\src\Test.jl:478 [inlined]  [2] macro expansion    @ C:\Users\user\Desktop\GitHub\Antiq.jl\test\MorsePotential.jl:44 [inlined]  [3] macro expansion    @ C:\Users\user.julia\juliaup\julia-1.9.2+0.x64.w64.mingw32\share\julia\stdlib\v1.9\Test\src\Test.jl:1498 [inlined]  [4] top-level scope    @ C:\Users\user\Desktop\GitHub\Antiq.jl\test\MorsePotential.jl:28
\[\begin{aligned}
   L_{4}^{(1)}(x)
    = \frac{\frac{1}{24} e^{x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} x^{5} e^{ - x}}{x}
-  &= 5 - 10 x + 5 x^{2} - \frac{5}{6} x^{3} + \frac{1}{24} x^{4} \\
-  &= 5 - 10 x + 5 x^{2} - \frac{5}{6} x^{3} + \frac{1}{24} x^{4}
-\end{aligned}\]
$n=4, α=2:$ ✔
\[\begin{aligned}
+  &= 5 - 10 x - \frac{5}{6} x^{3} + 5 x^{2} + \frac{1}{24} x^{4} \\
+  &= 5 - \frac{5}{6} x^{3} - 10 x + 5 x^{2} + \frac{1}{24} x^{4}
+\end{aligned}\]
$n=4, α=2:$ ✗ Lₙ⁽ᵅ⁾(x) = x⁻ᵅeˣ/n! dⁿ/dxⁿ xⁿ⁺ᵅe⁻ˣ: Test Failed at C:\Users\user\Desktop\GitHub\Antiq.jl\test\MorsePotential.jl:44   Expression: acceptance
Stacktrace:  [1] macro expansion    @ C:\Users\user.julia\juliaup\julia-1.9.2+0.x64.w64.mingw32\share\julia\stdlib\v1.9\Test\src\Test.jl:478 [inlined]  [2] macro expansion    @ C:\Users\user\Desktop\GitHub\Antiq.jl\test\MorsePotential.jl:44 [inlined]  [3] macro expansion    @ C:\Users\user.julia\juliaup\julia-1.9.2+0.x64.w64.mingw32\share\julia\stdlib\v1.9\Test\src\Test.jl:1498 [inlined]  [4] top-level scope    @ C:\Users\user\Desktop\GitHub\Antiq.jl\test\MorsePotential.jl:28
\[\begin{aligned}
   L_{4}^{(2)}(x)
-   = \frac{\frac{1}{24} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} x^{6} e^{ - x} e^{x}}{x^{2}}
-  &= 15 - 20 x + \frac{15}{2} x^{2} - x^{3} + \frac{1}{24} x^{4} \\
-  &= 15 - 20 x + \frac{15}{2} x^{2} - x^{3} + \frac{1}{24} x^{4}
-\end{aligned}\]
$n=4, α=3:$ ✔
\[\begin{aligned}
+   = \frac{\frac{1}{24} e^{x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} x^{6} e^{ - x}}{x^{2}}
+  &= 15 - 20 x - x^{3} + \frac{15}{2} x^{2} + \frac{1}{24} x^{4} \\
+  &= 15 - x^{3} - 20 x + \frac{15}{2} x^{2} + \frac{1}{24} x^{4}
+\end{aligned}\]
$n=4, α=3:$ ✗ Lₙ⁽ᵅ⁾(x) = x⁻ᵅeˣ/n! dⁿ/dxⁿ xⁿ⁺ᵅe⁻ˣ: Test Failed at C:\Users\user\Desktop\GitHub\Antiq.jl\test\MorsePotential.jl:44   Expression: acceptance
Stacktrace:  [1] macro expansion    @ C:\Users\user.julia\juliaup\julia-1.9.2+0.x64.w64.mingw32\share\julia\stdlib\v1.9\Test\src\Test.jl:478 [inlined]  [2] macro expansion    @ C:\Users\user\Desktop\GitHub\Antiq.jl\test\MorsePotential.jl:44 [inlined]  [3] macro expansion    @ C:\Users\user.julia\juliaup\julia-1.9.2+0.x64.w64.mingw32\share\julia\stdlib\v1.9\Test\src\Test.jl:1498 [inlined]  [4] top-level scope    @ C:\Users\user\Desktop\GitHub\Antiq.jl\test\MorsePotential.jl:28
\[\begin{aligned}
   L_{4}^{(3)}(x)
    = \frac{\frac{1}{24} e^{x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} x^{7} e^{ - x}}{x^{3}}
-  &= 35 - 35 x + \frac{21}{2} x^{2} - \frac{7}{6} x^{3} + \frac{1}{24} x^{4} \\
-  &= 35 - 35 x + \frac{21}{2} x^{2} - \frac{7}{6} x^{3} + \frac{1}{24} x^{4}
-\end{aligned}\]
$n=4, α=4:$ ✔
\[\begin{aligned}
+  &= 35 - 35 x - \frac{7}{6} x^{3} + \frac{21}{2} x^{2} + \frac{1}{24} x^{4} \\
+  &= 35 - \frac{7}{6} x^{3} - 35 x + \frac{21}{2} x^{2} + \frac{1}{24} x^{4}
+\end{aligned}\]
$n=4, α=4:$ ✗ Lₙ⁽ᵅ⁾(x) = x⁻ᵅeˣ/n! dⁿ/dxⁿ xⁿ⁺ᵅe⁻ˣ: Test Failed at C:\Users\user\Desktop\GitHub\Antiq.jl\test\MorsePotential.jl:44   Expression: acceptance
Stacktrace:  [1] macro expansion    @ C:\Users\user.julia\juliaup\julia-1.9.2+0.x64.w64.mingw32\share\julia\stdlib\v1.9\Test\src\Test.jl:478 [inlined]  [2] macro expansion    @ C:\Users\user\Desktop\GitHub\Antiq.jl\test\MorsePotential.jl:44 [inlined]  [3] macro expansion    @ C:\Users\user.julia\juliaup\julia-1.9.2+0.x64.w64.mingw32\share\julia\stdlib\v1.9\Test\src\Test.jl:1498 [inlined]  [4] top-level scope    @ C:\Users\user\Desktop\GitHub\Antiq.jl\test\MorsePotential.jl:28
\[\begin{aligned}
   L_{4}^{(4)}(x)
    = \frac{\frac{1}{24} e^{x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} x^{8} e^{ - x}}{x^{4}}
-  &= 70 - 56 x + 14 x^{2} - \frac{4}{3} x^{3} + \frac{1}{24} x^{4} \\
-  &= 70 - 56 x + 14 x^{2} - \frac{4}{3} x^{3} + \frac{1}{24} x^{4}
-\end{aligned}\]
Test Summary:                      | Pass  Total   Time
-Lₙ⁽ᵅ⁾(x) = x⁻ᵅeˣ/n! dⁿ/dxⁿ xⁿ⁺ᵅe⁻ˣ |   15     15  20.0s
Normalization & Orthogonality of $L_n^{(\alpha)}(x)$
\[\int_0^\infty L_i^{(\alpha)}(x) L_j^{(\alpha)}(x) x^\alpha \mathrm{e}^{-x} \mathrm{d}x = \frac{\Gamma(n+\alpha+1)}{n!} \delta_{ij}\]
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-7.0	  4	  9	0.0000000000000000	0.0000000000000000	0.0000000000000000%	✔
-7.0	  5	  0	0.0000000000000000	0.0000000000000000	0.0000000000000000%	✔
-7.0	  5	  1	0.0000000000000000	0.0000000000000000	0.0000000000000000%	✔
-7.0	  5	  2	0.0000000000000000	0.0000000000000000	0.0000000000000000%	✔
-7.0	  5	  3	0.0000000000000000	0.0000000000000000	0.0000000000000000%	✔
-7.0	  5	  4	0.0000000000000000	0.0000000000000000	0.0000000000000000%	✔
-7.0	  5	  5	6806.2227874771806455	6806.2227874771806455	0.0000000000000000%	✔
-7.0	  5	  6	0.0000000000000000	0.0000000000000000	0.0000000000000000%	✔
-7.0	  5	  7	0.0000000000000000	0.0000000000000000	0.0000000000000000%	✔
-7.0	  5	  8	0.0000000000000000	0.0000000000000000	0.0000000000000000%	✔
-7.0	  5	  9	0.0000000000000000	0.0000000000000000	0.0000000000000000%	✔
-7.0	  6	  0	0.0000000000000000	0.0000000000000000	0.0000000000000000%	✔
-7.0	  6	  1	0.0000000000000000	0.0000000000000000	0.0000000000000000%	✔
-7.0	  6	  2	0.0000000000000000	0.0000000000000000	0.0000000000000000%	✔
-7.0	  6	  3	0.0000000000000000	0.0000000000000000	0.0000000000000000%	✔
-7.0	  6	  4	0.0000000000000000	0.0000000000000000	0.0000000000000000%	✔
-7.0	  6	  5	0.0000000000000000	0.0000000000000000	0.0000000000000000%	✔
-7.0	  6	  6	81674.6734497261786601	81674.6734497261786601	0.0000000000000000%	✔
-7.0	  6	  7	0.0000000000000000	0.0000000000000000	0.0000000000000000%	✔
-7.0	  6	  8	0.0000000000000000	0.0000000000000000	0.0000000000000000%	✔
-7.0	  6	  9	0.0000000000000000	0.0000000000000000	0.0000000000000000%	✔
-7.0	  7	  0	0.0000000000000000	0.0000000000000000	0.0000000000000000%	✔
-7.0	  7	  1	0.0000000000000000	0.0000000000000000	0.0000000000000000%	✔
-7.0	  7	  2	0.0000000000000000	0.0000000000000000	0.0000000000000000%	✔
-7.0	  7	  3	0.0000000000000000	0.0000000000000000	0.0000000000000000%	✔
-7.0	  7	  4	0.0000000000000000	0.0000000000000000	0.0000000000000000%	✔
-7.0	  7	  5	0.0000000000000000	0.0000000000000000	0.0000000000000000%	✔
-7.0	  7	  6	0.0000000000000000	0.0000000000000000	0.0000000000000000%	✔
-7.0	  7	  7	1143445.4282961664721370	1143445.4282961664721370	0.0000000000000000%	✔
-7.0	  7	  8	0.0000000000000000	0.0000000000000000	0.0000000000000000%	✔
-7.0	  7	  9	0.0000000000000000	0.0000000000000000	0.0000000000000000%	✔
-7.0	  8	  0	0.0000000000000000	0.0000000000000000	0.0000000000000000%	✔
-7.0	  8	  1	0.0000000000000000	0.0000000000000000	0.0000000000000000%	✔
-7.0	  8	  2	0.0000000000000000	0.0000000000000000	0.0000000000000000%	✔
-7.0	  8	  3	0.0000000000000000	0.0000000000000000	0.0000000000000000%	✔
-7.0	  8	  4	0.0000000000000000	0.0000000000000000	0.0000000000000000%	✔
-7.0	  8	  5	0.0000000000000000	0.0000000000000000	0.0000000000000000%	✔
-7.0	  8	  6	0.0000000000000000	0.0000000000000000	0.0000000000000000%	✔
-7.0	  8	  7	0.0000000000000000	0.0000000000000000	0.0000000000000000%	✔
-7.0	  8	  8	18295126.8527386635541916	18295126.8527386635541916	0.0000000000000000%	✔
-7.0	  8	  9	0.0000000000000000	0.0000000000000000	0.0000000000000000%	✔
-7.0	  9	  0	0.0000000000000000	0.0000000000000000	0.0000000000000000%	✔
-7.0	  9	  1	0.0000000000000000	0.0000000000000000	0.0000000000000000%	✔
-7.0	  9	  2	0.0000000000000000	0.0000000000000000	0.0000000000000000%	✔
-7.0	  9	  3	0.0000000000000000	0.0000000000000000	0.0000000000000000%	✔
-7.0	  9	  4	0.0000000000000000	0.0000000000000000	0.0000000000000000%	✔
-7.0	  9	  5	0.0000000000000000	0.0000000000000000	0.0000000000000000%	✔
-7.0	  9	  6	0.0000000000000000	0.0000000000000000	0.0000000000000000%	✔
-7.0	  9	  7	0.0000000000000000	0.0000000000000000	0.0000000000000000%	✔
-7.0	  9	  8	0.0000000000000000	0.0000000000000000	0.0000000000000000%	✔
-7.0	  9	  9	329312283.3492959141731262	329312283.3492959141731262	0.0000000000000000%	✔
-Test Summary:                               | Pass  Total  Time
-∫Lᵢ⁽ᵅ⁾Lⱼ⁽ᵅ⁾(x)xᵅexp(-x)dx = Γ(i+α+1)/i! δᵢⱼ |  400    400  0.2s
Normalization & Orthogonality of $\psi_n(r)$
\[\int_0^\infty \psi_i^\ast(r) \psi_j(r) \mathrm{d}r = \delta_{ij}\]
  i	  j	numerical         	analytical        	|error|
-  0	  0	1.0000000000000000	1.0000000000000000	0.0000000000000000%	✔
-  0	  1	0.0000000000000010	0.0000000000000000	0.0000000000000000%	✔
-  0	  2	-0.0000000000000028	0.0000000000000000	0.0000000000000000%	✔
-  0	  3	0.0000000000000107	0.0000000000000000	0.0000000000000000%	✔
-  0	  4	0.0000000000001308	0.0000000000000000	0.0000000000000000%	✔
-  0	  5	-0.0000000000001160	0.0000000000000000	0.0000000000000000%	✔
-  0	  6	-0.0000000000000148	0.0000000000000000	0.0000000000000000%	✔
-  0	  7	0.0000000000024863	0.0000000000000000	0.0000000000000000%	✔
-  0	  8	-0.0000000000256450	0.0000000000000000	0.0000000000000000%	✔
-  0	  9	-0.0000000001036846	0.0000000000000000	0.0000000000000000%	✔
-  1	  0	0.0000000000000010	0.0000000000000000	0.0000000000000000%	✔
-  1	  1	0.9999999999999992	1.0000000000000000	0.0000000000000777%	✔
-  1	  2	-0.0000000000000010	0.0000000000000000	0.0000000000000000%	✔
-  1	  3	0.0000000000000054	0.0000000000000000	0.0000000000000000%	✔
-  1	  4	0.0000000000000185	0.0000000000000000	0.0000000000000000%	✔
-  1	  5	-0.0000000000002839	0.0000000000000000	0.0000000000000000%	✔
-  1	  6	0.0000000000000085	0.0000000000000000	0.0000000000000000%	✔
-  1	  7	0.0000000000014165	0.0000000000000000	0.0000000000000000%	✔
-  1	  8	-0.0000000000217631	0.0000000000000000	0.0000000000000000%	✔
-  1	  9	-0.0000000000667054	0.0000000000000000	0.0000000000000000%	✔
-  2	  0	-0.0000000000000028	0.0000000000000000	0.0000000000000000%	✔
-  2	  1	-0.0000000000000010	0.0000000000000000	0.0000000000000000%	✔
-  2	  2	1.0000000000000004	1.0000000000000000	0.0000000000000444%	✔
-  2	  3	-0.0000000000000019	0.0000000000000000	0.0000000000000000%	✔
-  2	  4	0.0000000000000070	0.0000000000000000	0.0000000000000000%	✔
-  2	  5	-0.0000000000001934	0.0000000000000000	0.0000000000000000%	✔
-  2	  6	0.0000000000002637	0.0000000000000000	0.0000000000000000%	✔
-  2	  7	0.0000000000000954	0.0000000000000000	0.0000000000000000%	✔
-  2	  8	-0.0000000000088982	0.0000000000000000	0.0000000000000000%	✔
-  2	  9	-0.0000000000304947	0.0000000000000000	0.0000000000000000%	✔
-  3	  0	0.0000000000000107	0.0000000000000000	0.0000000000000000%	✔
-  3	  1	0.0000000000000054	0.0000000000000000	0.0000000000000000%	✔
-  3	  2	-0.0000000000000019	0.0000000000000000	0.0000000000000000%	✔
-  3	  3	1.0000000000000002	1.0000000000000000	0.0000000000000222%	✔
-  3	  4	-0.0000000000000056	0.0000000000000000	0.0000000000000000%	✔
-  3	  5	-0.0000000000000417	0.0000000000000000	0.0000000000000000%	✔
-  3	  6	0.0000000000002512	0.0000000000000000	0.0000000000000000%	✔
-  3	  7	-0.0000000000006396	0.0000000000000000	0.0000000000000000%	✔
-  3	  8	-0.0000000000016690	0.0000000000000000	0.0000000000000000%	✔
-  3	  9	-0.0000000000058517	0.0000000000000000	0.0000000000000000%	✔
-  4	  0	0.0000000000001308	0.0000000000000000	0.0000000000000000%	✔
-  4	  1	0.0000000000000185	0.0000000000000000	0.0000000000000000%	✔
-  4	  2	0.0000000000000070	0.0000000000000000	0.0000000000000000%	✔
-  4	  3	-0.0000000000000056	0.0000000000000000	0.0000000000000000%	✔
-  4	  4	0.9999999999999821	1.0000000000000000	0.0000000000017875%	✔
-  4	  5	0.0000000000000129	0.0000000000000000	0.0000000000000000%	✔
-  4	  6	0.0000000000002184	0.0000000000000000	0.0000000000000000%	✔
-  4	  7	0.0000000000001733	0.0000000000000000	0.0000000000000000%	✔
-  4	  8	-0.0000000000012127	0.0000000000000000	0.0000000000000000%	✔
-  4	  9	0.0000000000009228	0.0000000000000000	0.0000000000000000%	✔
-  5	  0	-0.0000000000001160	0.0000000000000000	0.0000000000000000%	✔
-  5	  1	-0.0000000000002839	0.0000000000000000	0.0000000000000000%	✔
-  5	  2	-0.0000000000001934	0.0000000000000000	0.0000000000000000%	✔
-  5	  3	-0.0000000000000417	0.0000000000000000	0.0000000000000000%	✔
-  5	  4	0.0000000000000129	0.0000000000000000	0.0000000000000000%	✔
-  5	  5	1.0000000000000182	1.0000000000000000	0.0000000000018208%	✔
-  5	  6	-0.0000000000000032	0.0000000000000000	0.0000000000000000%	✔
-  5	  7	-0.0000000000007341	0.0000000000000000	0.0000000000000000%	✔
-  5	  8	0.0000000000002908	0.0000000000000000	0.0000000000000000%	✔
-  5	  9	-0.0000000000005260	0.0000000000000000	0.0000000000000000%	✔
-  6	  0	-0.0000000000000148	0.0000000000000000	0.0000000000000000%	✔
-  6	  1	0.0000000000000085	0.0000000000000000	0.0000000000000000%	✔
-  6	  2	0.0000000000002637	0.0000000000000000	0.0000000000000000%	✔
-  6	  3	0.0000000000002512	0.0000000000000000	0.0000000000000000%	✔
-  6	  4	0.0000000000002184	0.0000000000000000	0.0000000000000000%	✔
-  6	  5	-0.0000000000000032	0.0000000000000000	0.0000000000000000%	✔
-  6	  6	1.0000000000002365	1.0000000000000000	0.0000000000236478%	✔
-  6	  7	-0.0000000000001816	0.0000000000000000	0.0000000000000000%	✔
-  6	  8	-0.0000000000018256	0.0000000000000000	0.0000000000000000%	✔
-  6	  9	-0.0000000000029103	0.0000000000000000	0.0000000000000000%	✔
-  7	  0	0.0000000000024863	0.0000000000000000	0.0000000000000000%	✔
-  7	  1	0.0000000000014165	0.0000000000000000	0.0000000000000000%	✔
-  7	  2	0.0000000000000954	0.0000000000000000	0.0000000000000000%	✔
-  7	  3	-0.0000000000006396	0.0000000000000000	0.0000000000000000%	✔
-  7	  4	0.0000000000001733	0.0000000000000000	0.0000000000000000%	✔
-  7	  5	-0.0000000000007341	0.0000000000000000	0.0000000000000000%	✔
-  7	  6	-0.0000000000001816	0.0000000000000000	0.0000000000000000%	✔
-  7	  7	0.9999999999995498	1.0000000000000000	0.0000000000450195%	✔
-  7	  8	-0.0000000000001716	0.0000000000000000	0.0000000000000000%	✔
-  7	  9	0.0000000000041122	0.0000000000000000	0.0000000000000000%	✔
-  8	  0	-0.0000000000256450	0.0000000000000000	0.0000000000000000%	✔
-  8	  1	-0.0000000000217631	0.0000000000000000	0.0000000000000000%	✔
-  8	  2	-0.0000000000088982	0.0000000000000000	0.0000000000000000%	✔
-  8	  3	-0.0000000000016690	0.0000000000000000	0.0000000000000000%	✔
-  8	  4	-0.0000000000012127	0.0000000000000000	0.0000000000000000%	✔
-  8	  5	0.0000000000002908	0.0000000000000000	0.0000000000000000%	✔
-  8	  6	-0.0000000000018256	0.0000000000000000	0.0000000000000000%	✔
-  8	  7	-0.0000000000001716	0.0000000000000000	0.0000000000000000%	✔
-  8	  8	0.9999999999948437	1.0000000000000000	0.0000000005156320%	✔
-  8	  9	0.0000000000003093	0.0000000000000000	0.0000000000000000%	✔
-  9	  0	-0.0000000001036846	0.0000000000000000	0.0000000000000000%	✔
-  9	  1	-0.0000000000667054	0.0000000000000000	0.0000000000000000%	✔
-  9	  2	-0.0000000000304947	0.0000000000000000	0.0000000000000000%	✔
-  9	  3	-0.0000000000058517	0.0000000000000000	0.0000000000000000%	✔
-  9	  4	0.0000000000009228	0.0000000000000000	0.0000000000000000%	✔
-  9	  5	-0.0000000000005260	0.0000000000000000	0.0000000000000000%	✔
-  9	  6	-0.0000000000029103	0.0000000000000000	0.0000000000000000%	✔
-  9	  7	0.0000000000041122	0.0000000000000000	0.0000000000000000%	✔
-  9	  8	0.0000000000003093	0.0000000000000000	0.0000000000000000%	✔
-  9	  9	1.0000000000154354	1.0000000000000000	0.0000000015435431%	✔
-Test Summary: | Pass  Total  Time
-<ψᵢ|ψⱼ> = δᵢⱼ |  100    100  1.9s
Eigen Values
\[  \begin{aligned}
-    E_n
-    &=      \int \psi^\ast_n(r) \hat{H} \psi_n(r) \mathrm{d}x \\
-    &=      \int \psi^\ast_n(r) \left[ \hat{V} + \hat{T} \right] \psi(r) \mathrm{d}x \\
-    &=      \int \psi^\ast_n(r) \left[ V(r) - \frac{\hbar^2}{2m} \frac{\mathrm{d}^{2}}{\mathrm{d} r^{2}} \right] \psi(r) \mathrm{d}x \\
-    &\simeq \int \psi^\ast_n(r) \left[ V(r)\psi(r) -\frac{\hbar^2}{2m} \frac{\psi(r+\Delta r) - 2\psi(r) + \psi(r-\Delta r)}{\Delta r^{2}} \right] \mathrm{d}x.
-  \end{aligned}\]
Where, the difference formula for the 2nd-order derivative:
\[\begin{aligned}
-  % 2\psi(r)
-  % + \frac{\mathrm{d}^{2} \psi(r)}{\mathrm{d} r^{2}} \Delta r^{2}
-  % + O\left(\Delta r^{4}\right)
-  % &=
-  % \psi(r+\Delta r)
-  % + \psi(r-\Delta r)
-  % \\
-  % \frac{\mathrm{d}^{2} \psi(r)}{\mathrm{d} r^{2}} \Delta r^{2}
-  % &=
-  % \psi(r+\Delta r)
-  % - 2\psi(r)
-  % + \psi(r-\Delta r)
-  % - O\left(\Delta r^{4}\right)
-  % \\
-  % \frac{\mathrm{d}^{2} \psi(r)}{\mathrm{d} r^{2}}
-  % &=
-  % \frac{\psi(r+\Delta r) - 2\psi(r) + \psi(r-\Delta r)}{\Delta r^{2}}
-  % - \frac{O\left(\Delta r^{4}\right)}{\Delta r^{2}}
-  % \\
-  \frac{\mathrm{d}^{2} \psi(r)}{\mathrm{d} r^{2}}
-  &=
-  \frac{\psi(r+\Delta r) - 2\psi(r) + \psi(r-\Delta r)}{\Delta r^{2}}
-  + O\left(\Delta r^{2}\right)
-\end{aligned}\]
are given by the sum of 2 Taylor series:
\[\begin{aligned}
-\psi(r+\Delta r)
-&= \psi(r)
-+ \frac{\mathrm{d} \psi(r)}{\mathrm{d} r} \Delta r
-+ \frac{1}{2!} \frac{\mathrm{d}^{2} \psi(r)}{\mathrm{d} r^{2}} \Delta r^{2}
-+ \frac{1}{3!} \frac{\mathrm{d}^{3} \psi(r)}{\mathrm{d} r^{3}} \Delta r^{3}
-+ O\left(\Delta r^{4}\right),
-\\
-\psi(r-\Delta r)
-&= \psi(r)
-- \frac{\mathrm{d} \psi(r)}{\mathrm{d} r} \Delta r
-+ \frac{1}{2!} \frac{\mathrm{d}^{2} \psi(r)}{\mathrm{d} r^{2}} \Delta r^{2}
-- \frac{1}{3!} \frac{\mathrm{d}^{3} \psi(r)}{\mathrm{d} r^{3}} \Delta r^{3}
-+ O\left(\Delta r^{4}\right).
-\end{aligned}\]
       k	  n	numerical         	analytical        	|error|
-0.10000	  0	-0.0974826299431356	-0.0974826299043344	0.0000000398031920%	✔
-0.10000	  1	-0.0875766292083366	-0.0875766290728626	0.0000001546919476%	✔
-0.10000	  2	-0.0782012653591236	-0.0782012650053099	0.0000004524398330%	✔
-0.10000	  3	-0.0693565382661101	-0.0693565377016763	0.0000008138149329%	✔
-0.10000	  4	-0.0610424487768959	-0.0610424471619618	0.0000026455920072%	✔
-0.10000	  5	-0.0532589961311526	-0.0532589933861664	0.0000051540333741%	✔
-0.10000	  6	-0.0460061778289117	-0.0460061763742900	0.0000031617965576%	✔
-0.10000	  7	-0.0392839977431085	-0.0392839961263328	0.0000041156089608%	✔
-0.10000	  8	-0.0330924678509606	-0.0330924526422947	0.0000459581104973%	✔
-0.10000	  9	-0.0274314677918590	-0.0274315459221756	0.0002848192256583%	✔
-0.20000	  0	-0.0953874611439631	-0.0953874610810323	0.0000000659738844%	✔
-0.20000	  1	-0.0816891004273675	-0.0816891001758958	0.0000003078400078%	✔
-0.20000	  2	-0.0690520133797856	-0.0690520127985975	0.0000008416671348%	✔
-0.20000	  3	-0.0574761998667120	-0.0574761989491373	0.0000015964429232%	✔
-0.20000	  4	-0.0469616603170172	-0.0469616586275153	0.0000035976196186%	✔
-0.20000	  5	-0.0375083932021688	-0.0375083918337316	0.0000036483494226%	✔
-0.20000	  6	-0.0291164003400053	-0.0291163985677860	0.0000060866708238%	✔
-0.20000	  7	-0.0217856840623437	-0.0217856788296786	0.0000240188297604%	✔
-0.20000	  8	-0.0155162375387778	-0.0155162326194094	0.0000317046571552%	✔
-0.20000	  9	-0.0103080627546147	-0.0103080599369784	0.0000273343029443%	✔
-0.30000	  0	-0.0937952146950324	-0.0937952146053093	0.0000000956584648%	✔
-0.30000	  1	-0.0773103386936924	-0.0773103383216661	0.0000004812114878%	✔
-0.30000	  2	-0.0624173731668509	-0.0624173723297802	0.0000013410860668%	✔
-0.30000	  3	-0.0491163180291512	-0.0491163166296516	0.0000028493577729%	✔
-0.30000	  4	-0.0374071730726919	-0.0374071712212802	0.0000049493494821%	✔
-0.30000	  5	-0.0272899380273731	-0.0272899361046662	0.0000070454797707%	✔
-0.30000	  6	-0.0187646136934960	-0.0187646112798094	0.0000128629716864%	✔
-0.30000	  7	-0.0118311981022458	-0.0118311967467099	0.0000114573015073%	✔
-0.30000	  8	-0.0064896942752905	-0.0064896925053676	0.0000272728303901%	✔
-0.30000	  9	-0.0027401008929208	-0.0027400985557827	0.0000852939413987%	✔
-0.10273	  0	-0.0974137779674584	-0.0974137779441826	0.0000000238937717%	✔
-0.10273	  1	-0.0873809242053181	-0.0873809240676091	0.0000001575961943%	✔
-0.10273	  2	-0.0778931751450495	-0.0778931747885570	0.0000004576684890%	✔
-0.10273	  3	-0.0689505306599348	-0.0689505301070264	0.0000008018914211%	✔
-0.10273	  4	-0.0605529890953712	-0.0605529900230172	0.0000015319574295%	✔
-0.10273	  5	-0.0527005572549377	-0.0527005545365296	0.0000051582153937%	✔
-0.10273	  6	-0.0453932228176823	-0.0453932236475634	0.0000018282046030%	✔
-0.10273	  7	-0.0386310171570861	-0.0386309973561186	0.0000512566822197%	✔
-0.10273	  8	-0.0324138862460601	-0.0324138756621953	0.0000326522656802%	✔
-0.10273	  9	-0.0267420183757886	-0.0267418585657935	0.0005976024244533%	✔
-Test Summary:              | Pass  Total  Time
-<ψₙ|H|ψₙ> = ∫ψₙ*Hψₙdx = Eₙ |   40     40  4.5s
-

+  &= 70 - 56 x - \frac{4}{3} x^{3} + 14 x^{2} + \frac{1}{24} x^{4} \\
+  &= 70 - \frac{4}{3} x^{3} - 56 x + 14 x^{2} + \frac{1}{24} x^{4}
+\end{aligned}\]
Test Summary:                      | Pass  Fail  Total   Time
+Lₙ⁽ᵅ⁾(x) = x⁻ᵅeˣ/n! dⁿ/dxⁿ xⁿ⁺ᵅe⁻ˣ |    8     7     15  15.8s
+Error: LoadError: Some tests did not pass: 8 passed, 7 failed, 0 errored, 0 broken.
+in expression starting at C:\Users\user\Desktop\GitHub\Antiq.jl\test\MorsePotential.jl:27

diff --git a/dev/assets/fig/HarmonicOscillator_6_1.png b/dev/assets/fig/HarmonicOscillator_6_1.png
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diff --git a/dev/assets/fig/InfinitePotentialWell_5_1.png b/dev/assets/fig/InfinitePotentialWell_5_1.png
index ca2d25d..f64791d 100644
Binary files a/dev/assets/fig/InfinitePotentialWell_5_1.png and b/dev/assets/fig/InfinitePotentialWell_5_1.png differ
diff --git a/dev/assets/fig/MorsePotential_6_1.png b/dev/assets/fig/MorsePotential_6_1.png
index 776f9d6..b724c27 100644
Binary files a/dev/assets/fig/MorsePotential_6_1.png and b/dev/assets/fig/MorsePotential_6_1.png differ
diff --git a/dev/index.html b/dev/index.html
index d80ba69..721fcb2 100644
--- a/dev/index.html
+++ b/dev/index.html
@@ -30,4 +30,4 @@
     
     HydrogenAtom
   
-
Future Works
List of quantum-mechanical systems with analytical solutions
Acknowledgment
This package was named by @KB-satou and @ultimatile:
Analytical soulutions of Schrödinger equations.

+
Future Works
List of quantum-mechanical systems with analytical solutions
Acknowledgment
This package was named by @KB-satou and @ultimatile:
Analytical soulutions of Schrödinger equations.

diff --git a/dev/jmd/HarmonicOscillator.jmd b/dev/jmd/HarmonicOscillator.jmd
index ff1052b..1b6f3a4 100644
--- a/dev/jmd/HarmonicOscillator.jmd
+++ b/dev/jmd/HarmonicOscillator.jmd
@@ -133,7 +133,7 @@ Potential energy curve, Energy levels, Wave functions:
 
 ```julia
 using Plots
-plot(xlim=(-5.5,5.5), ylim=(-0.2,5.4), xlabel="\$x\$", ylabel="\$V(x),~E_n,~\\psi_n(x)\\times0.5+E_n\$", size=(480,400), dpi=400)
+plot(xlim=(-5.5,5.5), ylim=(-0.2,5.4), xlabel="\$x\$", ylabel="\$V(x),~E_n,~\\psi_n(x)\\times0.5+E_n\$", size=(480,400), dpi=300)
 for n in 0:4
   # energy
   hline!([HO.E(n=n)], lc=:black, ls=:dash, label="")
diff --git a/dev/jmd/HydrogenAtom.jmd b/dev/jmd/HydrogenAtom.jmd
index fe0792a..d1d7336 100644
--- a/dev/jmd/HydrogenAtom.jmd
+++ b/dev/jmd/HydrogenAtom.jmd
@@ -212,7 +212,7 @@ Potential energy curve:
 
 ```julia
 using Plots
-plot(xlims=(0.0,15.0), ylims=(-0.6,0.05), xlabel="\$r~/~a_0\$", ylabel="\$V(r)/E_\\mathrm{h},~E_n/E_\\mathrm{h}\$", legend=:bottomright, size=(480,400), dpi=400)
+plot(xlims=(0.0,15.0), ylims=(-0.6,0.05), xlabel="\$r~/~a_0\$", ylabel="\$V(r)/E_\\mathrm{h},~E_n/E_\\mathrm{h}\$", legend=:bottomright, size=(480,400))
 plot!(0.1:0.01:15, r -> H.V(r), lc=:black, lw=2, label="") # potential
 ```
 
@@ -220,7 +220,7 @@ Potential energy curve, Energy levels:
 
 ```julia
 using Plots
-plot(xlims=(0.0,15.0), ylims=(-0.6,0.05), xlabel="\$r~/~a_0\$", ylabel="\$V(r)/E_\\mathrm{h}\$", legend=:bottomright, size=(480,400), dpi=400)
+plot(xlims=(0.0,15.0), ylims=(-0.6,0.05), xlabel="\$r~/~a_0\$", ylabel="\$V(r)/E_\\mathrm{h}\$", legend=:bottomright, size=(480,400))
 for n in 0:10
   plot!(0.0:0.01:15, r -> H.E(n=n) > H.V(r) ? H.E(n=n) : NaN, lc=n, lw=1, label="") # energy level
 end
@@ -231,7 +231,7 @@ Radial functions:
 
 ```julia
 using Plots
-plot(xlabel="\$r~/~a_0\$", ylabel="\$r^2|R_{nl}(r)|^2~/~a_0^{-1}\$", ylims=(-0.01,0.55), xticks=0:1:20, size=(480,400), dpi=400)
+plot(xlabel="\$r~/~a_0\$", ylabel="\$r^2|R_{nl}(r)|^2~/~a_0^{-1}\$", ylims=(-0.01,0.55), xticks=0:1:20, size=(480,400), dpi=300)
 for n in 1:3
   for l in 0:n-1
     plot!(0:0.01:20, r->r^2*H.R(r,n=n,l=l)^2, lc=n, lw=2, ls=[:solid,:dash,:dot,:dashdot,:dashdotdot][l+1], label="\$n = $n, l=$l\$")
diff --git a/dev/jmd/InfinitePotentialWell.jmd b/dev/jmd/InfinitePotentialWell.jmd
index a71cba3..3c0b509 100644
--- a/dev/jmd/InfinitePotentialWell.jmd
+++ b/dev/jmd/InfinitePotentialWell.jmd
@@ -89,7 +89,7 @@ Potential energy curve, Energy levels, Wave functions:
 ```julia
 L = 1
 using Plots
-plot(xlim=(-0.5,1.5), ylim=(-5,140), xlabel="\$x\$", ylabel="\$V(x),~E_n,~\\psi_n(x)\\times5+E_n\$", size=(480,400), dpi=400)
+plot(xlim=(-0.5,1.5), ylim=(-5,140), xlabel="\$x\$", ylabel="\$V(x),~E_n,~\\psi_n(x)\\times5+E_n\$", size=(480,400), dpi=300)
 for n in 1:5
   # energy
   plot!([0,L], fill(IPW.E(n=n,L=L),2), lc=:black, lw=2, label="")
diff --git a/dev/jmd/MorsePotential.jmd b/dev/jmd/MorsePotential.jmd
index e03765c..d7005ca 100644
--- a/dev/jmd/MorsePotential.jmd
+++ b/dev/jmd/MorsePotential.jmd
@@ -130,7 +130,7 @@ Potential energy curve, Energy levels, Comparison with harmonic oscillator:
 MP = antiq(:MorsePotential)
 HO = antiq(:HarmonicOscillator, k=MP.k, m=MP.μ)
 using Plots
-plot(xlims=(0.1,9.1), ylims=(-0.11,0.01), xlabel="\$r\$", ylabel="\$V(r), E_n\$", legend=:bottomright, size=(480,400), dpi=400)
+plot(xlims=(0.1,9.1), ylims=(-0.11,0.01), xlabel="\$r\$", ylabel="\$V(r), E_n\$", legend=:bottomright, size=(480,400), dpi=300)
 for n in 0:MP.nₘₐₓ()
   # energy
   EM = MP.E(n=n)
diff --git a/dev/search_index.js b/dev/search_index.js
index 03d0395..aa7f2bd 100644
--- a/dev/search_index.js
+++ b/dev/search_index.js
@@ -1,3 +1,3 @@
 var documenterSearchIndex = {"docs":
-[{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"CurrentModule = Antiq","category":"page"},{"location":"InfinitePotentialWell/#Infinite-Potential-Well-(Particle-in-a-Box)","page":"Infinite Potential Well","title":"Infinite Potential Well (Particle in a Box)","text":"","category":"section"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"The infinite potential well (particle in a box) is the simplest model for quantum mechanical system.","category":"page"},{"location":"InfinitePotentialWell/#Definitions","page":"Infinite Potential Well","title":"Definitions","text":"","category":"section"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"L is the length of the 1D-box, m is the mass of particle.","category":"page"},{"location":"InfinitePotentialWell/#Schrödinger-Equation","page":"Infinite Potential Well","title":"Schrödinger Equation","text":"","category":"section"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"  hatH psi(x) = E psi(x)","category":"page"},{"location":"InfinitePotentialWell/#Hamiltonian","page":"Infinite Potential Well","title":"Hamiltonian","text":"","category":"section"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"  hatH = frachbar^22m fracmathrmd^2mathrmdx ^2 + V(x)","category":"page"},{"location":"InfinitePotentialWell/#Potential","page":"Infinite Potential Well","title":"Potential","text":"","category":"section"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"V(x; L=L)","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"  V(x) =\n  left\n    beginarrayll\n    infty  x lt 0 L lt x \n    0       0 leq x leq L\n    endarray\n  right","category":"page"},{"location":"InfinitePotentialWell/#Eigen-Values","page":"Infinite Potential Well","title":"Eigen Values","text":"","category":"section"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"E(; n=0, L=L, m=m, ℏ=ℏ)","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"  E_n = frachbar^2 n^2 pi^22 m L^2","category":"page"},{"location":"InfinitePotentialWell/#Eigen-Functions","page":"Infinite Potential Well","title":"Eigen Functions","text":"","category":"section"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"ψ(x; n=0, L=L)","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"   psi_n(x) = sqrtfrac2L sin fracnpi xL","category":"page"},{"location":"InfinitePotentialWell/#Proofs","page":"Infinite Potential Well","title":"Proofs","text":"","category":"section"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"Eigen Functions & Eigen Values\nNormalization","category":"page"},{"location":"InfinitePotentialWell/#Usage-and-Examples","page":"Infinite Potential Well","title":"Usage & Examples","text":"","category":"section"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"Install Antiq.jl for the first run and run using Antiq before each use. The function antiq(model, parameters...) returns a module that has E(), ψ(x), V(x) and some other functions. In this system, the model name is specified by :InfinitePotentialWell and several parameters L, m and ℏ are set as optional arguments.","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"using Antiq\nIPW = antiq(:InfinitePotentialWell, L=1.0, m=1.0, ℏ=1.0)","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"Parameters:","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"julia> IPW.L\n1.0\n\njulia> IPW.m\n1.0\n\njulia> IPW.ℏ\n1.0","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"Eigen values:","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"julia> IPW.E(n=1)\n4.934802200544679\n\njulia> IPW.E(n=2)\n19.739208802178716","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"Wave functions:","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"using Plots\nplot(xlim=(0,1), xlabel=\"x\", ylabel=\"ψ(x)\")\nplot!(x -> IPW.ψ(x, n=1), label=\"n=1\", lw=2)\nplot!(x -> IPW.ψ(x, n=2), label=\"n=2\", lw=2)\nplot!(x -> IPW.ψ(x, n=3), label=\"n=3\", lw=2)\nplot!(x -> IPW.ψ(x, n=4), label=\"n=4\", lw=2)\nplot!(x -> IPW.ψ(x, n=5), label=\"n=5\", lw=2)","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"(Image: )","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"Potential energy curve, Energy levels, Wave functions:","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"L = 1\nusing Plots\nplot(xlim=(-0.5,1.5), ylim=(-5,140), xlabel=\"\\$x\\$\", ylabel=\"\\$V(x),~E_n,~\\\\psi_n(x)\\\\times5+E_n\\$\", size=(480,400), dpi=400)\nfor n in 1:5\n  # energy\n  plot!([0,L], fill(IPW.E(n=n,L=L),2), lc=:black, lw=2, label=\"\")\n  # wave function\n  plot!(0:0.01:L, x->IPW.E(n=n,L=L)+5*IPW.ψ(x,n=n,L=L), lc=n, lw=2, label=\"\")\nend\n# potential\nplot!([0,0,L,L], [140,0,0,140], lc=:black, lw=2, label=\"\")","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"(Image: )","category":"page"},{"location":"InfinitePotentialWell/#Testing","page":"Infinite Potential Well","title":"Testing","text":"","category":"section"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"Unit testing and Integration testing were done using numerical integration (QuadGK.jl). The test script is here.","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"include(\"../../../test/InfinitePotentialWell.jl\")","category":"page"},{"location":"InfinitePotentialWell/#Normalization-and-Orthogonality-of-\\psi_n(x)","page":"Infinite Potential Well","title":"Normalization & Orthogonality of psi_n(x)","text":"","category":"section"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"int_0^L psi_i^ast(x) psi_j(x) mathrmdx = delta_ij","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"  i\t  j\tnumerical         \tanalytical        \t|error|\n  1\t  1\t1.0000000000000002\t1.0000000000000000\t0.0000000000000222%\t✔\n  1\t  2\t0.0000000000000000\t0.0000000000000000\t0.0000000000000000%\t✔\n  1\t  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4\t0.0000000000000000\t0.0000000000000000\t0.0000000000000000%\t✔\n 10\t  5\t0.0000000000000000\t0.0000000000000000\t0.0000000000000000%\t✔\n 10\t  6\t-0.0000000000000000\t0.0000000000000000\t0.0000000000000000%\t✔\n 10\t  7\t-0.0000000000000000\t0.0000000000000000\t0.0000000000000000%\t✔\n 10\t  8\t0.0000000000000000\t0.0000000000000000\t0.0000000000000000%\t✔\n 10\t  9\t0.0000000000000001\t0.0000000000000000\t0.0000000000000000%\t✔\n 10\t 10\t1.0000000000000002\t1.0000000000000000\t0.0000000000000222%\t✔\nTest Summary:            | Pass  Total  Time\n<ψᵢ|ψⱼ> = ∫ψₙ*ψₙdx = δᵢⱼ |  100    100  1.5s","category":"page"},{"location":"InfinitePotentialWell/#Eigen-Values-2","page":"Infinite Potential Well","title":"Eigen Values","text":"","category":"section"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"  beginaligned\n    E_n\n    =      int_0^L psi^ast_n(x) hatH psi_n(x) mathrmdx \n    =      int_0^L psi^ast_n(x) left hatV + hatT right psi(x) mathrmdx \n    =      int_0^L psi^ast_n(x) left 0 - frachbar^22m fracmathrmd^2mathrmd x^2 right psi(x) mathrmdx \n    simeq int_0^L psi^ast_n(x) left -frachbar^22m fracpsi(x+Delta x) - 2psi(x) + psi(x-Delta x)Delta x^2 right mathrmdx\n  endaligned","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"Where, the difference formula for the 2nd-order derivative:","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"beginaligned\n   2psi(x)\n   + fracmathrmd^2 psi(x)mathrmd x^2 Delta x^2\n   + Oleft(Delta x^4right)\n   =\n   psi(x+Delta x)\n   + psi(x-Delta x)\n   \n   fracmathrmd^2 psi(x)mathrmd x^2 Delta x^2\n   =\n   psi(x+Delta x)\n   - 2psi(x)\n   + psi(x-Delta x)\n   - Oleft(Delta x^4right)\n   \n   fracmathrmd^2 psi(x)mathrmd x^2\n   =\n   fracpsi(x+Delta x) - 2psi(x) + psi(x-Delta x)Delta x^2\n   - fracOleft(Delta x^4right)Delta x^2\n   \n  fracmathrmd^2 psi(x)mathrmd x^2\n  =\n  fracpsi(x+Delta x) - 2psi(x) + psi(x-Delta x)Delta x^2\n  + Oleft(Delta x^2right)\nendaligned","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"are given by the sum of 2 Taylor series:","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"beginaligned\npsi(x+Delta x)\n= psi(x)\n+ fracmathrmd psi(x)mathrmd x Delta x\n+ frac12 fracmathrmd^2 psi(x)mathrmd x^2 Delta x^2\n+ frac13 fracmathrmd^3 psi(x)mathrmd x^3 Delta x^3\n+ Oleft(Delta x^4right)\n\npsi(x-Delta x)\n= psi(x)\n- fracmathrmd psi(x)mathrmd x Delta x\n+ frac12 fracmathrmd^2 psi(x)mathrmd x^2 Delta x^2\n- frac13 fracmathrmd^3 psi(x)mathrmd x^3 Delta x^3\n+ Oleft(Delta x^4right)\nendaligned","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"  L    m    ℏ   n  numerical     \tanalytical    \t|error|\n0.1  0.1  0.1   1  49.348021579139\t49.348022005447\t0.000000863880%\t✔\n0.1  0.1  0.1   2  197.392081461942\t197.392088021787\t0.000003323256%\t✔\n0.1  0.1  0.1   3  444.132165131018\t444.132198049021\t0.000007411758%\t✔\n0.1  0.1  0.1   4  789.568248175979\t789.568352087149\t0.000013160503%\t✔\n0.1  0.1  0.1   5  1233.700296336187\t1233.700550136170\t0.000020572252%\t✔\n0.1  0.1  0.1   6  1776.528266243334\t1776.528792196084\t0.000029605642%\t✔\n0.1  0.1  0.1   7  2418.052103857080\t2418.053078266893\t0.000040297288%\t✔\n0.1  0.1  0.1   8  3158.271745875927\t3158.273408348594\t0.000052638656%\t✔\n0.1  0.1  0.1   9  3997.187119264267\t3997.189782441189\t0.000066626232%\t✔\n0.1  0.1  0.1  10  4934.798141994514\t4934.802200544678\t0.000082243421%\t✔\n0.1  0.1  1.0   1  4934.802157913905\t4934.802200544678\t0.000000863880%\t✔\n0.1  0.1  1.0   2  19739.208146194214\t19739.208802178713\t0.000003323256%\t✔\n0.1  0.1  1.0   3  44413.216513101754\t44413.219804902103\t0.000007411758%\t✔\n0.1  0.1  1.0   4  78956.824817597895\t78956.835208714852\t0.000013160503%\t✔\n0.1  0.1  1.0   5  123370.029633618717\t123370.055013616948\t0.000020572252%\t✔\n0.1  0.1  1.0   6  177652.826624333364\t177652.879219608410\t0.000029605642%\t✔\n0.1  0.1  1.0   7  241805.210385707964\t241805.307826689212\t0.000040297288%\t✔\n0.1  0.1  1.0   8  315827.174587592541\t315827.340834859409\t0.000052638656%\t✔\n0.1  0.1  1.0   9  399718.711926426622\t399718.978244118916\t0.000066626232%\t✔\n0.1  0.1  1.0  10  493479.814199451241\t493480.220054467791\t0.000082243421%\t✔\n0.1  1.0  0.1   1  4.934802157914\t4.934802200545\t0.000000863880%\t✔\n0.1  1.0  0.1   2  19.739208146194\t19.739208802179\t0.000003323256%\t✔\n0.1  1.0  0.1   3  44.413216513102\t44.413219804902\t0.000007411758%\t✔\n0.1  1.0  0.1   4  78.956824817598\t78.956835208715\t0.000013160503%\t✔\n0.1  1.0  0.1   5  123.370029633619\t123.370055013617\t0.000020572252%\t✔\n0.1  1.0  0.1   6  177.652826624333\t177.652879219608\t0.000029605642%\t✔\n0.1  1.0  0.1   7  241.805210385708\t241.805307826689\t0.000040297288%\t✔\n0.1  1.0  0.1   8  315.827174587593\t315.827340834859\t0.000052638656%\t✔\n0.1  1.0  0.1   9  399.718711926427\t399.718978244119\t0.000066626232%\t✔\n0.1  1.0  0.1  10  493.479814199451\t493.480220054468\t0.000082243421%\t✔\n0.1  1.0  1.0   1  493.480215791390\t493.480220054468\t0.000000863880%\t✔\n0.1  1.0  1.0   2  1973.920814619422\t1973.920880217871\t0.000003323256%\t✔\n0.1  1.0  1.0   3  4441.321651310176\t4441.321980490210\t0.000007411758%\t✔\n0.1  1.0  1.0   4  7895.682481759791\t7895.683520871485\t0.000013160503%\t✔\n0.1  1.0  1.0   5  12337.002963361872\t12337.005501361695\t0.000020572252%\t✔\n0.1  1.0  1.0   6  17765.282662433339\t17765.287921960840\t0.000029605642%\t✔\n0.1  1.0  1.0   7  24180.521038570794\t24180.530782668924\t0.000040297288%\t✔\n0.1  1.0  1.0   8  31582.717458759253\t31582.734083485939\t0.000052638656%\t✔\n0.1  1.0  1.0   9  39971.871192642662\t39971.897824411892\t0.000066626232%\t✔\n0.1  1.0  1.0  10  49347.981419945128\t49348.022005446779\t0.000082243421%\t✔\n1.0  0.1  0.1   1  0.493480215948\t0.493480220054\t0.000000832112%\t✔\n1.0  0.1  0.1   2  1.973920815419\t1.973920880218\t0.000003282767%\t✔\n1.0  0.1  0.1   3  4.441321651944\t4.441321980490\t0.000007397478%\t✔\n1.0  0.1  0.1   4  7.895682481265\t7.895683520871\t0.000013166775%\t✔\n1.0  0.1  0.1   5  12.337002965030\t12.337005501362\t0.000020558729%\t✔\n1.0  0.1  0.1   6  17.765282661715\t17.765287921961\t0.000029609687%\t✔\n1.0  0.1  0.1   7  24.180521036064\t24.180530782669\t0.000040307654%\t✔\n1.0  0.1  0.1   8  31.582717460023\t31.582734083486\t0.000052634655%\t✔\n1.0  0.1  0.1   9  39.971871195191\t39.971897824412\t0.000066619856%\t✔\n1.0  0.1  0.1  10  49.347981417827\t49.348022005447\t0.000082247714%\t✔\n1.0  0.1  1.0   1  49.348021594816\t49.348022005447\t0.000000832112%\t✔\n1.0  0.1  1.0   2  197.392081541864\t197.392088021787\t0.000003282767%\t✔\n1.0  0.1  1.0   3  444.132165194438\t444.132198049021\t0.000007397478%\t✔\n1.0  0.1  1.0   4  789.568248126463\t789.568352087149\t0.000013166775%\t✔\n1.0  0.1  1.0   5  1233.700296503016\t1233.700550136170\t0.000020558729%\t✔\n1.0  0.1  1.0   6  1776.528266171473\t1776.528792196084\t0.000029609687%\t✔\n1.0  0.1  1.0   7  2418.052103606433\t2418.053078266892\t0.000040307654%\t✔\n1.0  0.1  1.0   8  3158.271746002275\t3158.273408348594\t0.000052634655%\t✔\n1.0  0.1  1.0   9  3997.187119519121\t3997.189782441190\t0.000066619856%\t✔\n1.0  0.1  1.0  10  4934.798141782662\t4934.802200544679\t0.000082247714%\t✔\n1.0  1.0  0.1   1  0.049348021595\t0.049348022005\t0.000000832112%\t✔\n1.0  1.0  0.1   2  0.197392081542\t0.197392088022\t0.000003282767%\t✔\n1.0  1.0  0.1   3  0.444132165194\t0.444132198049\t0.000007397478%\t✔\n1.0  1.0  0.1   4  0.789568248126\t0.789568352087\t0.000013166775%\t✔\n1.0  1.0  0.1   5  1.233700296503\t1.233700550136\t0.000020558729%\t✔\n1.0  1.0  0.1   6  1.776528266171\t1.776528792196\t0.000029609687%\t✔\n1.0  1.0  0.1   7  2.418052103606\t2.418053078267\t0.000040307654%\t✔\n1.0  1.0  0.1   8  3.158271746002\t3.158273408349\t0.000052634655%\t✔\n1.0  1.0  0.1   9  3.997187119519\t3.997189782441\t0.000066619856%\t✔\n1.0  1.0  0.1  10  4.934798141783\t4.934802200545\t0.000082247714%\t✔\n1.0  1.0  1.0   1  4.934802159482\t4.934802200545\t0.000000832112%\t✔\n1.0  1.0  1.0   2  19.739208154186\t19.739208802179\t0.000003282767%\t✔\n1.0  1.0  1.0   3  44.413216519444\t44.413219804902\t0.000007397478%\t✔\n1.0  1.0  1.0   4  78.956824812646\t78.956835208715\t0.000013166775%\t✔\n1.0  1.0  1.0   5  123.370029650302\t123.370055013617\t0.000020558729%\t✔\n1.0  1.0  1.0   6  177.652826617147\t177.652879219608\t0.000029609687%\t✔\n1.0  1.0  1.0   7  241.805210360643\t241.805307826689\t0.000040307654%\t✔\n1.0  1.0  1.0   8  315.827174600228\t315.827340834859\t0.000052634655%\t✔\n1.0  1.0  1.0   9  399.718711951912\t399.718978244119\t0.000066619856%\t✔\n1.0  1.0  1.0  10  493.479814178266\t493.480220054468\t0.000082247714%\t✔\nTest Summary:               | Pass  Total  Time\n<ψₙ|H|ψₙ>  = ∫ψₙ*Tψₙdx = Eₙ |   80     80  1.2s","category":"page"},{"location":"InfinitePotentialWell/#Expected-Value-of-x","page":"Infinite Potential Well","title":"Expected Value of x","text":"","category":"section"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"langle x rangle_n=1\n= int_0^L psi_1^ast(x) hatx psi_1(x) mathrmdx\n= frac2(2a)^2pi^3 left( fracpi^36 - fracpi4 right)","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"for only n=1.","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"Reference:","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"LibreTexts PHYSICS, 6.4: Expectation Values, Observables, and Uncertainty","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"   L\t  n\tnumerical         \tanalytical        \t|error|\n0.1\t  1\t0.0500000000000000\t0.0500000000000000\t0.0000000000000278%\t✔\n0.5\t  1\t0.2500000000000000\t0.2500000000000000\t0.0000000000000000%\t✔\n1.0\t  1\t0.5000000000000002\t0.5000000000000000\t0.0000000000000444%\t✔\n7.0\t  1\t3.5000000000000000\t3.5000000000000000\t0.0000000000000000%\t✔\nTest Summary:    | Pass  Total  Time\n<ψₙ|x|ψₙ>  = L/2 |    4      4  0.7s","category":"page"},{"location":"InfinitePotentialWell/#Expected-Value-of-x2","page":"Infinite Potential Well","title":"Expected Value of x^2","text":"","category":"section"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"langle x^2 rangle_n=1\n= int_0^L psi_1^ast(x) hatx^2 psi_1(x) mathrmdx\n= frac2(2a)^2pi^3 left( fracpi^36 - fracpi4 right)","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"Reference:","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"LibreTexts PHYSICS, 6.4: Expectation Values, Observables, and Uncertainty","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"   L\t  n\tnumerical         \tanalytical        \t|error|\n0.1\t  1\t0.0028267274151216\t0.0028267274151216\t0.0000000000000460%\t✔\n0.5\t  1\t0.0706681853780411\t0.0706681853780411\t0.0000000000000000%\t✔\n1.0\t  1\t0.2826727415121645\t0.2826727415121645\t0.0000000000000196%\t✔\n7.0\t  1\t13.8509643340960604\t13.8509643340960569\t0.0000000000000256%\t✔\nTest Summary:                 | Pass  Total  Time\n<ψₙ|x²|ψₙ> = 2L²/π³(π³/6-π/4) |    4      4  0.5s","category":"page"},{"location":"InfinitePotentialWell/#Expected-Value-of-p","page":"Infinite Potential Well","title":"Expected Value of p","text":"","category":"section"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"langle p rangle_n=1\n= int_0^L psi_1^ast(x) hatp psi_1(x) mathrmdx\n= 0","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"Reference:","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"LibreTexts PHYSICS, 6.4: Expectation Values, Observables, and Uncertainty","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"  beginaligned\n    langle p rangle_n=1\n    = int_0^L psi^ast_n(x) hatp psi_n(x) mathrmdx \n    = int_0^L psi^ast_n(x) left -ihbarfracmathrmdmathrmd x right psi(x) mathrmdx \n    simeq int_0^L psi^ast_n(x) left -ihbar fracpsi(x+Delta x) - psi(x-Delta x)2Delta x right mathrmdx\n  endaligned","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"Where, the difference formula for the 2nd-order derivative:","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"beginaligned\n   2fracmathrmd psi(x)mathrmdx Delta x\n   + Oleft(Delta x^3right)\n   = \n   psi(x+Delta x)\n   - psi(x-Delta x)\n   \n   2fracmathrmd psi(x)mathrmdx Delta x\n   = \n   psi(x+Delta x)\n   - psi(x-Delta x)\n   - Oleft(Delta x^3right)\n   \n   fracmathrmd psi(x)mathrmdx\n   = \n   fracpsi(x+Delta x)- psi(x-Delta x)2Delta x\n   - fracOleft(Delta x^3right)2Delta x\n   \n  fracmathrmd psi(x)mathrmdx\n  = \n  fracpsi(x+Delta x)- psi(x-Delta x)2Delta x\n  + Oleft(Delta x^2right)\nendaligned","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"are given by the sum of 2 Taylor series:","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"beginaligned\n  psi(x+Delta x)\n  =\n  psi(x)\n  + fracmathrmd psi(x)mathrmdx Delta x\n  + frac12 fracmathrmd^2 psi(x)mathrmdx^2 Delta x^2\n  + Oleft(Delta x^3right)\n  \n  psi(x-Delta x)\n  =\n  psi(x)\n  - fracmathrmd psi(x)mathrmdx Delta x\n  + frac12 fracmathrmd^2 psi(x)mathrmdx^2 Delta x^2\n  + Oleft(Delta x^3right)\nendaligned","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"   L\t  n\tnumerical         \tanalytical        \t|error|\n0.1\t  1\t0.0000000000033684\t0.0000000000000000\t0.0000000000000000%\t✔\n0.5\t  1\t0.0000000000002308\t0.0000000000000000\t0.0000000000000000%\t✔\n1.0\t  1\t0.0000000000001066\t0.0000000000000000\t0.0000000000000000%\t✔\n7.0\t  1\t0.0000000000000252\t0.0000000000000000\t0.0000000000000000%\t✔\nTest Summary:                      | Pass  Total  Time\n<ψₙ|p|ψₙ>  = ∫ψₙ*(-iℏd/dx)ψₙdx = 0 |    4      4  0.9s","category":"page"},{"location":"InfinitePotentialWell/#Expected-Value-of-p2","page":"Infinite Potential Well","title":"Expected Value of p^2","text":"","category":"section"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"langle p^2 rangle\n= int_0^L psi_1^ast(x) hatp^2 psi_1(x) mathrmdx\n= fracpi^2hbar^2L^2","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"Reference:","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"LibreTexts PHYSICS, 6.4: Expectation Values, Observables, and Uncertainty","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"  beginaligned\n    langle p^2 rangle\n    = int_0^L psi^ast_n(x) hatp psi_n(x) mathrmdx \n    = int_0^L psi^ast_n(x) left -hbar^2fracmathrmd^2mathrmdx^2 right psi(x) mathrmdx \n    simeq int_0^L psi^ast_n(x) left -hbar^2 fracpsi(x+Delta x) - 2psi(x) + psi(x-Delta x)Delta x^2 right mathrmdx\n  endaligned","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"Where, the difference formula for the 2nd-order derivative:","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"beginaligned\n   2psi(x)\n   + fracmathrmd^2 psi(x)mathrmd x^2 Delta x^2\n   + Oleft(Delta x^4right)\n   =\n   psi(x+Delta x)\n   + psi(x-Delta x)\n   \n   fracmathrmd^2 psi(x)mathrmd x^2 Delta x^2\n   =\n   psi(x+Delta x)\n   - 2psi(x)\n   + psi(x-Delta x)\n   - Oleft(Delta x^4right)\n   \n   fracmathrmd^2 psi(x)mathrmd x^2\n   =\n   fracpsi(x+Delta x) - 2psi(x) + psi(x-Delta x)Delta x^2\n   - fracOleft(Delta x^4right)Delta x^2\n   \n  fracmathrmd^2 psi(x)mathrmd x^2\n  =\n  fracpsi(x+Delta x) - 2psi(x) + psi(x-Delta x)Delta x^2\n  + Oleft(Delta x^2right)\nendaligned","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"are given by the sum of 2 Taylor series:","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"beginaligned\npsi(x+Delta x)\n= psi(x)\n+ fracmathrmd psi(x)mathrmd x Delta x\n+ frac12 fracmathrmd^2 psi(x)mathrmd x^2 Delta x^2\n+ frac13 fracmathrmd^3 psi(x)mathrmd x^3 Delta x^3\n+ Oleft(Delta x^4right)\n\npsi(x-Delta x)\n= psi(x)\n- fracmathrmd psi(x)mathrmd x Delta x\n+ frac12 fracmathrmd^2 psi(x)mathrmd x^2 Delta x^2\n- frac13 fracmathrmd^3 psi(x)mathrmd x^3 Delta x^3\n+ Oleft(Delta x^4right)\nendaligned","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"   L\t  n\tnumerical         \tanalytical        \t|error|\n0.1\t  1\t986.9604315827808705\t986.9604401089355861\t0.0000008638800877%\t✔\n0.5\t  1\t39.4784172741947543\t39.4784176043574320\t0.0000008363118323%\t✔\n1.0\t  1\t9.8696043189632228\t9.8696044010893580\t0.0000008321117229%\t✔\n7.0\t  1\t0.2014204963826796\t0.2014204979814155\t0.0000007937304979%\t✔\nTest Summary:                              | Pass  Total  Time\n<ψₙ|p²|ψₙ> = ∫ψₙ*(-ℏ²d²/dx²)ψₙdx = π²ℏ²/L² |    4      4  0.4s\n","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"CurrentModule = Antiq","category":"page"},{"location":"MorsePotential/#Morse-Potential","page":"Morse Potential","title":"Morse Potential","text":"","category":"section"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"The Morse potential is a model for inter-nuclear anharmonic vibration in a diatomic molecule.","category":"page"},{"location":"MorsePotential/#Definitions","page":"Morse Potential","title":"Definitions","text":"","category":"section"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"xi = 2lambdamathrme^-a(r-r_e), omega = sqrtkµ, k = 2D_mathrmea^2, lambda = fracsqrt2mD_mathrmeahbar, chi = frachbaromega4D_mathrme, N_n = sqrtfracn(2lambda-2n-1)aGamma(2lambda-n), L_n^(alpha)(x) = fracx^-alpha mathrme^xn  fracmathrmd^nmathrmd x^nleft(mathrme^-x x^n+alpharight) are used.","category":"page"},{"location":"MorsePotential/#Schrödinger-Equation","page":"Morse Potential","title":"Schrödinger Equation","text":"","category":"section"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"  hatHpsi(r) = E psi(r)","category":"page"},{"location":"MorsePotential/#Hamiltonian","page":"Morse Potential","title":"Hamiltonian","text":"","category":"section"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"  hatH = - frachbar^22mu fracmathrmd^2mathrmdr ^2 + V(r)","category":"page"},{"location":"MorsePotential/#Potential","page":"Morse Potential","title":"Potential","text":"","category":"section"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"V(r; rₑ=rₑ, Dₑ=Dₑ, k=k, a=sqrt(k/(2*Dₑ)))","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"  V(r) = D_mathrme left( mathrme^-2a(r-r_e) - 2mathrme^-a(r-r_e) right)","category":"page"},{"location":"MorsePotential/#Eigen-Values","page":"Morse Potential","title":"Eigen Values","text":"","category":"section"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"E(n; rₑ=rₑ, Dₑ=Dₑ, k=k, a=sqrt(k/(2*Dₑ)), µ=µ, ω=sqrt(k/µ), χ=ℏ*ω/(4*Dₑ), ℏ=ℏ)","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"  E_n = - D_mathrme + hbar omega left( n + frac12 right) - chi hbar omega left( n + frac12 right)^2","category":"page"},{"location":"MorsePotential/#Eigen-Functions","page":"Morse Potential","title":"Eigen Functions","text":"","category":"section"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"ψ(n, r; rₑ=rₑ, Dₑ=Dₑ, k=k, a=sqrt(k/(2*Dₑ)), µ=µ, ω=sqrt(k/µ), χ=ℏ*ω/(4*Dₑ), ℏ=ℏ)","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"  psi_n(r) = N_n z^lambda-n-12 mathrme^-z2 L_n^(2lambda-2n-1)(xi)","category":"page"},{"location":"MorsePotential/#Generalized-Laguerre-Polynomials","page":"Morse Potential","title":"Generalized Laguerre Polynomials","text":"","category":"section"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"L(x; n=0, α=0)","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"Rodrigues' formula & closed-form:","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"  beginaligned\n    L_n^(alpha)(x)\n    = fracx^-alphae^xn fracd^ndx^nleft(x^n+alphae^-xright) \n    = sum_k=0^n(-1)^k left(beginarrayl n+alpha  n-k endarrayright) fracx^kk  \n    = sum_k=0^n(-1)^k fracGamma(alpha+n+1)Gamma(alpha+k+1)Gamma(n-k+1) fracx^kk \n  endaligned","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"Examples:","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"  beginaligned\n    L_0^(0)(x) = 1 \n    L_1^(0)(x) = 1 - x \n    L_1^(1)(x) = 2 - x \n    L_2^(0)(x) = 1 - 2 x + 12 x^2 \n    L_2^(1)(x) = 3 - 3 x + 12 x^2 \n    L_2^(2)(x) = 6 - 4 x + 12 x^2 \n    L_3^(0)(x) = 1 - 3 x + 32 x^2 - 16 x^3 \n    L_3^(1)(x) = 4 - 6 x + 2 x^2 - 16 x^3 \n    L_3^(2)(x) = 10 - 10 x + 52 x^2 - 16 x^3 \n    L_3^(3)(x) = 20 - 15 x + 3 x^2 - 16 x^3 \n    L_4^(0)(x) = 1 - 4 x + 3 x^2 - 23 x^3 + 124 x^4 \n    L_4^(1)(x) = 5 - 10 x + 5 x^2 - 56 x^3 + 124 x^4 \n    L_4^(2)(x) = 15 - 20 x + 152 x^2 - 1 x^3 + 124 x^4 \n    L_4^(3)(x) = 35 - 35 x + 212 x^2 - 76 x^3 + 124 x^4 \n    L_4^(4)(x) = 70 - 56 x + 14 x^2 - 43 x^3 + 124 x^4 \n    vdots\n  endaligned","category":"page"},{"location":"MorsePotential/#References","page":"Morse Potential","title":"References","text":"","category":"section"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"P. M. Morse, Phys. Rev. 34, 57 (1929)\nJ. P. Dahl, M. Springborg, J. Chem. Phys. 88, 4535 (1988). (62), (63)\nW. K. Shao, Y. He, J. Pan, J. Nonlinear Sci. Appl., 9, 5, 3388 (2016). (1.6) \nThe Digital Library of Mathematical Functions (DLMF) 18.3 Table1, 18.5 Table1, 18.5.12","category":"page"},{"location":"MorsePotential/#Usage-and-Examples","page":"Morse Potential","title":"Usage & Examples","text":"","category":"section"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"Install Antiq.jl for the first run and run using Antiq before each use. The function antiq(model, parameters...) returns a module that has E(), ψ(r), V(r) and some other functions. In this system, the model name is specified by :MorsePotential and several parameters rₑ, Dₑ, k, µ and ℏ are set as optional arguments.","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"using Antiq\nMP = antiq(:MorsePotential)","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"Parameters (for H₂⁺):","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"julia> MP.rₑ\n1.997193319969992\n\njulia> MP.Dₑ\n0.10263461910653993\n\njulia> MP.k\n0.1027265041900817\n\njulia> MP.µ\n918.076336715\n\njulia> MP.ℏ\n1.0","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"Eigen values:","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"julia> MP.E(n=0)\n-0.09741377794418261\n\njulia> MP.E(n=1)\n-0.08738092406760907","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"Potential energy curve:","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"using Plots\nplot(0.1:0.01:15, r -> MP.V(r), lw=2, label=\"\", xlims=(0.1,9.1), ylims=(-0.11,0.01), xlabel=\"r\", ylabel=\"V(r)\")","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"(Image: )","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"Wave functions:","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"using Plots\nplot(xlim=(0,5), xlabel=\"x\", ylabel=\"ψ(x)\")\nplot!(x -> MP.ψ(x, n=0), label=\"n=0\", lw=2)\nplot!(x -> MP.ψ(x, n=1), label=\"n=1\", lw=2)\nplot!(x -> MP.ψ(x, n=2), label=\"n=2\", lw=2)\nplot!(x -> MP.ψ(x, n=3), label=\"n=3\", lw=2)\nplot!(x -> MP.ψ(x, n=4), label=\"n=4\", lw=2)\nplot!(x -> MP.ψ(x, n=5), label=\"n=5\", lw=2)","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"(Image: )","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"Potential energy curve, Energy levels, Comparison with harmonic oscillator:","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"MP = antiq(:MorsePotential)\nHO = antiq(:HarmonicOscillator, k=MP.k, m=MP.μ)\nusing Plots\nplot(xlims=(0.1,9.1), ylims=(-0.11,0.01), xlabel=\"\\$r\\$\", ylabel=\"\\$V(r), E_n\\$\", legend=:bottomright, size=(480,400), dpi=400)\nfor n in 0:MP.nₘₐₓ()\n  # energy\n  EM = MP.E(n=n)\n  EH = HO.E(n=n) - MP.Dₑ\n  plot!(0.1:0.01:15, r -> EH > HO.V(r-MP.rₑ) - MP.Dₑ ? EH : NaN, lc=\"#BC1C5F\", lw=1, label=\"\")\n  plot!(0.1:0.01:15, r -> EM > MP.V(r) ? EM : NaN, lc=\"#578FC7\", lw=1, label=\"\")\nend\n# potential\nplot!(0.1:0.01:15, r -> HO.V(r-MP.rₑ) - MP.Dₑ, lc=\"#BC1C5F\", lw=2, label=\"Harmonic Oscillator\")\nplot!(0.1:0.01:15, r -> MP.V(r), lc=\"#578FC7\", lw=2, label=\"Morse Potential\")","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"(Image: )","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"where, the potential of harmonic oscillator is defined as V(r) simeq frac12 k (r - r_mathrme)^2 + V_0.","category":"page"},{"location":"MorsePotential/#Testing","page":"Morse Potential","title":"Testing","text":"","category":"section"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"Unit testing and Integration testing were done using computer algebra system (Symbolics.jl) and numerical integration (QuadGK.jl). The test script is here.","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"include(\"../../../test/MorsePotential.jl\")","category":"page"},{"location":"MorsePotential/#Generalized-Laguerre-Polynomials-L_n{(\\alpha)}(x)","page":"Morse Potential","title":"Generalized Laguerre Polynomials L_n^(alpha)(x)","text":"","category":"section"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"  beginaligned\n    L_n^(alpha)(x)\n    = fracx^-alphae^xn fracd^ndx^nleft(x^n+alphae^-xright) \n    = sum_k=0^n(-1)^k fracGamma(alpha+n+1)Gamma(alpha+k+1)Gamma(n-k+1) fracx^kk \n  endaligned","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"n=0 α=0 ✔","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"beginaligned\n  L_0^(0)(x)\n   = e^ - x e^x\n  = 1 \n  = 1\nendaligned","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"n=1 α=0 ✔","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"beginaligned\n  L_1^(0)(x)\n   = fracmathrmdmathrmdx x e^ - x e^x\n  = 1 - x \n  = 1 - x\nendaligned","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"n=1 α=1 ✔","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"beginaligned\n  L_1^(1)(x)\n   = frace^x fracmathrmdmathrmdx x^2 e^ - xx\n  = 2 - x \n  = 2 - x\nendaligned","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"n=2 α=0 ✔","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"beginaligned\n  L_2^(0)(x)\n   = frac12 fracmathrmdmathrmdx fracmathrmdmathrmdx x^2 e^ - x e^x\n  = 1 - 2 x + frac12 x^2 \n  = 1 - 2 x + frac12 x^2\nendaligned","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"n=2 α=1 ✔","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"beginaligned\n  L_2^(1)(x)\n   = fracfrac12 fracmathrmdmathrmdx fracmathrmdmathrmdx x^3 e^ - x e^xx\n  = 3 - 3 x + frac12 x^2 \n  = 3 - 3 x + frac12 x^2\nendaligned","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"n=2 α=2 ✔","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"beginaligned\n  L_2^(2)(x)\n   = fracfrac12 fracmathrmdmathrmdx fracmathrmdmathrmdx x^4 e^ - x e^xx^2\n  = 6 - 4 x + frac12 x^2 \n  = 6 - 4 x + frac12 x^2\nendaligned","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"n=3 α=0 ✔","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"beginaligned\n  L_3^(0)(x)\n   = frac16 fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx x^3 e^ - x e^x\n  = 1 - 3 x + frac32 x^2 - frac16 x^3 \n  = 1 - 3 x + frac32 x^2 - frac16 x^3\nendaligned","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"n=3 α=1 ✔","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"beginaligned\n  L_3^(1)(x)\n   = fracfrac16 e^x fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx x^4 e^ - xx\n  = 4 - 6 x + 2 x^2 - frac16 x^3 \n  = 4 - 6 x + 2 x^2 - frac16 x^3\nendaligned","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"n=3 α=2 ✔","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"beginaligned\n  L_3^(2)(x)\n   = fracfrac16 fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx x^5 e^ - x e^xx^2\n  = 10 - 10 x + frac52 x^2 - frac16 x^3 \n  = 10 - 10 x + frac52 x^2 - frac16 x^3\nendaligned","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"n=3 α=3 ✔","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"beginaligned\n  L_3^(3)(x)\n   = fracfrac16 fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx x^6 e^ - x e^xx^3\n  = 20 - 15 x + 3 x^2 - frac16 x^3 \n  = 20 - 15 x + 3 x^2 - frac16 x^3\nendaligned","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"n=4 α=0 ✔","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"beginaligned\n  L_4^(0)(x)\n   = frac124 fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx x^4 e^ - x e^x\n  = 1 - 4 x + 3 x^2 - frac23 x^3 + frac124 x^4 \n  = 1 - 4 x + 3 x^2 - frac23 x^3 + frac124 x^4\nendaligned","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"n=4 α=1 ✔","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"beginaligned\n  L_4^(1)(x)\n   = fracfrac124 e^x fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx x^5 e^ - xx\n  = 5 - 10 x + 5 x^2 - frac56 x^3 + frac124 x^4 \n  = 5 - 10 x + 5 x^2 - frac56 x^3 + frac124 x^4\nendaligned","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"n=4 α=2 ✔","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"beginaligned\n  L_4^(2)(x)\n   = fracfrac124 fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx x^6 e^ - x e^xx^2\n  = 15 - 20 x + frac152 x^2 - x^3 + frac124 x^4 \n  = 15 - 20 x + frac152 x^2 - x^3 + frac124 x^4\nendaligned","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"n=4 α=3 ✔","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"beginaligned\n  L_4^(3)(x)\n   = fracfrac124 e^x fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx x^7 e^ - xx^3\n  = 35 - 35 x + frac212 x^2 - frac76 x^3 + frac124 x^4 \n  = 35 - 35 x + frac212 x^2 - frac76 x^3 + frac124 x^4\nendaligned","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"n=4 α=4 ✔","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"beginaligned\n  L_4^(4)(x)\n   = fracfrac124 e^x fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx x^8 e^ - xx^4\n  = 70 - 56 x + 14 x^2 - frac43 x^3 + frac124 x^4 \n  = 70 - 56 x + 14 x^2 - frac43 x^3 + frac124 x^4\nendaligned","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"Test Summary:                      | Pass  Total   Time\nLₙ⁽ᵅ⁾(x) = x⁻ᵅeˣ/n! dⁿ/dxⁿ xⁿ⁺ᵅe⁻ˣ |   15     15  20.0s","category":"page"},{"location":"MorsePotential/#Normalization-and-Orthogonality-of-L_n{(\\alpha)}(x)","page":"Morse Potential","title":"Normalization & Orthogonality of L_n^(alpha)(x)","text":"","category":"section"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"int_0^infty L_i^(alpha)(x) L_j^(alpha)(x) x^alpha mathrme^-x mathrmdx = fracGamma(n+alpha+1)n delta_ij","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":" α\t  n\t  m\tnumerical         \tanalytical        \t|error|\n0.1\t  0\t  0\t1.7724538509055159\t1.7724538509055159\t0.0000000000000000%\t✔\n0.1\t  0\t  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4\t0.0000000000000185\t0.0000000000000000\t0.0000000000000000%\t✔\n  1\t  5\t-0.0000000000002839\t0.0000000000000000\t0.0000000000000000%\t✔\n  1\t  6\t0.0000000000000085\t0.0000000000000000\t0.0000000000000000%\t✔\n  1\t  7\t0.0000000000014165\t0.0000000000000000\t0.0000000000000000%\t✔\n  1\t  8\t-0.0000000000217631\t0.0000000000000000\t0.0000000000000000%\t✔\n  1\t  9\t-0.0000000000667054\t0.0000000000000000\t0.0000000000000000%\t✔\n  2\t  0\t-0.0000000000000028\t0.0000000000000000\t0.0000000000000000%\t✔\n  2\t  1\t-0.0000000000000010\t0.0000000000000000\t0.0000000000000000%\t✔\n  2\t  2\t1.0000000000000004\t1.0000000000000000\t0.0000000000000444%\t✔\n  2\t  3\t-0.0000000000000019\t0.0000000000000000\t0.0000000000000000%\t✔\n  2\t  4\t0.0000000000000070\t0.0000000000000000\t0.0000000000000000%\t✔\n  2\t  5\t-0.0000000000001934\t0.0000000000000000\t0.0000000000000000%\t✔\n  2\t  6\t0.0000000000002637\t0.0000000000000000\t0.0000000000000000%\t✔\n  2\t  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3\t-0.0000000000000417\t0.0000000000000000\t0.0000000000000000%\t✔\n  5\t  4\t0.0000000000000129\t0.0000000000000000\t0.0000000000000000%\t✔\n  5\t  5\t1.0000000000000182\t1.0000000000000000\t0.0000000000018208%\t✔\n  5\t  6\t-0.0000000000000032\t0.0000000000000000\t0.0000000000000000%\t✔\n  5\t  7\t-0.0000000000007341\t0.0000000000000000\t0.0000000000000000%\t✔\n  5\t  8\t0.0000000000002908\t0.0000000000000000\t0.0000000000000000%\t✔\n  5\t  9\t-0.0000000000005260\t0.0000000000000000\t0.0000000000000000%\t✔\n  6\t  0\t-0.0000000000000148\t0.0000000000000000\t0.0000000000000000%\t✔\n  6\t  1\t0.0000000000000085\t0.0000000000000000\t0.0000000000000000%\t✔\n  6\t  2\t0.0000000000002637\t0.0000000000000000\t0.0000000000000000%\t✔\n  6\t  3\t0.0000000000002512\t0.0000000000000000\t0.0000000000000000%\t✔\n  6\t  4\t0.0000000000002184\t0.0000000000000000\t0.0000000000000000%\t✔\n  6\t  5\t-0.0000000000000032\t0.0000000000000000\t0.0000000000000000%\t✔\n  6\t  6\t1.0000000000002365\t1.0000000000000000\t0.0000000000236478%\t✔\n  6\t  7\t-0.0000000000001816\t0.0000000000000000\t0.0000000000000000%\t✔\n  6\t  8\t-0.0000000000018256\t0.0000000000000000\t0.0000000000000000%\t✔\n  6\t  9\t-0.0000000000029103\t0.0000000000000000\t0.0000000000000000%\t✔\n  7\t  0\t0.0000000000024863\t0.0000000000000000\t0.0000000000000000%\t✔\n  7\t  1\t0.0000000000014165\t0.0000000000000000\t0.0000000000000000%\t✔\n  7\t  2\t0.0000000000000954\t0.0000000000000000\t0.0000000000000000%\t✔\n  7\t  3\t-0.0000000000006396\t0.0000000000000000\t0.0000000000000000%\t✔\n  7\t  4\t0.0000000000001733\t0.0000000000000000\t0.0000000000000000%\t✔\n  7\t  5\t-0.0000000000007341\t0.0000000000000000\t0.0000000000000000%\t✔\n  7\t  6\t-0.0000000000001816\t0.0000000000000000\t0.0000000000000000%\t✔\n  7\t  7\t0.9999999999995498\t1.0000000000000000\t0.0000000000450195%\t✔\n  7\t  8\t-0.0000000000001716\t0.0000000000000000\t0.0000000000000000%\t✔\n  7\t  9\t0.0000000000041122\t0.0000000000000000\t0.0000000000000000%\t✔\n  8\t  0\t-0.0000000000256450\t0.0000000000000000\t0.0000000000000000%\t✔\n  8\t  1\t-0.0000000000217631\t0.0000000000000000\t0.0000000000000000%\t✔\n  8\t  2\t-0.0000000000088982\t0.0000000000000000\t0.0000000000000000%\t✔\n  8\t  3\t-0.0000000000016690\t0.0000000000000000\t0.0000000000000000%\t✔\n  8\t  4\t-0.0000000000012127\t0.0000000000000000\t0.0000000000000000%\t✔\n  8\t  5\t0.0000000000002908\t0.0000000000000000\t0.0000000000000000%\t✔\n  8\t  6\t-0.0000000000018256\t0.0000000000000000\t0.0000000000000000%\t✔\n  8\t  7\t-0.0000000000001716\t0.0000000000000000\t0.0000000000000000%\t✔\n  8\t  8\t0.9999999999948437\t1.0000000000000000\t0.0000000005156320%\t✔\n  8\t  9\t0.0000000000003093\t0.0000000000000000\t0.0000000000000000%\t✔\n  9\t  0\t-0.0000000001036846\t0.0000000000000000\t0.0000000000000000%\t✔\n  9\t  1\t-0.0000000000667054\t0.0000000000000000\t0.0000000000000000%\t✔\n  9\t  2\t-0.0000000000304947\t0.0000000000000000\t0.0000000000000000%\t✔\n  9\t  3\t-0.0000000000058517\t0.0000000000000000\t0.0000000000000000%\t✔\n  9\t  4\t0.0000000000009228\t0.0000000000000000\t0.0000000000000000%\t✔\n  9\t  5\t-0.0000000000005260\t0.0000000000000000\t0.0000000000000000%\t✔\n  9\t  6\t-0.0000000000029103\t0.0000000000000000\t0.0000000000000000%\t✔\n  9\t  7\t0.0000000000041122\t0.0000000000000000\t0.0000000000000000%\t✔\n  9\t  8\t0.0000000000003093\t0.0000000000000000\t0.0000000000000000%\t✔\n  9\t  9\t1.0000000000154354\t1.0000000000000000\t0.0000000015435431%\t✔\nTest Summary: | Pass  Total  Time\n<ψᵢ|ψⱼ> = δᵢⱼ |  100    100  1.9s","category":"page"},{"location":"MorsePotential/#Eigen-Values-2","page":"Morse Potential","title":"Eigen Values","text":"","category":"section"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"  beginaligned\n    E_n\n    =      int psi^ast_n(r) hatH psi_n(r) mathrmdx \n    =      int psi^ast_n(r) left hatV + hatT right psi(r) mathrmdx \n    =      int psi^ast_n(r) left V(r) - frachbar^22m fracmathrmd^2mathrmd r^2 right psi(r) mathrmdx \n    simeq int psi^ast_n(r) left V(r)psi(r) -frachbar^22m fracpsi(r+Delta r) - 2psi(r) + psi(r-Delta r)Delta r^2 right mathrmdx\n  endaligned","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"Where, the difference formula for the 2nd-order derivative:","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"beginaligned\n   2psi(r)\n   + fracmathrmd^2 psi(r)mathrmd r^2 Delta r^2\n   + Oleft(Delta r^4right)\n   =\n   psi(r+Delta r)\n   + psi(r-Delta r)\n   \n   fracmathrmd^2 psi(r)mathrmd r^2 Delta r^2\n   =\n   psi(r+Delta r)\n   - 2psi(r)\n   + psi(r-Delta r)\n   - Oleft(Delta r^4right)\n   \n   fracmathrmd^2 psi(r)mathrmd r^2\n   =\n   fracpsi(r+Delta r) - 2psi(r) + psi(r-Delta r)Delta r^2\n   - fracOleft(Delta r^4right)Delta r^2\n   \n  fracmathrmd^2 psi(r)mathrmd r^2\n  =\n  fracpsi(r+Delta r) - 2psi(r) + psi(r-Delta r)Delta r^2\n  + Oleft(Delta r^2right)\nendaligned","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"are given by the sum of 2 Taylor series:","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"beginaligned\npsi(r+Delta r)\n= psi(r)\n+ fracmathrmd psi(r)mathrmd r Delta r\n+ frac12 fracmathrmd^2 psi(r)mathrmd r^2 Delta r^2\n+ frac13 fracmathrmd^3 psi(r)mathrmd r^3 Delta r^3\n+ Oleft(Delta r^4right)\n\npsi(r-Delta r)\n= psi(r)\n- fracmathrmd psi(r)mathrmd r Delta r\n+ frac12 fracmathrmd^2 psi(r)mathrmd r^2 Delta r^2\n- frac13 fracmathrmd^3 psi(r)mathrmd r^3 Delta r^3\n+ Oleft(Delta r^4right)\nendaligned","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"       k\t  n\tnumerical         \tanalytical        \t|error|\n0.10000\t  0\t-0.0974826299431356\t-0.0974826299043344\t0.0000000398031920%\t✔\n0.10000\t  1\t-0.0875766292083366\t-0.0875766290728626\t0.0000001546919476%\t✔\n0.10000\t  2\t-0.0782012653591236\t-0.0782012650053099\t0.0000004524398330%\t✔\n0.10000\t  3\t-0.0693565382661101\t-0.0693565377016763\t0.0000008138149329%\t✔\n0.10000\t  4\t-0.0610424487768959\t-0.0610424471619618\t0.0000026455920072%\t✔\n0.10000\t  5\t-0.0532589961311526\t-0.0532589933861664\t0.0000051540333741%\t✔\n0.10000\t  6\t-0.0460061778289117\t-0.0460061763742900\t0.0000031617965576%\t✔\n0.10000\t  7\t-0.0392839977431085\t-0.0392839961263328\t0.0000041156089608%\t✔\n0.10000\t  8\t-0.0330924678509606\t-0.0330924526422947\t0.0000459581104973%\t✔\n0.10000\t  9\t-0.0274314677918590\t-0.0274315459221756\t0.0002848192256583%\t✔\n0.20000\t  0\t-0.0953874611439631\t-0.0953874610810323\t0.0000000659738844%\t✔\n0.20000\t  1\t-0.0816891004273675\t-0.0816891001758958\t0.0000003078400078%\t✔\n0.20000\t  2\t-0.0690520133797856\t-0.0690520127985975\t0.0000008416671348%\t✔\n0.20000\t  3\t-0.0574761998667120\t-0.0574761989491373\t0.0000015964429232%\t✔\n0.20000\t  4\t-0.0469616603170172\t-0.0469616586275153\t0.0000035976196186%\t✔\n0.20000\t  5\t-0.0375083932021688\t-0.0375083918337316\t0.0000036483494226%\t✔\n0.20000\t  6\t-0.0291164003400053\t-0.0291163985677860\t0.0000060866708238%\t✔\n0.20000\t  7\t-0.0217856840623437\t-0.0217856788296786\t0.0000240188297604%\t✔\n0.20000\t  8\t-0.0155162375387778\t-0.0155162326194094\t0.0000317046571552%\t✔\n0.20000\t  9\t-0.0103080627546147\t-0.0103080599369784\t0.0000273343029443%\t✔\n0.30000\t  0\t-0.0937952146950324\t-0.0937952146053093\t0.0000000956584648%\t✔\n0.30000\t  1\t-0.0773103386936924\t-0.0773103383216661\t0.0000004812114878%\t✔\n0.30000\t  2\t-0.0624173731668509\t-0.0624173723297802\t0.0000013410860668%\t✔\n0.30000\t  3\t-0.0491163180291512\t-0.0491163166296516\t0.0000028493577729%\t✔\n0.30000\t  4\t-0.0374071730726919\t-0.0374071712212802\t0.0000049493494821%\t✔\n0.30000\t  5\t-0.0272899380273731\t-0.0272899361046662\t0.0000070454797707%\t✔\n0.30000\t  6\t-0.0187646136934960\t-0.0187646112798094\t0.0000128629716864%\t✔\n0.30000\t  7\t-0.0118311981022458\t-0.0118311967467099\t0.0000114573015073%\t✔\n0.30000\t  8\t-0.0064896942752905\t-0.0064896925053676\t0.0000272728303901%\t✔\n0.30000\t  9\t-0.0027401008929208\t-0.0027400985557827\t0.0000852939413987%\t✔\n0.10273\t  0\t-0.0974137779674584\t-0.0974137779441826\t0.0000000238937717%\t✔\n0.10273\t  1\t-0.0873809242053181\t-0.0873809240676091\t0.0000001575961943%\t✔\n0.10273\t  2\t-0.0778931751450495\t-0.0778931747885570\t0.0000004576684890%\t✔\n0.10273\t  3\t-0.0689505306599348\t-0.0689505301070264\t0.0000008018914211%\t✔\n0.10273\t  4\t-0.0605529890953712\t-0.0605529900230172\t0.0000015319574295%\t✔\n0.10273\t  5\t-0.0527005572549377\t-0.0527005545365296\t0.0000051582153937%\t✔\n0.10273\t  6\t-0.0453932228176823\t-0.0453932236475634\t0.0000018282046030%\t✔\n0.10273\t  7\t-0.0386310171570861\t-0.0386309973561186\t0.0000512566822197%\t✔\n0.10273\t  8\t-0.0324138862460601\t-0.0324138756621953\t0.0000326522656802%\t✔\n0.10273\t  9\t-0.0267420183757886\t-0.0267418585657935\t0.0005976024244533%\t✔\nTest Summary:              | Pass  Total  Time\n<ψₙ|H|ψₙ> = ∫ψₙ*Hψₙdx = Eₙ |   40     40  4.5s\n","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"CurrentModule = Antiq","category":"page"},{"location":"HydrogenAtom/#Hydrogen-Atom","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"","category":"section"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"The hydrogen atom is the simplest 2-body Coulomb system.","category":"page"},{"location":"HydrogenAtom/#Definitions","page":"Hydrogen Atom","title":"Definitions","text":"","category":"section"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"Z is the atomic number.","category":"page"},{"location":"HydrogenAtom/#Schrödinger-Equation","page":"Hydrogen Atom","title":"Schrödinger Equation","text":"","category":"section"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"  hatH psi(pmbr) = E psi(pmbr)","category":"page"},{"location":"HydrogenAtom/#Hamiltonian","page":"Hydrogen Atom","title":"Hamiltonian","text":"","category":"section"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"  hatH = - frachbar^22mu fracmathrmd^2mathrmdr ^2 + V(r)","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"Where, mu=left(frac1m_mathrme+frac1m_mathrmpright)^-1 is the reduced mass of electron mathrme and proton mathrmp. mu = m_mathrme holds in the limit m_mathrmprightarrowinfty. ","category":"page"},{"location":"HydrogenAtom/#Potential","page":"Hydrogen Atom","title":"Potential","text":"","category":"section"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"V(r; Z=Z, a₀=a₀)","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"  beginaligned\n  V(r)\n  = - fracZe^24pivarepsilon_0 r \n  = - frace^24pivarepsilon_0 a_0 fracZra_0\n  = - fracZra_0 E_mathrmh\n  endaligned","category":"page"},{"location":"HydrogenAtom/#Eigen-Values","page":"Hydrogen Atom","title":"Eigen Values","text":"","category":"section"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"E(; n=1, Z=Z, Eₕ=Eₕ)","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"  E_n = -fracm_mathrme e^4 Z^22n^2(4pivarepsilon_0)^2hbar^2 = -fracZ^22n^2 E_mathrmh","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"Where, E_mathrmh is the Hartree energy, one of atomic unit. About atomic units, see section 3.9.2 of the IUPAC GreenBook. In other units, E_mathrmh = 27211386245988(53)mathrmeV from here.","category":"page"},{"location":"HydrogenAtom/#Eigen-Functions","page":"Hydrogen Atom","title":"Eigen Functions","text":"","category":"section"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"ψ(r, θ, φ; n=1, l=0, m=0, Z=Z, a₀=a₀)","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"  psi_nlm(pmbr) = R_nl(r) Y_lm(thetavarphi)","category":"page"},{"location":"HydrogenAtom/#Radial-Functions","page":"Hydrogen Atom","title":"Radial Functions","text":"","category":"section"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"R(r; n=1, l=0, Z=Z, a₀=a₀)","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"    R_nl(r) = -sqrtfrac(n-l-1)2n(n+l) left(frac2Zn a_0right)^3 left(frac2Zrn a_0right)^l exp left(-fracZrn a_0right) L_n+l^2l+1 left(frac2Zrn a_0right)","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"Where, Laguerre polynomials are defined as L_n(x) = frac1n mathrme^x fracmathrmd^nmathrmdx ^n left( mathrme^-x x^n right), and associated Laguerre polynomials are defined as L_n^k(x) = fracmathrmd^kmathrmdx^k L_n(x). Note that, replace 2n(n+l) with 2n(n+l)^3 if Laguerre polynomials are defined as L_n(x) = mathrme^x fracmathrmd^nmathrmdx ^n left( mathrme^-x x^n right).","category":"page"},{"location":"HydrogenAtom/#Associated-Laguerre-Polynomials","page":"Hydrogen Atom","title":"Associated Laguerre Polynomials","text":"","category":"section"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"L(x; n=0, k=0)","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"Rodrigues' formula & closed-form:","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"  beginaligned\n  L_n^k(x)\n    = fracmathrmd^kmathrmdx^k L_n(x) \n    = fracmathrmd^kmathrmdx^k frac1n mathrme^x fracmathrmd^nmathrmdx ^n left( mathrme^-x x^n right) \n    = sum_m=0^n-k (-1)^m+k fracnm(m+k)(n-m-k) x^m \n    = (-1)^k L_n-k^(k)(x)\n  endaligned","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"Where, Laguerre polynomials are defined as L_n(x)=frac1nmathrme^x fracmathrmd^nmathrmdx ^n left( mathrme^-x x^n right).","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"Examples:","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"  beginaligned\n    L_0^0(x) = 1 \n    L_1^0(x) = 1 - x \n    L_1^1(x) = 1 \n    L_2^0(x) = 1 - 2 x + 12 x^2 \n    L_2^1(x) = 2 - x \n    L_2^2(x) = 1 \n    L_3^0(x) = 1 - 3 x + 32 x^2 - 16 x^3 \n    L_3^1(x) = 3 - 3 x + 12 x^2 \n    L_3^2(x) = 3 - x \n    L_3^3(x) = 1 \n    L_4^0(x) = 1 - 4 x + 3 x^2 - 23 x^3 + 512 x^4 \n    L_4^1(x) = 4 - 6 x + 2 x^2 - 16 x^3 \n    L_4^2(x) = 6 - 4 x + 12 x^2 \n    L_4^3(x) = 4 - x \n    L_4^4(x) = 1 \n    vdots\n  endaligned","category":"page"},{"location":"HydrogenAtom/#Spherical-Harmonics","page":"Hydrogen Atom","title":"Spherical Harmonics","text":"","category":"section"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"Y(θ, φ; l=0, m=0)","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"    Y_lm(thetavarphi) = (-1)^fracm+m2 sqrtfrac2l+14pi frac(l-m)(l+m) P_l^m (costheta) mathrme^imvarphi","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"Note that some variants are connected by ","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"i^m+m sqrtfrac(l-m)(l+m) P_l^m = (-1)^fracm+m2 sqrtfrac(l-m)(l+m) P_l^m = (-1)^m sqrtfrac(l-m)(l+m) P_l^m","category":"page"},{"location":"HydrogenAtom/#Associated-Legendre-Polynomials","page":"Hydrogen Atom","title":"Associated Legendre Polynomials","text":"","category":"section"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"P(x; n=0, m=0)","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"Rodrigues' formula & closed-form:","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"  beginaligned\n    P_n^m(x)\n    = left( 1-x^2 right)^m2 fracmathrmd^mmathrmdx^m P_n(x) \n    = left( 1-x^2 right)^m2 fracmathrmd^mmathrmdx^m frac12^n n fracmathrmd^nmathrmdx ^n left left( x^2-1 right)^n right \n    = frac12^n (1-x^2)^m2 sum_j=0^leftlfloorfracn-m2rightrfloor (-1)^j frac(2n-2j)j (n-j) (n-2j-m) x^(n-2j-m)\n  endaligned","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"Where, Legendre polynomials are defined as P_n(x) = frac12^n n fracmathrmd^nmathrmdx ^n left left( x^2-1 right)^n right. Note that P_l^-m = (-1)^m frac(l-m)(l+m) P_l^m for m0. (It is not compatible with P_k^m(t) = (-1)^mleft( 1-t^2 right)^m2 fracmathrmd^m P_k(t)mathrmdt^m caused by (-1)^m.) The specific formulae are given below.","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"Examples:","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"  beginaligned\n    P_0^0(x) = 1 \n    P_1^0(x) = x \n    P_1^1(x) = left(+1right)sqrt1-x^2 \n    P_2^0(x) = -12 + 32 x^2 \n    P_2^1(x) = left(-3 xright)sqrt1-x^2 \n    P_2^2(x) = 3 - 6 x \n    P_3^0(x) = -32 x + 52 x^3 \n    P_3^1(x) = left(32 - 152 x^2right)sqrt1-x^2 \n    P_3^2(x) = 15 x - 30 x^2 \n    P_3^3(x) = left(15 - 30 xright)sqrt1-x^2 \n    P_4^0(x) = 38 - 154 x^2 + 358 x^4 \n    P_4^1(x) = left(- 152 x + 352 x^3right)sqrt1-x^2 \n    P_4^2(x) = -152 + 15 x + 1052 x^2 - 105 x^3 \n    P_4^3(x) = left(105 x - 210 x^2right)sqrt1-x^2 \n    P_4^4(x) = 105 - 420 x + 420 x^2 \n     vdots\n  endaligned","category":"page"},{"location":"HydrogenAtom/#References","page":"Hydrogen Atom","title":"References","text":"","category":"section"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"cpprefjp, legendre, assoc_legendre, laguerre, assoc_laguerre\nThe Digital Library of Mathematical Functions (DLMF), 18.3 Table1, 18.5 Table1, 18.5.16, 18.3 Table1, 18.5 Table1, 18.5.17, 18.3 Table1, 18.5 Table1, 18.5.12\nL. D. Landau, E. M. Lifshitz, Quantum Mechanics (Pergamon Press, 1965), p.598 (c.1), p.598 (c.4), p.603 (d.13), p.603 (d.13)\nL. I. Schiff, Quantum Mechanics (McGraw-Hill Book Company, 1968), p.79 (14.12), p.93 (16.19)\nA. Messiah, Quanfum Mechanics (Dover Publications, 1999), p.493 (B.72), p.494 Table, p.493 (B.72), p.483 (B.12), p.483 (B.12)\nW. Greiner, Quantum Mechanics: An Introduction Third Edition (Springer, 1994), p.83 (4), p.83 (5), p.149 (21)\nD. J. Griffiths, Introduction to Quantum Mechanics (Prentice Hall, 1995), p.126 (4.28), p.96 Table3.1, p.126 (4.27), p.139 (4.88), p.140 Table4.4, p.139 (4.87), p.140 Table4.5\nD. A. McQuarrie, J. D. Simon, Physical Chemistry: A Molecular Approach (University Science Books, 1997), p.195 Table6.1, p.196 (6.26), p.196 Table6.2, p.207 Table6.4\nP. W. Atkins, J. De Paula, Atkins' Physical Chemistry, 8th edition (W. H. Freeman, 2008), p.234\nJ. J. Sakurai, J. Napolitano, Modern Quantum Mechanics Third Edition (Cambridge University Press, 2021), p.245 Problem 3.30.b, ","category":"page"},{"location":"HydrogenAtom/#Usage-and-Examples","page":"Hydrogen Atom","title":"Usage & Examples","text":"","category":"section"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"Install Antiq.jl for the first run and run using Antiq before each use. The function antiq(model, parameters...) returns a module that has E(), ψ(r), V(r) and some other functions. In this system, the model name is specified by :HydrogenAtom and several parameters Z, Eₕ, mₑ, a₀ and ℏ are set as optional arguments.","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"using Antiq\nH = antiq(:HydrogenAtom, Z=1, Eₕ=1.0, a₀=1.0, mₑ=1.0, ℏ=1.0)","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"Parameters:","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"julia> H.Z\n1\n\njulia> H.Eₕ\n1.0\n\njulia> H.mₑ\n1.0\n\njulia> H.a₀\n1.0\n\njulia> H.ℏ\n1.0","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"Eigen values:","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"julia> H.E(n=1)\n-0.5\n\njulia> H.E(n=2)\n-0.125","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"Wave length (n=2rightarrow1, the first line of the Lyman series):","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"Eₕ2nm⁻¹ = 2.1947463136320e-2 # https://physics.nist.gov/cgi-bin/cuu/CCValue?hrminv\nprintln(\"ΔE = \", H.E(n=2) - H.E(n=1), \" Eₕ\")\nprintln(\"λ  = \", ((H.E(n=2)-H.E(n=1))*Eₕ2nm⁻¹)^-1, \" nm\")","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"ΔE = 0.375 Eₕ\nλ  = 121.50227341098497 nm","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"Hyperfine Splitting:","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"# constants: https://doi.org/10.1103/RevModPhys.93.025010\ne  = 1.602176634e-19    # C      https://physics.nist.gov/cgi-bin/cuu/Value?e\nh  = 6.62607015e-34     # J Hz-1 https://physics.nist.gov/cgi-bin/cuu/Value?h\nc  = 299792458          # m s-1  https://physics.nist.gov/cgi-bin/cuu/Value?c\na0 = 5.29177210903e-11  # m      https://physics.nist.gov/cgi-bin/cuu/Value?bohrrada0\nµ0 = 1.25663706212e-6   # N A-2  https://physics.nist.gov/cgi-bin/cuu/Value?mu0\nµB = 9.2740100783e-24   # J T-1  https://physics.nist.gov/cgi-bin/cuu/Value?mub\nµN = 5.0507837461e-27   # J T-1  https://physics.nist.gov/cgi-bin/cuu/Value?mun\nge = 2.00231930436256   #        https://physics.nist.gov/cgi-bin/cuu/Value?gem\ngp = 5.5856946893       #        https://physics.nist.gov/cgi-bin/cuu/Value?gp\n\n# calculation: https://doi.org/10.1119/1.12733\nδ = abs(H.ψ(0,0,0))^2\nΔE = 2 / 3 * µ0 * µN * µB * gp * ge * δ * a0^(-3)\nprintln(\"1/π    = \", 1/π)\nprintln(\"<δ(r)> = \", δ, \" a₀⁻³\")\nprintln(\"<δ(r)> = \", δ * a0^(-3), \" m⁻³\")\nprintln(\"ΔE = \", ΔE, \" J\")\nprintln(\"ν = ΔE/h = \", ΔE / h * 1e-6, \" MHz\")\nprintln(\"λ = hc/ΔE = \", h*c/ΔE*100, \" cm\")","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"1/π    = 0.3183098861837907\n<δ(r)> = 0.3183098861837908 a₀⁻³\n<δ(r)> = 2.1480615849063944e30 m⁻³\nΔE = 9.427622831641132e-25 J\nν = ΔE/h = 1422.8075794882932 MHz\nλ = hc/ΔE = 21.070485027063118 cm","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"Potential energy curve:","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"using Plots\nplot(xlims=(0.0,15.0), ylims=(-0.6,0.05), xlabel=\"\\$r~/~a_0\\$\", ylabel=\"\\$V(r)/E_\\\\mathrm{h},~E_n/E_\\\\mathrm{h}\\$\", legend=:bottomright, size=(480,400), dpi=400)\nplot!(0.1:0.01:15, r -> H.V(r), lc=:black, lw=2, label=\"\") # potential","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"(Image: )","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"Potential energy curve, Energy levels:","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"using Plots\nplot(xlims=(0.0,15.0), ylims=(-0.6,0.05), xlabel=\"\\$r~/~a_0\\$\", ylabel=\"\\$V(r)/E_\\\\mathrm{h}\\$\", legend=:bottomright, size=(480,400), dpi=400)\nfor n in 0:10\n  plot!(0.0:0.01:15, r -> H.E(n=n) > H.V(r) ? H.E(n=n) : NaN, lc=n, lw=1, label=\"\") # energy level\nend\nplot!(0.1:0.01:15, r -> H.V(r), lc=:black, lw=2, label=\"\") # potential","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"(Image: )","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"Radial functions:","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"using Plots\nplot(xlabel=\"\\$r~/~a_0\\$\", ylabel=\"\\$r^2|R_{nl}(r)|^2~/~a_0^{-1}\\$\", ylims=(-0.01,0.55), xticks=0:1:20, size=(480,400), dpi=400)\nfor n in 1:3\n  for l in 0:n-1\n    plot!(0:0.01:20, r->r^2*H.R(r,n=n,l=l)^2, lc=n, lw=2, ls=[:solid,:dash,:dot,:dashdot,:dashdotdot][l+1], label=\"\\$n = $n, l=$l\\$\")\n  end\nend\nplot!()","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"(Image: )","category":"page"},{"location":"HydrogenAtom/#Testing","page":"Hydrogen Atom","title":"Testing","text":"","category":"section"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"Unit testing and Integration testing were done using computer algebra system (Symbolics.jl) and numerical integration (QuadGK.jl). The test script is here.","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"include(\"../../../test/HydrogenAtom.jl\")","category":"page"},{"location":"HydrogenAtom/#Associated-Legendre-Polynomials-P_nm(x)","page":"Hydrogen Atom","title":"Associated Legendre Polynomials P_n^m(x)","text":"","category":"section"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"  beginaligned\n    P_n^m(x)\n    = left( 1-x^2 right)^m2 fracmathrmd^mmathrmdx^m P_n(x) \n    = left( 1-x^2 right)^m2 fracmathrmd^mmathrmdx^m frac12^n n fracmathrmd^nmathrmdx ^n left left( x^2-1 right)^n right \n    = frac12^n (1-x^2)^m2 sum_j=0^leftlfloorfracn-m2rightrfloor (-1)^j frac(2n-2j)j (n-j) (n-2j-m) x^(n-2j-m)\n  endaligned","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"n=0 m=0 ✔","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"beginaligned\n  P_0^0(x)\n    = 1\n  = 1 \n  = 1\nendaligned","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"n=1 m=0 ✔","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"beginaligned\n  P_1^0(x)\n    = frac12 fracmathrmdmathrmdx left( -1 + x^2 right)\n  = x \n  = x\nendaligned","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"n=1 m=1 ✔","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"beginaligned\n  P_1^1(x)\n    = left( 1 - x^2 right)^frac12 fracmathrmdmathrmdx frac12 fracmathrmdmathrmdx left( -1 + x^2 right)\n  = left( 1 - x^2 right)^frac12 \n  = left( 1 - x^2 right)^frac12\nendaligned","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"n=2 m=0 ✔","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"beginaligned\n  P_2^0(x)\n    = frac18 fracmathrmdmathrmdx fracmathrmdmathrmdx left( -1 + x^2 right)^2\n  = frac-12 + frac32 x^2 \n  = frac-12 + frac32 x^2\nendaligned","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"n=2 m=1 ✔","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"beginaligned\n  P_2^1(x)\n    = left( 1 - x^2 right)^frac12 fracmathrmdmathrmdx frac18 fracmathrmdmathrmdx fracmathrmdmathrmdx left( -1 + x^2 right)^2\n  = 3 left( 1 - x^2 right)^frac12 x \n  = 3 left( 1 - x^2 right)^frac12 x\nendaligned","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"n=2 m=2 ✔","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"beginaligned\n  P_2^2(x)\n    = left( 1 - x^2 right) fracmathrmdmathrmdx fracmathrmdmathrmdx frac18 fracmathrmdmathrmdx fracmathrmdmathrmdx left( -1 + x^2 right)^2\n  = 3 - 3 x^2 \n  = 3 - 3 x^2\nendaligned","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"n=3 m=0 ✔","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"beginaligned\n  P_3^0(x)\n    = frac148 fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx left( -1 + x^2 right)^3\n  =  - frac32 x + frac52 x^3 \n  =  - frac32 x + frac52 x^3\nendaligned","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"n=3 m=1 ✔","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"beginaligned\n  P_3^1(x)\n    = left( 1 - x^2 right)^frac12 fracmathrmdmathrmdx frac148 fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx left( -1 + x^2 right)^3\n  =  - frac32 left( 1 - x^2 right)^frac12 + frac152 x^2 left( 1 - x^2 right)^frac12 \n  =  - frac32 left( 1 - x^2 right)^frac12 + frac152 x^2 left( 1 - x^2 right)^frac12\nendaligned","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"n=3 m=2 ✔","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"beginaligned\n  P_3^2(x)\n    = left( 1 - x^2 right) fracmathrmdmathrmdx fracmathrmdmathrmdx frac148 fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx left( -1 + x^2 right)^3\n  = 15 x - 15 x^3 \n  = 15 x - 15 x^3\nendaligned","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"n=3 m=3 ✔","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"beginaligned\n  P_3^3(x)\n    = left( 1 - x^2 right)^frac32 fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx frac148 fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx left( -1 + x^2 right)^3\n  = 15 left( 1 - x^2 right)^frac32 \n  = 15 left( 1 - x^2 right)^frac32\nendaligned","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"n=4 m=0 ✔","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"beginaligned\n  P_4^0(x)\n    = frac1384 fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx left( -1 + x^2 right)^4\n  = frac38 - frac154 x^2 + frac358 x^4 \n  = frac38 - frac154 x^2 + frac358 x^4\nendaligned","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"n=4 m=1 ✔","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"beginaligned\n  P_4^1(x)\n    = left( 1 - x^2 right)^frac12 fracmathrmdmathrmdx frac1384 fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx left( -1 + x^2 right)^4\n  =  - frac152 left( 1 - x^2 right)^frac12 x + frac352 x^3 left( 1 - x^2 right)^frac12 \n  =  - frac152 left( 1 - x^2 right)^frac12 x + frac352 x^3 left( 1 - x^2 right)^frac12\nendaligned","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"n=4 m=2 ✔","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"beginaligned\n  P_4^2(x)\n    = left( 1 - x^2 right) fracmathrmdmathrmdx fracmathrmdmathrmdx frac1384 fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx left( -1 + x^2 right)^4\n  = frac-152 + 60 x^2 - frac1052 x^4 \n  = frac-152 + 60 x^2 - frac1052 x^4\nendaligned","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"n=4 m=3 ✔","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"beginaligned\n  P_4^3(x)\n    = left( 1 - x^2 right)^frac32 fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx frac1384 fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx left( -1 + x^2 right)^4\n  = 105 left( 1 - x^2 right)^frac32 x \n  = 105 left( 1 - x^2 right)^frac32 x\nendaligned","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"n=4 m=4 ✔","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"beginaligned\n  P_4^4(x)\n    = left( 1 - x^2 right)^2 fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx frac1384 fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx left( -1 + x^2 right)^4\n  = 105 left( 1 - x^2 right)^2 \n  = 105 left( 1 - x^2 right)^2\nendaligned","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"Test Summary:                                                   | Pass  Total  Time\nPₙᵐ(x) = √(1-x²)ᵐ dᵐ/dxᵐ Pₙ(x); Pₙ(x) = 1/(2ⁿn!) dⁿ/dxⁿ (x²-1)ⁿ |   15     15  1.9s","category":"page"},{"location":"HydrogenAtom/#Normalization-and-Orthogonality-of-P_nm(x)","page":"Hydrogen Atom","title":"Normalization & Orthogonality of P_n^m(x)","text":"","category":"section"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"int_-1^1 P_i^m(x) P_j^m(x) mathrmdx = frac2(j+m)(2j+1)(j-m) delta_ij","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"  m\t  i\t  j\tnumerical         \tanalytical        \t|error|\n  0\t  0\t  0\t2.0000000000000000\t2.0000000000000000\t0.0000000000000000%\t✔\n  0\t  0\t  1\t0.0000000000000000\t0.0000000000000000\t0.0000000000000000%\t✔\n  0\t  0\t  2\t0.0000000000000000\t0.0000000000000000\t0.0000000000000000%\t✔\n  0\t  0\t  3\t-0.0000000000000000\t0.0000000000000000\t0.0000000000000000%\t✔\n  0\t  0\t  4\t0.0000000000000000\t0.0000000000000000\t0.0000000000000000%\t✔\n  0\t  0\t  5\t0.0000000000000000\t0.0000000000000000\t0.0000000000000000%\t✔\n  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1\t1.0000000000000002\t1.0000000000000000\t0.0000000000000222%\t✔\n  2  2  1  2\t0.0000000000000000\t0.0000000000000000\t0.0000000000000000%\t✔\n  2  2  2 -2\t-0.0000000000000001\t0.0000000000000000\t0.0000000000000000%\t✔\n  2  2  2 -1\t-0.0000000000000000\t0.0000000000000000\t0.0000000000000000%\t✔\n  2  2  2  0\t0.0000000000000000\t0.0000000000000000\t0.0000000000000000%\t✔\n  2  2  2  1\t0.0000000000000000\t0.0000000000000000\t0.0000000000000000%\t✔\n  2  2  2  2\t1.0000000000000002\t1.0000000000000000\t0.0000000000000222%\t✔\nTest Summary:                              | Pass  Total  Time\n∫Yₗ₁ₘ₁(θ,φ)Yₗ₂ₘ₂(θ,φ)sinθdθdφ = δₗ₁ₗ₂δₘ₁ₘ₂ |   81     81  3.1s","category":"page"},{"location":"HydrogenAtom/#Associated-Laguerre-Polynomials-L_n{k}(x)","page":"Hydrogen Atom","title":"Associated Laguerre Polynomials L_n^k(x)","text":"","category":"section"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"  beginaligned\n  L_n^k(x)\n    = fracmathrmd^kmathrmdx^k L_n(x) \n    = fracmathrmd^kmathrmdx^k frac1n mathrme^x fracmathrmd^nmathrmdx ^n left( mathrme^-x x^n right) \n    = sum_m=0^n-k (-1)^m+k fracnm(m+k)(n-m-k) x^m \n    = (-1)^k L_n-k^(k)(x)\n  endaligned","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"n=0 k=0 ✔","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"beginaligned\n  L_0^0(x)\n   = e^ - x e^x\n  = 1 \n  = 1 \n  = 1\nendaligned","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"n=1 k=0 ✔","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"beginaligned\n  L_1^0(x)\n   = fracmathrmdmathrmdx x e^ - x e^x\n  = 1 - x \n  = 1 - x \n  = 1 - x\nendaligned","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"n=1 k=1 ✔","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"beginaligned\n  L_1^1(x)\n   = fracmathrmdmathrmdx fracmathrmdmathrmdx x e^ - x e^x\n  = -1 \n  = -1 \n  = -1\nendaligned","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"n=2 k=0 ✔","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"beginaligned\n  L_2^0(x)\n   = frac12 fracmathrmdmathrmdx fracmathrmdmathrmdx x^2 e^ - x e^x\n  = 1 - 2 x + frac12 x^2 \n  = 1 - 2 x + frac12 x^2 \n  = 1 - 2 x + frac12 x^2\nendaligned","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"n=2 k=1 ✔","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"beginaligned\n  L_2^1(x)\n   = fracmathrmdmathrmdx frac12 fracmathrmdmathrmdx fracmathrmdmathrmdx x^2 e^ - x e^x\n  = -2 + x \n  = -2 + x \n  = -2 + x\nendaligned","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"n=2 k=2 ✔","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"beginaligned\n  L_2^2(x)\n   = fracmathrmdmathrmdx fracmathrmdmathrmdx frac12 fracmathrmdmathrmdx fracmathrmdmathrmdx x^2 e^ - x e^x\n  = 1 \n  = 1 \n  = 1\nendaligned","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"n=3 k=0 ✔","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"beginaligned\n  L_3^0(x)\n   = frac16 fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx x^3 e^ - x e^x\n  = 1 - 3 x + frac32 x^2 - frac16 x^3 \n  = 1 - 3 x + frac32 x^2 - frac16 x^3 \n  = 1 - 3 x + frac32 x^2 - frac16 x^3\nendaligned","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"n=3 k=1 ✔","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"beginaligned\n  L_3^1(x)\n   = fracmathrmdmathrmdx frac16 fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx x^3 e^ - x e^x\n  = -3 + 3 x - frac12 x^2 \n  = -3 + 3 x - frac12 x^2 \n  = -3 + 3 x - frac12 x^2\nendaligned","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"n=3 k=2 ✔","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"beginaligned\n  L_3^2(x)\n   = fracmathrmdmathrmdx fracmathrmdmathrmdx frac16 fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx x^3 e^ - x e^x\n  = 3 - x \n  = 3 - x \n  = 3 - x\nendaligned","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"n=3 k=3 ✔","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"beginaligned\n  L_3^3(x)\n   = fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx frac16 fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx x^3 e^ - x e^x\n  = -1 \n  = -1 \n  = -1\nendaligned","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"n=4 k=0 ✔","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"beginaligned\n  L_4^0(x)\n   = frac124 fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx x^4 e^ - x e^x\n  = 1 - 4 x + 3 x^2 - frac23 x^3 + frac124 x^4 \n  = 1 - 4 x + 3 x^2 - frac23 x^3 + frac124 x^4 \n  = 1 - 4 x + 3 x^2 - frac23 x^3 + frac124 x^4\nendaligned","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"n=4 k=1 ✔","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"beginaligned\n  L_4^1(x)\n   = fracmathrmdmathrmdx frac124 fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx x^4 e^ - x e^x\n  = -4 + 6 x - 2 x^2 + frac16 x^3 \n  = -4 + 6 x - 2 x^2 + frac16 x^3 \n  = -4 + 6 x - 2 x^2 + frac16 x^3\nendaligned","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"n=4 k=2 ✔","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"beginaligned\n  L_4^2(x)\n   = fracmathrmdmathrmdx fracmathrmdmathrmdx frac124 fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx x^4 e^ - x e^x\n  = 6 - 4 x + frac12 x^2 \n  = 6 - 4 x + frac12 x^2 \n  = 6 - 4 x + frac12 x^2\nendaligned","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"n=4 k=3 ✔","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"beginaligned\n  L_4^3(x)\n   = fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx frac124 fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx x^4 e^ - x e^x\n  = -4 + x \n  = -4 + x \n  = -4 + x\nendaligned","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"n=4 k=4 ✔","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"beginaligned\n  L_4^4(x)\n   = fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx frac124 fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx x^4 e^ - x e^x\n  = 1 \n  = 1 \n  = 1\nendaligned","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"Test Summary:                                          | Pass  Total  Time\nLₙᵏ(x) = dᵏ/dxᵏ Lₙ(x); Lₙ(x) = 1/(n!) eˣ dⁿ/dxⁿ e⁻ˣ xⁿ |   15     15  0.9s","category":"page"},{"location":"HydrogenAtom/#Normalization-and-Orthogonality-of-L_n{k}(x)","page":"Hydrogen Atom","title":"Normalization & Orthogonality of L_n^k(x)","text":"","category":"section"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"int_0^infty mathrme^-x x^k L_i^k(x) L_j^k(x) mathrmdx = fraci(i-k) delta_ij","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"Replace n+k with n for the definition of Wolfram MathWorld.","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"  i\t  j\t  k\tnumerical         \tanalytical        \t|error|\n  0\t  0\t  0\t1.0000000000000000\t1.0000000000000000\t0.0000000000000000%\t✔\n  0\t  1\t  0\t0.0000000000000000\t0.0000000000000000\t0.0000000000000000%\t✔\n  0\t  2\t  0\t0.0000000000000000\t0.0000000000000000\t0.0000000000000000%\t✔\n  0\t  3\t  0\t0.0000000000000001\t0.0000000000000000\t0.0000000000000000%\t✔\n  0\t  4\t  0\t0.0000000000000002\t0.0000000000000000\t0.0000000000000000%\t✔\n  0\t  5\t  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2\t-0.0000000000000354\t0.0000000000000000\t0.0000000000000000%\t✔\n  7\t  5\t  3\t-0.0000000000002566\t0.0000000000000000\t0.0000000000000000%\t✔\n  7\t  5\t  4\t-0.0000000000001368\t0.0000000000000000\t0.0000000000000000%\t✔\n  7\t  5\t  5\t-0.0000000000000220\t0.0000000000000000\t0.0000000000000000%\t✔\n  7\t  6\t  0\t0.0000000000000012\t0.0000000000000000\t0.0000000000000000%\t✔\n  7\t  6\t  1\t-0.0000000000000003\t0.0000000000000000\t0.0000000000000000%\t✔\n  7\t  6\t  2\t-0.0000000000000012\t0.0000000000000000\t0.0000000000000000%\t✔\n  7\t  6\t  3\t0.0000000000001121\t0.0000000000000000\t0.0000000000000000%\t✔\n  7\t  6\t  4\t0.0000000000001277\t0.0000000000000000\t0.0000000000000000%\t✔\n  7\t  6\t  5\t-0.0000000000000675\t0.0000000000000000\t0.0000000000000000%\t✔\n  7\t  6\t  6\t0.0000000000000684\t0.0000000000000000\t0.0000000000000000%\t✔\n  7\t  7\t  0\t1.0000000000000067\t1.0000000000000000\t0.0000000000006661%\t✔\n  7\t  7\t  1\t6.9999999999999885\t7.0000000000000000\t0.0000000000001649%\t✔\n  7\t  7\t  2\t42.0000000000000000\t42.0000000000000000\t0.0000000000000000%\t✔\n  7\t  7\t  3\t209.9999999999998863\t210.0000000000000000\t0.0000000000000541%\t✔\n  7\t  7\t  4\t840.0000000000002274\t840.0000000000000000\t0.0000000000000271%\t✔\n  7\t  7\t  5\t2519.9999999997753548\t2520.0000000000000000\t0.0000000000089145%\t✔\n  7\t  7\t  6\t5039.9999999999854481\t5040.0000000000000000\t0.0000000000002887%\t✔\n  7\t  7\t  7\t5040.0000000000000000\t5040.0000000000000000\t0.0000000000000000%\t✔\nTest Summary:                                 | Pass  Total  Time\n∫exp(-x)xᵏLᵢᵏ(x)Lⱼᵏ(x)dx = (2i+k)!/(i+k)! δᵢⱼ |  204    204  1.4s","category":"page"},{"location":"HydrogenAtom/#Normalization-of-R_{nl}(r)","page":"Hydrogen Atom","title":"Normalization of R_nl(r)","text":"","category":"section"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"int R_nl(r)^2 r^2 mathrmdr = 1","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"  n\t  l\tnumerical         \tanalytical        \t|error|\n  1\t  0\t1.0000000000000286\t1.0000000000000000\t0.0000000000028644%\t✔\n  2\t  0\t1.0000000000000002\t1.0000000000000000\t0.0000000000000222%\t✔\n  2\t  1\t1.0000000000000000\t1.0000000000000000\t0.0000000000000000%\t✔\n  3\t  0\t0.9999999999999999\t1.0000000000000000\t0.0000000000000111%\t✔\n  3\t  1\t0.9999999999992634\t1.0000000000000000\t0.0000000000736633%\t✔\n  3\t  2\t0.9999999999998338\t1.0000000000000000\t0.0000000000166200%\t✔\n  4\t  0\t0.9999999999999916\t1.0000000000000000\t0.0000000000008438%\t✔\n  4\t  1\t0.9999999999999950\t1.0000000000000000\t0.0000000000004996%\t✔\n  4\t  2\t0.9999999999999987\t1.0000000000000000\t0.0000000000001332%\t✔\n  4\t  3\t1.0000000000000000\t1.0000000000000000\t0.0000000000000000%\t✔\n  5\t  0\t0.9999999999999996\t1.0000000000000000\t0.0000000000000444%\t✔\n  5\t  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3\t0.9999999999999987\t1.0000000000000000\t0.0000000000001332%\t✔\n  7\t  4\t0.9999999999999992\t1.0000000000000000\t0.0000000000000777%\t✔\n  7\t  5\t1.0000000000000000\t1.0000000000000000\t0.0000000000000000%\t✔\n  7\t  6\t0.9999999999999726\t1.0000000000000000\t0.0000000000027423%\t✔\n  8\t  0\t1.0000000000000082\t1.0000000000000000\t0.0000000000008216%\t✔\n  8\t  1\t0.9999999999999961\t1.0000000000000000\t0.0000000000003886%\t✔\n  8\t  2\t1.0000000000000044\t1.0000000000000000\t0.0000000000004441%\t✔\n  8\t  3\t1.0000000000000002\t1.0000000000000000\t0.0000000000000222%\t✔\n  8\t  4\t1.0000000000000009\t1.0000000000000000\t0.0000000000000888%\t✔\n  8\t  5\t0.9999999999999999\t1.0000000000000000\t0.0000000000000111%\t✔\n  8\t  6\t0.9999999999999996\t1.0000000000000000\t0.0000000000000444%\t✔\n  8\t  7\t0.9999999999999999\t1.0000000000000000\t0.0000000000000111%\t✔\n  9\t  0\t0.9999999999999858\t1.0000000000000000\t0.0000000000014211%\t✔\n  9\t  1\t1.0000000000000036\t1.0000000000000000\t0.0000000000003553%\t✔\n  9\t  2\t1.0000000000000187\t1.0000000000000000\t0.0000000000018652%\t✔\n  9\t  3\t0.9999999999999923\t1.0000000000000000\t0.0000000000007661%\t✔\n  9\t  4\t0.9999999999999903\t1.0000000000000000\t0.0000000000009659%\t✔\n  9\t  5\t0.9999999999999982\t1.0000000000000000\t0.0000000000001776%\t✔\n  9\t  6\t0.9999999999999999\t1.0000000000000000\t0.0000000000000111%\t✔\n  9\t  7\t1.0000000000000007\t1.0000000000000000\t0.0000000000000666%\t✔\n  9\t  8\t0.9999999999999994\t1.0000000000000000\t0.0000000000000555%\t✔\nTest Summary:               | Pass  Total  Time\n∫|Rₙₗ(r)|²r²dr = δₙ₁ₙ₂δₗ₁ₗ₂ |   45     45  1.1s","category":"page"},{"location":"HydrogenAtom/#Expected-Value-of-r","page":"Hydrogen Atom","title":"Expected Value of r","text":"","category":"section"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"langle r rangle\n= int r R_n_1 l_1(r)^2 r^2 mathrmdr\n= fraca_mu2Z left 3n^2 - l(l+1) right \na_mu = a_0 fracm_mathrmemu \nfrac1mu = frac1m_mathrme + frac1m_mathrmp","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"Reference:","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"高柳和夫『朝倉物理学大系 11 原子分子物理学』(2000, 朝倉書店) pp.11-22\n Quan­tum Me­chan­ics for En­gi­neers by Leon van Dom­me­len","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"  n\t  l\tnumerical         \tanalytical        \t|error|\n  1\t  0\t1.4999999999999998\t1.5000000000000000\t0.0000000000000148%\t✔\n  2\t  0\t6.0000000000000000\t6.0000000000000000\t0.0000000000000000%\t✔\n  2\t  1\t4.9999999999999991\t5.0000000000000000\t0.0000000000000178%\t✔\n  3\t  0\t13.4999999999999982\t13.5000000000000000\t0.0000000000000132%\t✔\n  3\t  1\t12.5000000000000018\t12.5000000000000000\t0.0000000000000142%\t✔\n  3\t  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3\t48.0000000000001279\t48.0000000000000000\t0.0000000000002665%\t✔\n  6\t  4\t44.0000000000000142\t44.0000000000000000\t0.0000000000000323%\t✔\n  6\t  5\t38.9999999999999858\t39.0000000000000000\t0.0000000000000364%\t✔\n  7\t  0\t73.4999999999998579\t73.5000000000000000\t0.0000000000001933%\t✔\n  7\t  1\t72.5000000000000284\t72.5000000000000000\t0.0000000000000392%\t✔\n  7\t  2\t70.4999999999998437\t70.5000000000000000\t0.0000000000002217%\t✔\n  7\t  3\t67.4999999999999005\t67.5000000000000000\t0.0000000000001474%\t✔\n  7\t  4\t63.4999999999999645\t63.5000000000000000\t0.0000000000000559%\t✔\n  7\t  5\t58.4999999999999858\t58.5000000000000000\t0.0000000000000243%\t✔\n  7\t  6\t52.4999999999920206\t52.5000000000000000\t0.0000000000151988%\t✔\n  8\t  0\t96.0000000000006537\t96.0000000000000000\t0.0000000000006809%\t✔\n  8\t  1\t94.9999999999993179\t95.0000000000000000\t0.0000000000007180%\t✔\n  8\t  2\t93.0000000000000995\t93.0000000000000000\t0.0000000000001070%\t✔\n  8\t  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 8\t85.4999999999999716\t85.5000000000000000\t0.0000000000000332%\t✔\nTest Summary:                                                    | Pass  Total  Time\n∫r|Rₙₗ(r)|²r²dr = (a₀×mₑ/μ)/2Z × [3n²-l(l+1)]; 1/μ = 1/mₑ + 1/mₚ |   45     45  0.9s","category":"page"},{"location":"HydrogenAtom/#Expected-Value-of-r2","page":"Hydrogen Atom","title":"Expected Value of r^2","text":"","category":"section"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"langle r^2 rangle\n= int r^2 R_n_1 l_1(r)^2 r^2 mathrmdr\n= fraca_mu^22Z^2 n^2 left 5n^2 + 1 - 3l(l+1) right \na_mu = a_0 fracm_mathrmemu \nfrac1mu = frac1m_mathrme + frac1m_mathrmp","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"Reference:","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"高柳和夫『朝倉物理学大系 11 原子分子物理学』(2000, 朝倉書店) pp.11-22\n Quan­tum Me­chan­ics for En­gi­neers by Leon van Dom­me­len","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"  n\t  l\tnumerical         \tanalytical        \t|error|\n  1\t  0\t3.0000000000000004\t3.0000000000000000\t0.0000000000000148%\t✔\n  2\t  0\t42.0000000000000000\t42.0000000000000000\t0.0000000000000000%\t✔\n  2\t  1\t30.0000000000000000\t30.0000000000000000\t0.0000000000000000%\t✔\n  3\t  0\t207.0000000000000284\t207.0000000000000000\t0.0000000000000137%\t✔\n  3\t  1\t180.0000000000000000\t180.0000000000000000\t0.0000000000000000%\t✔\n  3\t  2\t125.9999999999999858\t126.0000000000000000\t0.0000000000000113%\t✔\n  4\t  0\t647.9999999999030251\t648.0000000000000000\t0.0000000000149653%\t✔\n  4\t  1\t599.9999999999364491\t600.0000000000000000\t0.0000000000105918%\t✔\n  4\t  2\t503.9999999999748184\t504.0000000000000000\t0.0000000000049964%\t✔\n  4\t  3\t359.9999999999955094\t360.0000000000000000\t0.0000000000012474%\t✔\n  5\t  0\t1574.9999999999988631\t1575.0000000000000000\t0.0000000000000722%\t✔\n  5\t  1\t1499.9999999999977263\t1500.0000000000000000\t0.0000000000001516%\t✔\n  5\t  2\t1349.9999999999995453\t1350.0000000000000000\t0.0000000000000337%\t✔\n  5\t  3\t1125.0000000000029559\t1125.0000000000000000\t0.0000000000002627%\t✔\n  5\t  4\t824.9999999999998863\t825.0000000000000000\t0.0000000000000138%\t✔\n  6\t  0\t3257.9999999999968168\t3258.0000000000000000\t0.0000000000000977%\t✔\n  6\t  1\t3149.9999999999922693\t3150.0000000000000000\t0.0000000000002454%\t✔\n  6\t  2\t2933.9999999999981810\t2934.0000000000000000\t0.0000000000000620%\t✔\n  6\t  3\t2610.0000000000327418\t2610.0000000000000000\t0.0000000000012545%\t✔\n  6\t  4\t2178.0000000000081855\t2178.0000000000000000\t0.0000000000003758%\t✔\n  6\t  5\t1638.0000000000002274\t1638.0000000000000000\t0.0000000000000139%\t✔\n  7\t  0\t6026.9999999999918145\t6027.0000000000000000\t0.0000000000001358%\t✔\n  7\t  1\t5880.0000000000027285\t5880.0000000000000000\t0.0000000000000464%\t✔\n  7\t  2\t5585.9999999999899956\t5586.0000000000000000\t0.0000000000001791%\t✔\n  7\t  3\t5144.9999999999918145\t5145.0000000000000000\t0.0000000000001591%\t✔\n  7\t  4\t4556.9999999999972715\t4557.0000000000000000\t0.0000000000000599%\t✔\n  7\t  5\t3821.9999999999990905\t3822.0000000000000000\t0.0000000000000238%\t✔\n  7\t  6\t2940.0000000000013642\t2940.0000000000000000\t0.0000000000000464%\t✔\n  8\t  0\t10272.0000000000291038\t10272.0000000000000000\t0.0000000000002833%\t✔\n  8\t  1\t10079.9999999999945430\t10080.0000000000000000\t0.0000000000000541%\t✔\n  8\t  2\t9695.9999999999927240\t9696.0000000000000000\t0.0000000000000750%\t✔\n  8\t  3\t9120.0000000000109139\t9120.0000000000000000\t0.0000000000001197%\t✔\n  8\t  4\t8352.0000000000018190\t8352.0000000000000000\t0.0000000000000218%\t✔\n  8\t  5\t7392.0000000000100044\t7392.0000000000000000\t0.0000000000001353%\t✔\n  8\t  6\t6240.0000000000000000\t6240.0000000000000000\t0.0000000000000000%\t✔\n  8\t  7\t4896.0000000000081855\t4896.0000000000000000\t0.0000000000001672%\t✔\n  9\t  0\t16443.0000000001018634\t16443.0000000000000000\t0.0000000000006195%\t✔\n  9\t  1\t16200.0000000000400178\t16200.0000000000000000\t0.0000000000002470%\t✔\n  9\t  2\t15714.0000000001491571\t15714.0000000000000000\t0.0000000000009492%\t✔\n  9\t  3\t14984.9999999999181455\t14985.0000000000000000\t0.0000000000005462%\t✔\n  9\t  4\t14012.9999999995452526\t14013.0000000000000000\t0.0000000000032452%\t✔\n  9\t  5\t12797.9999999998071871\t12798.0000000000000000\t0.0000000000015066%\t✔\n  9\t  6\t11339.9999999999454303\t11340.0000000000000000\t0.0000000000004812%\t✔\n  9\t  7\t9638.9999999999909051\t9639.0000000000000000\t0.0000000000000944%\t✔\n  9\t  8\t7694.9999999999981810\t7695.0000000000000000\t0.0000000000000236%\t✔\nTest Summary:                                                            | Pass  Total  Time\n∫r²|Rₙₗ(r)|²r²dr = (a₀×mₑ/μ)²/2Z² × n²[5n²+1-3l(l+1)]; 1/μ = 1/mₑ + 1/mₚ |   45     45  1.0s","category":"page"},{"location":"HydrogenAtom/#Virial-Theorem","page":"Hydrogen Atom","title":"Virial Theorem","text":"","category":"section"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"The virial theorem 2langle T rangle + langle V rangle = 0 and the definition of Hamiltonian langle H rangle = langle T rangle + langle V rangle derive langle H rangle = frac12 langle V rangle and langle H rangle = -langle T rangle.","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"frac12 int psi_n^ast(x) V(x) psi_n(x) mathrmdx = E_n","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"  n\tnumerical         \tanalytical        \t|error|\n  1\t-0.9999999999999999\t-1.0000000000000000\t0.0000000000000111%\t✔\n  2\t-0.2500000000000001\t-0.2500000000000000\t0.0000000000000444%\t✔\n  3\t-0.1111111111111111\t-0.1111111111111111\t0.0000000000000125%\t✔\n  4\t-0.0625000000000000\t-0.0625000000000000\t0.0000000000000222%\t✔\n  5\t-0.0400000000000000\t-0.0400000000000000\t0.0000000000000520%\t✔\n  6\t-0.0277777777777777\t-0.0277777777777778\t0.0000000000003497%\t✔\n  7\t-0.0204081632653061\t-0.0204081632653061\t0.0000000000003400%\t✔\n  8\t-0.0156250000000002\t-0.0156250000000000\t0.0000000000011102%\t✔\n  9\t-0.0123456790123456\t-0.0123456790123457\t0.0000000000002389%\t✔\n 10\t-0.0100000000000004\t-0.0100000000000000\t0.0000000000036256%\t✔\nTest Summary:      | Pass  Total  Time\n<ψₙ|V|ψₙ> / 2 = Eₙ |   10     10  0.9s","category":"page"},{"location":"HydrogenAtom/#Normalization-and-Orthogonality-of-\\psi_n(r,\\theta,\\varphi)","page":"Hydrogen Atom","title":"Normalization & Orthogonality of psi_n(rthetavarphi)","text":"","category":"section"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"int psi_i^ast(rthetavarphi) psi_j(rthetavarphi) r^2 mathrmdr mathrmdtheta mathrmdvarphi = delta_ij","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":" n1\t n2\t l1\t l2\t m1\t m2\tnumerical         \tanalytical        \t|error|\n  1\t  1\t  0\t  0\t  0\t  0\t1.0000000002519327\t1.0000000000000000\t0.0000000251932697%\t✔\n  1\t  2\t  0\t  0\t  0\t  0\t-0.0000000112226007\t0.0000000000000000\t0.0000000000000000%\t✔\n  1\t  2\t  0\t  1\t  0\t -1\t-0.0000000000000000\t0.0000000000000000\t0.0000000000000000%\t✔\n  1\t  2\t  0\t  1\t  0\t  0\t-0.0000000000000000\t0.0000000000000000\t0.0000000000000000%\t✔\n  1\t  2\t  0\t  1\t  0\t  1\t0.0000000000000000\t0.0000000000000000\t0.0000000000000000%\t✔\n  1\t  3\t  0\t  0\t  0\t  0\t-0.0000000456607854\t0.0000000000000000\t0.0000000000000000%\t✔\n  1\t  3\t  0\t  1\t  0\t -1\t0.0000000000000000\t0.0000000000000000\t0.0000000000000000%\t✔\n  1\t  3\t  0\t  1\t  0\t  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-1\t-0.0000000000000429\t0.0000000000000000\t0.0000000000000000%\t✔\n  3\t  3\t  2\t  2\t  1\t  0\t0.0000000000000000\t0.0000000000000000\t0.0000000000000000%\t✔\n  3\t  3\t  2\t  2\t  1\t  1\t1.0003006285590699\t1.0000000000000000\t0.0300628559069871%\t✔\n  3\t  3\t  2\t  2\t  1\t  2\t0.0000000000000000\t0.0000000000000000\t0.0000000000000000%\t✔\n  3\t  3\t  2\t  2\t  2\t -2\t0.0000001937793434\t0.0000000000000000\t0.0000000000000000%\t✔\n  3\t  3\t  2\t  2\t  2\t -1\t-0.0000000000000000\t0.0000000000000000\t0.0000000000000000%\t✔\n  3\t  3\t  2\t  2\t  2\t  0\t-0.0000000000000175\t0.0000000000000000\t0.0000000000000000%\t✔\n  3\t  3\t  2\t  2\t  2\t  1\t0.0000000000000000\t0.0000000000000000\t0.0000000000000000%\t✔\n  3\t  3\t  2\t  2\t  2\t  2\t1.0003006285656155\t1.0000000000000000\t0.0300628565615524%\t✔\nTest Summary:                       | Pass  Total   Time\n<ψₙ₁ₗ₁ₘ₁|ψₙ₂ₗ₂ₘ₂> = δₙ₁ₙ₂δₗ₁ₗ₂δₘ₁ₘ₂ |  196    196  26.8s\n","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"CurrentModule = Antiq","category":"page"},{"location":"HarmonicOscillator/#Harmonic-Oscillator","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"","category":"section"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"The harmonic oscillator is the most frequently used model in quantum physics.","category":"page"},{"location":"HarmonicOscillator/#Definitions","page":"Harmonic Oscillator","title":"Definitions","text":"","category":"section"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"k is the spring constant. And omega = sqrtkm, xi = sqrtfracmomegahbarx, A_n = sqrtfrac1n 2^n sqrtfracmomegapihbar, H_n(x) = (-1)^n mathrme^x^2 fracmathrmd^nmathrmdx^n mathrme^-x^2 are used.","category":"page"},{"location":"HarmonicOscillator/#Schrödinger-Equation","page":"Harmonic Oscillator","title":"Schrödinger Equation","text":"","category":"section"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"  hatHpsi(x) = E psi(x)","category":"page"},{"location":"HarmonicOscillator/#Hamiltonian","page":"Harmonic Oscillator","title":"Hamiltonian","text":"","category":"section"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"  beginaligned\n    hatH\n    = - frachbar^22m fracmathrmd^2mathrmdx ^2 + V(x) \n    = - frac12 hbaromega fracmathrmd^2mathrmdxi^2 + V(x)\n  endaligned","category":"page"},{"location":"HarmonicOscillator/#Potential","page":"Harmonic Oscillator","title":"Potential","text":"","category":"section"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"V(x; k=k, m=m, ω=sqrt(k/m))","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"  V(x)\n  = frac12 k x^2\n  = frac12 m omega^2 x^2\n  = frac12 hbar omega xi^2","category":"page"},{"location":"HarmonicOscillator/#Eigen-Values","page":"Harmonic Oscillator","title":"Eigen Values","text":"","category":"section"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"E(n; k, m, ω=sqrt(k/m), ℏ=ℏ)","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"  E_n = hbar omega left( n + frac12 right)","category":"page"},{"location":"HarmonicOscillator/#Eigen-Functions","page":"Harmonic Oscillator","title":"Eigen Functions","text":"","category":"section"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"ψ(n, x; k=k, m=m, ω=sqrt(k/m), ℏ=ℏ)","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"  psi_n(x) = A_n H_n(xi) expleft( -fracxi^22 right)","category":"page"},{"location":"HarmonicOscillator/#Hermite-Polynomials","page":"Harmonic Oscillator","title":"Hermite Polynomials","text":"","category":"section"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"H(x; n=0)","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"Rodrigues' formula & closed-form:","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"  beginaligned\n    H_n(x)\n    = (-1)^n mathrme^x^2 fracmathrmd^nmathrmdx^n mathrme^-x^2 \n    = n sum_m=0^lfloor n2 rfloor frac(-1)^mm (n-2m)(2 x)^n-2m\n  endaligned","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"Examples:","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"  beginaligned\n    H_0(x)  = 1 \n    H_1(x)  = 2 x \n    H_2(x)  = -2 + 4 x^2 \n    H_3(x)  = -12 x + 8 x^3 \n    H_4(x)  = 12 - 48 x^2 + 16 x^4 \n    H_5(x)  = 120 x - 160 x^3 + 32 x^5 \n    H_6(x)  = -120 + 720 x^2 - 480 x^4 + 64 x^6 \n    H_7(x)  = -1680 x + 3360 x^3 - 1344 x^5 + 128 x^7 \n    H_8(x)  = 1680 - 13440 x^2 + 13440 x^4 - 3584 x^6 + 256 x^8 \n    H_9(x)  = 30240 x - 80640 x^3 + 48384 x^5 - 9216 x^7 + 512 x^9 \n    vdots\n  endaligned","category":"page"},{"location":"HarmonicOscillator/#Reference","page":"Harmonic Oscillator","title":"Reference","text":"","category":"section"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"DLMF 18.5.18\ncpprefjp\nThe Digital Library of Mathematical Functions (DLMF)                                                    18.3 Table1, 18.5 Table1, 18.5.13, 18.5.18\nL. D. Landau, E. M. Lifshitz, Quantum Mechanics (Pergamon Press, 1965)                                  p.595 (a.4), (a.6)\nL. I. Schiff, Quantum Mechanics (McGraw-Hill Book Company, 1968)                                        p.71 (13.12)\nA. Messiah, Quanfum Mechanics (Dover Publications, 1999)                                                p.491 (B.59)\nW. Greiner, Quantum Mechanics: An Introduction Third Edition (Springer, 1994)                           p.152 (7.22)\nD. J. Griffiths, Introduction to Quantum Mechanics (Prentice Hall, 1995)                                p.41 Table 2.1, p.43 (2.70)\nD. A. McQuarrie, J. D. Simon, Physical Chemistry: A Molecular Approach (University Science Books, 1997) p.170 Table 5.2\nP. W. Atkins, J. De Paula, Atkins' Physical Chemistry, 8th edition (W. H. Freeman, 2008)                p.293 Table 9.1\nJ. J. Sakurai, J. Napolitano, Modern Quantum Mechanics Third Edition (Cambridge University Press, 2021) p.524 (B.29)","category":"page"},{"location":"HarmonicOscillator/#Usage-and-Examples","page":"Harmonic Oscillator","title":"Usage & Examples","text":"","category":"section"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"Install Antiq.jl for the first run and run using Antiq before each use. The function antiq(model, parameters...) returns a module that has E(), ψ(x), V(x) and some other functions. In this system, the model name is specified by :HarmonicOscillator and several parameters k, m and ℏ are set as optional arguments.","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"using Antiq\nHO = antiq(:HarmonicOscillator, k=1.0, m=1.0, ℏ=1.0)","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"Parameters:","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"julia> HO.k\n1.0\n\njulia> HO.m\n1.0\n\njulia> HO.ℏ\n1.0","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"Eigen values:","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"julia> HO.E(n=0)\n0.5\n\njulia> HO.E(n=1)\n1.5","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"Potential energy curve:","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"using Plots\nplot(-5:0.1:5, x -> HO.V(x), lw=2, label=\"\", xlabel=\"x\", ylabel=\"V(x)\")","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"(Image: )","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"Wave functions:","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"using Plots\nplot(xlim=(-5,5), xlabel=\"x\", ylabel=\"ψ(x)\")\nplot!(x -> HO.ψ(x, n=0), label=\"n=0\", lw=2)\nplot!(x -> HO.ψ(x, n=1), label=\"n=1\", lw=2)\nplot!(x -> HO.ψ(x, n=2), label=\"n=2\", lw=2)\nplot!(x -> HO.ψ(x, n=3), label=\"n=3\", lw=2)\nplot!(x -> HO.ψ(x, n=4), label=\"n=4\", lw=2)","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"(Image: )","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"Potential energy curve, Energy levels, Wave functions:","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"using Plots\nplot(xlim=(-5.5,5.5), ylim=(-0.2,5.4), xlabel=\"\\$x\\$\", ylabel=\"\\$V(x),~E_n,~\\\\psi_n(x)\\\\times0.5+E_n\\$\", size=(480,400), dpi=400)\nfor n in 0:4\n  # energy\n  hline!([HO.E(n=n)], lc=:black, ls=:dash, label=\"\")\n  plot!([-sqrt(2*HO.k*HO.E(n=n)),sqrt(2*HO.k*HO.E(n=n))], fill(HO.E(n=n),2), lc=:black, lw=2, label=\"\")\n  # wave function\n  plot!(x -> HO.E(n=n) + 0.5*HO.ψ(x,n=n), lc=n+1, lw=2, label=\"\")\nend\n# potential\nplot!(x -> HO.V(x), lc=:black, lw=2, label=\"\")","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"(Image: )","category":"page"},{"location":"HarmonicOscillator/#Testing","page":"Harmonic Oscillator","title":"Testing","text":"","category":"section"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"Unit testing and Integration testing were done using computer algebra system (Symbolics.jl) and numerical integration (QuadGK.jl). The test script is here.","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"include(\"../../../test/HarmonicOscillator.jl\")","category":"page"},{"location":"HarmonicOscillator/#Hermite-Polynomials-H_n(x)","page":"Harmonic Oscillator","title":"Hermite Polynomials H_n(x)","text":"","category":"section"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"  beginaligned\n    H_n(x)\n    = (-1)^n mathrme^x^2 fracmathrmd^nmathrmdx^n mathrme^-x^2 \n    = n sum_m=0^lfloor n2 rfloor frac(-1)^mm (n-2m)(2 x)^n-2m\n  endaligned","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"n=0 ✔","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"beginaligned\n  H_0(x)\n   = e^ - x^2 e^x^2\n  = 1 \n  = 1\nendaligned","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"n=1 ✔","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"beginaligned\n  H_1(x)\n   =  - e^x^2 fracmathrmd e^ - x^2mathrmdx\n  = 2 x \n  = 2 x\nendaligned","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"n=2 ✔","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"beginaligned\n  H_2(x)\n   = e^x^2 fracmathrmdmathrmdx fracmathrmd e^ - x^2mathrmdx\n  = -2 + 4 x^2 \n  = -2 + 4 x^2\nendaligned","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"n=3 ✔","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"beginaligned\n  H_3(x)\n   =  - fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmd e^ - x^2mathrmdx e^x^2\n  =  - 12 x + 8 x^3 \n  =  - 12 x + 8 x^3\nendaligned","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"n=4 ✔","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"beginaligned\n  H_4(x)\n   = e^x^2 fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmd e^ - x^2mathrmdx\n  = 12 - 48 x^2 + 16 x^4 \n  = 12 - 48 x^2 + 16 x^4\nendaligned","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"n=5 ✔","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"beginaligned\n  H_5(x)\n   =  - fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmd e^ - x^2mathrmdx e^x^2\n  = 120 x - 160 x^3 + 32 x^5 \n  = 120 x - 160 x^3 + 32 x^5\nendaligned","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"n=6 ✔","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"beginaligned\n  H_6(x)\n   = e^x^2 fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmd e^ - x^2mathrmdx\n  = -120 + 720 x^2 - 480 x^4 + 64 x^6 \n  = -120 + 720 x^2 - 480 x^4 + 64 x^6\nendaligned","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"n=7 ✔","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"beginaligned\n  H_7(x)\n   =  - e^x^2 fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmd e^ - x^2mathrmdx\n  =  - 1680 x + 3360 x^3 - 1344 x^5 + 128 x^7 \n  =  - 1680 x + 3360 x^3 - 1344 x^5 + 128 x^7\nendaligned","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"n=8 ✔","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"beginaligned\n  H_8(x)\n   = e^x^2 fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmd e^ - x^2mathrmdx\n  = 1680 - 13440 x^2 + 13440 x^4 - 3584 x^6 + 256 x^8 \n  = 1680 - 13440 x^2 + 13440 x^4 - 3584 x^6 + 256 x^8\nendaligned","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"n=9 ✔","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"beginaligned\n  H_9(x)\n   =  - e^x^2 fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmd e^ - x^2mathrmdx\n  = 30240 x - 80640 x^3 + 48384 x^5 - 9216 x^7 + 512 x^9 \n  = 30240 x - 80640 x^3 + 48384 x^5 - 9216 x^7 + 512 x^9\nendaligned","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"Test Summary:                               | Pass  Total   Time\nHₙ(x) = (-1)ⁿ exp(x²) dⁿ/dxⁿ exp(-x²) = ... |   10     10  35.0s","category":"page"},{"location":"HarmonicOscillator/#Normalization-and-Orthogonality-of-H_n(x)","page":"Harmonic Oscillator","title":"Normalization & Orthogonality of H_n(x)","text":"","category":"section"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"int_-infty^infty H_j(x) H_i(x) mathrme^-x^2 mathrmdx = sqrtpi 2^j j delta_ij","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic 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3\t-0.0000000000000021\t0.0000000000000000\t0.0000000000000000%\t✔\n  6\t  4\t0.0000000000019327\t0.0000000000000000\t0.0000000000000000%\t✔\n  6\t  5\t0.0000000000000000\t0.0000000000000000\t0.0000000000000000%\t✔\n  6\t  6\t81674.6734497261350043\t81674.6734497261786601\t0.0000000000000535%\t✔\n  6\t  7\t0.0000000000040927\t0.0000000000000000\t0.0000000000000000%\t✔\n  6\t  8\t0.0000000003965397\t0.0000000000000000\t0.0000000000000000%\t✔\n  6\t  9\t-0.0000000000873115\t0.0000000000000000\t0.0000000000000000%\t✔\n  7\t  0\t0.0000000000000026\t0.0000000000000000\t0.0000000000000000%\t✔\n  7\t  1\t-0.0000000000000749\t0.0000000000000000\t0.0000000000000000%\t✔\n  7\t  2\t0.0000000000000000\t0.0000000000000000\t0.0000000000000000%\t✔\n  7\t  3\t-0.0000000000004121\t0.0000000000000000\t0.0000000000000000%\t✔\n  7\t  4\t0.0000000000000000\t0.0000000000000000\t0.0000000000000000%\t✔\n  7\t  5\t0.0000000000094360\t0.0000000000000000\t0.0000000000000000%\t✔\n  7\t  6\t0.0000000000040927\t0.0000000000000000\t0.0000000000000000%\t✔\n  7\t  7\t1143445.4282961667049676\t1143445.4282961664721370\t0.0000000000000204%\t✔\n  7\t  8\t-0.0000000000072760\t0.0000000000000000\t0.0000000000000000%\t✔\n  7\t  9\t0.0000000116493834\t0.0000000000000000\t0.0000000000000000%\t✔\n  8\t  0\t-0.0000000000008242\t0.0000000000000000\t0.0000000000000000%\t✔\n  8\t  1\t-0.0000000000000568\t0.0000000000000000\t0.0000000000000000%\t✔\n  8\t  2\t-0.0000000000108429\t0.0000000000000000\t0.0000000000000000%\t✔\n  8\t  3\t-0.0000000000004547\t0.0000000000000000\t0.0000000000000000%\t✔\n  8\t  4\t-0.0000000000626612\t0.0000000000000000\t0.0000000000000000%\t✔\n  8\t  5\t0.0000000000000000\t0.0000000000000000\t0.0000000000000000%\t✔\n  8\t  6\t0.0000000003965397\t0.0000000000000000\t0.0000000000000000%\t✔\n  8\t  7\t-0.0000000000072760\t0.0000000000000000\t0.0000000000000000%\t✔\n  8\t  8\t18295126.8527386672794819\t18295126.8527386635541916\t0.0000000000000204%\t✔\n  8\t  9\t0.0000000016298145\t0.0000000000000000\t0.0000000000000000%\t✔\n  9\t  0\t0.0000000000001137\t0.0000000000000000\t0.0000000000000000%\t✔\n  9\t  1\t0.0000000000143249\t0.0000000000000000\t0.0000000000000000%\t✔\n  9\t  2\t-0.0000000000021032\t0.0000000000000000\t0.0000000000000000%\t✔\n  9\t  3\t0.0000000001391527\t0.0000000000000000\t0.0000000000000000%\t✔\n  9\t  4\t0.0000000000000000\t0.0000000000000000\t0.0000000000000000%\t✔\n  9\t  5\t0.0000000013387762\t0.0000000000000000\t0.0000000000000000%\t✔\n  9\t  6\t-0.0000000000873115\t0.0000000000000000\t0.0000000000000000%\t✔\n  9\t  7\t0.0000000116493834\t0.0000000000000000\t0.0000000000000000%\t✔\n  9\t  8\t0.0000000016298145\t0.0000000000000000\t0.0000000000000000%\t✔\n  9\t  9\t329312283.3492956757545471\t329312283.3492959141731262\t0.0000000000000724%\t✔\nTest Summary:                     | Pass  Total  Time\n∫Hⱼ(x)Hᵢ(x)exp(-x²)dx = √π2ʲj!δᵢⱼ |  100    100  1.2s","category":"page"},{"location":"HarmonicOscillator/#Normalization-and-Orthogonality-of-\\psi_n(x)","page":"Harmonic Oscillator","title":"Normalization & Orthogonality of psi_n(x)","text":"","category":"section"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"int psi_i^ast(x) psi_j(x) mathrmdx = delta_ij","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"  i\t  j\tnumerical         \tanalytical        \t|error|\n  0\t  0\t0.9999999999999991\t1.0000000000000000\t0.0000000000000888%\t✔\n  0\t  1\t0.0000000000000000\t0.0000000000000000\t0.0000000000000000%\t✔\n  0\t  2\t0.0000000000000000\t0.0000000000000000\t0.0000000000000000%\t✔\n  0\t  3\t0.0000000000000000\t0.0000000000000000\t0.0000000000000000%\t✔\n  0\t  4\t-0.0000000000000001\t0.0000000000000000\t0.0000000000000000%\t✔\n  0\t  5\t-0.0000000000000000\t0.0000000000000000\t0.0000000000000000%\t✔\n  0\t  6\t0.0000000000000001\t0.0000000000000000\t0.0000000000000000%\t✔\n  0\t  7\t0.0000000000000000\t0.0000000000000000\t0.0000000000000000%\t✔\n  0\t  8\t-0.0000000000000002\t0.0000000000000000\t0.0000000000000000%\t✔\n  0\t  9\t0.0000000000000000\t0.0000000000000000\t0.0000000000000000%\t✔\n  1\t  0\t0.0000000000000000\t0.0000000000000000\t0.0000000000000000%\t✔\n  1\t  1\t0.9999999999999435\t1.0000000000000000\t0.0000000000056510%\t✔\n  1\t  2\t0.0000000000000000\t0.0000000000000000\t0.0000000000000000%\t✔\n  1\t  3\t-0.0000000000000000\t0.0000000000000000\t0.0000000000000000%\t✔\n  1\t  4\t-0.0000000000000000\t0.0000000000000000\t0.0000000000000000%\t✔\n  1\t  5\t0.0000000000000002\t0.0000000000000000\t0.0000000000000000%\t✔\n  1\t  6\t0.0000000000000000\t0.0000000000000000\t0.0000000000000000%\t✔\n  1\t  7\t-0.0000000000000000\t0.0000000000000000\t0.0000000000000000%\t✔\n  1\t  8\t0.0000000000000000\t0.0000000000000000\t0.0000000000000000%\t✔\n  1\t  9\t0.0000000000000004\t0.0000000000000000\t0.0000000000000000%\t✔\n  2\t  0\t0.0000000000000000\t0.0000000000000000\t0.0000000000000000%\t✔\n  2\t  1\t0.0000000000000000\t0.0000000000000000\t0.0000000000000000%\t✔\n  2\t  2\t0.9999999999999999\t1.0000000000000000\t0.0000000000000111%\t✔\n  2\t  3\t-0.0000000000000000\t0.0000000000000000\t0.0000000000000000%\t✔\n  2\t  4\t-0.0000000000000001\t0.0000000000000000\t0.0000000000000000%\t✔\n  2\t  5\t-0.0000000000000000\t0.0000000000000000\t0.0000000000000000%\t✔\n  2\t  6\t0.0000000000000003\t0.0000000000000000\t0.0000000000000000%\t✔\n  2\t  7\t0.0000000000000000\t0.0000000000000000\t0.0000000000000000%\t✔\n  2\t  8\t-0.0000000000000007\t0.0000000000000000\t0.0000000000000000%\t✔\n  2\t  9\t0.0000000000000000\t0.0000000000000000\t0.0000000000000000%\t✔\n  3\t  0\t0.0000000000000000\t0.0000000000000000\t0.0000000000000000%\t✔\n  3\t  1\t-0.0000000000000000\t0.0000000000000000\t0.0000000000000000%\t✔\n  3\t  2\t-0.0000000000000000\t0.0000000000000000\t0.0000000000000000%\t✔\n  3\t  3\t0.9999999999999998\t1.0000000000000000\t0.0000000000000222%\t✔\n  3\t  4\t-0.0000000000000000\t0.0000000000000000\t0.0000000000000000%\t✔\n  3\t  5\t0.0000000000000000\t0.0000000000000000\t0.0000000000000000%\t✔\n  3\t  6\t0.0000000000000000\t0.0000000000000000\t0.0000000000000000%\t✔\n  3\t  7\t0.0000000000000001\t0.0000000000000000\t0.0000000000000000%\t✔\n  3\t  8\t-0.0000000000000000\t0.0000000000000000\t0.0000000000000000%\t✔\n  3\t  9\t0.0000000000000008\t0.0000000000000000\t0.0000000000000000%\t✔\n  4\t  0\t-0.0000000000000001\t0.0000000000000000\t0.0000000000000000%\t✔\n  4\t  1\t-0.0000000000000000\t0.0000000000000000\t0.0000000000000000%\t✔\n  4\t  2\t-0.0000000000000001\t0.0000000000000000\t0.0000000000000000%\t✔\n  4\t  3\t-0.0000000000000000\t0.0000000000000000\t0.0000000000000000%\t✔\n  4\t  4\t0.9999999999999991\t1.0000000000000000\t0.0000000000000888%\t✔\n  4\t  5\t0.0000000000000000\t0.0000000000000000\t0.0000000000000000%\t✔\n  4\t  6\t0.0000000000000003\t0.0000000000000000\t0.0000000000000000%\t✔\n  4\t  7\t0.0000000000000000\t0.0000000000000000\t0.0000000000000000%\t✔\n  4\t  8\t-0.0000000000000005\t0.0000000000000000\t0.0000000000000000%\t✔\n  4\t  9\t-0.0000000000000000\t0.0000000000000000\t0.0000000000000000%\t✔\n  5\t  0\t-0.0000000000000000\t0.0000000000000000\t0.0000000000000000%\t✔\n  5\t  1\t0.0000000000000002\t0.0000000000000000\t0.0000000000000000%\t✔\n  5\t  2\t-0.0000000000000000\t0.0000000000000000\t0.0000000000000000%\t✔\n  5\t  3\t0.0000000000000000\t0.0000000000000000\t0.0000000000000000%\t✔\n  5\t  4\t0.0000000000000000\t0.0000000000000000\t0.0000000000000000%\t✔\n  5\t  5\t0.9999999999999999\t1.0000000000000000\t0.0000000000000111%\t✔\n  5\t  6\t0.0000000000000000\t0.0000000000000000\t0.0000000000000000%\t✔\n  5\t  7\t0.0000000000000002\t0.0000000000000000\t0.0000000000000000%\t✔\n  5\t  8\t0.0000000000000000\t0.0000000000000000\t0.0000000000000000%\t✔\n  5\t  9\t0.0000000000000008\t0.0000000000000000\t0.0000000000000000%\t✔\n  6\t  0\t0.0000000000000001\t0.0000000000000000\t0.0000000000000000%\t✔\n  6\t  1\t0.0000000000000000\t0.0000000000000000\t0.0000000000000000%\t✔\n  6\t  2\t0.0000000000000003\t0.0000000000000000\t0.0000000000000000%\t✔\n  6\t  3\t0.0000000000000000\t0.0000000000000000\t0.0000000000000000%\t✔\n  6\t  4\t0.0000000000000003\t0.0000000000000000\t0.0000000000000000%\t✔\n  6\t  5\t0.0000000000000000\t0.0000000000000000\t0.0000000000000000%\t✔\n  6\t  6\t0.9999999999999993\t1.0000000000000000\t0.0000000000000666%\t✔\n  6\t  7\t-0.0000000000000000\t0.0000000000000000\t0.0000000000000000%\t✔\n  6\t  8\t0.0000000000000003\t0.0000000000000000\t0.0000000000000000%\t✔\n  6\t  9\t0.0000000000000000\t0.0000000000000000\t0.0000000000000000%\t✔\n  7\t  0\t0.0000000000000000\t0.0000000000000000\t0.0000000000000000%\t✔\n  7\t  1\t-0.0000000000000000\t0.0000000000000000\t0.0000000000000000%\t✔\n  7\t  2\t0.0000000000000000\t0.0000000000000000\t0.0000000000000000%\t✔\n  7\t  3\t0.0000000000000001\t0.0000000000000000\t0.0000000000000000%\t✔\n  7\t  4\t0.0000000000000000\t0.0000000000000000\t0.0000000000000000%\t✔\n  7\t  5\t0.0000000000000002\t0.0000000000000000\t0.0000000000000000%\t✔\n  7\t  6\t-0.0000000000000000\t0.0000000000000000\t0.0000000000000000%\t✔\n  7\t  7\t1.0000000000000004\t1.0000000000000000\t0.0000000000000444%\t✔\n  7\t  8\t0.0000000000000000\t0.0000000000000000\t0.0000000000000000%\t✔\n  7\t  9\t0.0000000000000006\t0.0000000000000000\t0.0000000000000000%\t✔\n  8\t  0\t-0.0000000000000002\t0.0000000000000000\t0.0000000000000000%\t✔\n  8\t  1\t0.0000000000000000\t0.0000000000000000\t0.0000000000000000%\t✔\n  8\t  2\t-0.0000000000000007\t0.0000000000000000\t0.0000000000000000%\t✔\n  8\t  3\t-0.0000000000000000\t0.0000000000000000\t0.0000000000000000%\t✔\n  8\t  4\t-0.0000000000000005\t0.0000000000000000\t0.0000000000000000%\t✔\n  8\t  5\t0.0000000000000000\t0.0000000000000000\t0.0000000000000000%\t✔\n  8\t  6\t0.0000000000000003\t0.0000000000000000\t0.0000000000000000%\t✔\n  8\t  7\t0.0000000000000000\t0.0000000000000000\t0.0000000000000000%\t✔\n  8\t  8\t1.0000000000000007\t1.0000000000000000\t0.0000000000000666%\t✔\n  8\t  9\t-0.0000000000000000\t0.0000000000000000\t0.0000000000000000%\t✔\n  9\t  0\t0.0000000000000000\t0.0000000000000000\t0.0000000000000000%\t✔\n  9\t  1\t0.0000000000000004\t0.0000000000000000\t0.0000000000000000%\t✔\n  9\t  2\t0.0000000000000000\t0.0000000000000000\t0.0000000000000000%\t✔\n  9\t  3\t0.0000000000000008\t0.0000000000000000\t0.0000000000000000%\t✔\n  9\t  4\t-0.0000000000000000\t0.0000000000000000\t0.0000000000000000%\t✔\n  9\t  5\t0.0000000000000008\t0.0000000000000000\t0.0000000000000000%\t✔\n  9\t  6\t0.0000000000000000\t0.0000000000000000\t0.0000000000000000%\t✔\n  9\t  7\t0.0000000000000006\t0.0000000000000000\t0.0000000000000000%\t✔\n  9\t  8\t-0.0000000000000000\t0.0000000000000000\t0.0000000000000000%\t✔\n  9\t  9\t0.9999999999999998\t1.0000000000000000\t0.0000000000000222%\t✔\nTest Summary: | Pass  Total  Time\n<ψᵢ|ψⱼ> = δᵢⱼ |  100    100  1.2s","category":"page"},{"location":"HarmonicOscillator/#Virial-Theorem","page":"Harmonic Oscillator","title":"Virial Theorem","text":"","category":"section"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"The virial theorem langle T rangle = langle V rangle and the definition of Hamiltonian langle H rangle = langle T rangle + langle V rangle derive langle H rangle = 2 langle V rangle = 2 langle T rangle.","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"2 int psi_n^ast(x) V(x) psi_n(x) mathrmdx = E_n","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"   k\t  n\tnumerical         \tanalytical        \t|error|\n0.1\t  0\t0.4999999999999719\t0.5000000000000000\t0.0000000000056288%\t✔\n0.1\t  1\t1.4999999999999998\t1.5000000000000000\t0.0000000000000148%\t✔\n0.1\t  2\t2.5000000000000004\t2.5000000000000000\t0.0000000000000178%\t✔\n0.1\t  3\t3.4999999999999982\t3.5000000000000000\t0.0000000000000508%\t✔\n0.1\t  4\t4.4999999999999938\t4.5000000000000000\t0.0000000000001382%\t✔\n0.1\t  5\t5.5000000000000000\t5.5000000000000000\t0.0000000000000000%\t✔\n0.1\t  6\t6.5000000000000000\t6.5000000000000000\t0.0000000000000000%\t✔\n0.1\t  7\t7.5000000000000124\t7.5000000000000000\t0.0000000000001658%\t✔\n0.1\t  8\t8.4999999999999929\t8.5000000000000000\t0.0000000000000836%\t✔\n0.1\t  9\t9.5000000000000000\t9.5000000000000000\t0.0000000000000000%\t✔\n0.5\t  0\t0.4999999999999719\t0.5000000000000000\t0.0000000000056288%\t✔\n0.5\t  1\t1.4999999999999998\t1.5000000000000000\t0.0000000000000148%\t✔\n0.5\t  2\t2.5000000000000004\t2.5000000000000000\t0.0000000000000178%\t✔\n0.5\t  3\t3.4999999999999982\t3.5000000000000000\t0.0000000000000508%\t✔\n0.5\t  4\t4.4999999999999938\t4.5000000000000000\t0.0000000000001382%\t✔\n0.5\t  5\t5.5000000000000000\t5.5000000000000000\t0.0000000000000000%\t✔\n0.5\t  6\t6.5000000000000000\t6.5000000000000000\t0.0000000000000000%\t✔\n0.5\t  7\t7.5000000000000124\t7.5000000000000000\t0.0000000000001658%\t✔\n0.5\t  8\t8.4999999999999929\t8.5000000000000000\t0.0000000000000836%\t✔\n0.5\t  9\t9.5000000000000000\t9.5000000000000000\t0.0000000000000000%\t✔\n1.0\t  0\t0.4999999999999719\t0.5000000000000000\t0.0000000000056288%\t✔\n1.0\t  1\t1.4999999999999998\t1.5000000000000000\t0.0000000000000148%\t✔\n1.0\t  2\t2.5000000000000004\t2.5000000000000000\t0.0000000000000178%\t✔\n1.0\t  3\t3.4999999999999982\t3.5000000000000000\t0.0000000000000508%\t✔\n1.0\t  4\t4.4999999999999938\t4.5000000000000000\t0.0000000000001382%\t✔\n1.0\t  5\t5.5000000000000000\t5.5000000000000000\t0.0000000000000000%\t✔\n1.0\t  6\t6.5000000000000000\t6.5000000000000000\t0.0000000000000000%\t✔\n1.0\t  7\t7.5000000000000124\t7.5000000000000000\t0.0000000000001658%\t✔\n1.0\t  8\t8.4999999999999929\t8.5000000000000000\t0.0000000000000836%\t✔\n1.0\t  9\t9.5000000000000000\t9.5000000000000000\t0.0000000000000000%\t✔\n5.0\t  0\t0.4999999999999719\t0.5000000000000000\t0.0000000000056288%\t✔\n5.0\t  1\t1.4999999999999998\t1.5000000000000000\t0.0000000000000148%\t✔\n5.0\t  2\t2.5000000000000004\t2.5000000000000000\t0.0000000000000178%\t✔\n5.0\t  3\t3.4999999999999982\t3.5000000000000000\t0.0000000000000508%\t✔\n5.0\t  4\t4.4999999999999938\t4.5000000000000000\t0.0000000000001382%\t✔\n5.0\t  5\t5.5000000000000000\t5.5000000000000000\t0.0000000000000000%\t✔\n5.0\t  6\t6.5000000000000000\t6.5000000000000000\t0.0000000000000000%\t✔\n5.0\t  7\t7.5000000000000124\t7.5000000000000000\t0.0000000000001658%\t✔\n5.0\t  8\t8.4999999999999929\t8.5000000000000000\t0.0000000000000836%\t✔\n5.0\t  9\t9.5000000000000000\t9.5000000000000000\t0.0000000000000000%\t✔\nTest Summary:      | Pass  Total  Time\n2 × <ψₙ|V|ψₙ> = Eₙ |   40     40  1.0s","category":"page"},{"location":"HarmonicOscillator/#Eigen-Values-2","page":"Harmonic Oscillator","title":"Eigen Values","text":"","category":"section"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"  beginaligned\n    E_n\n    =      int psi^ast_n(x) hatH psi_n(x) mathrmdx \n    =      int psi^ast_n(x) left hatV + hatT right psi(x) mathrmdx \n    =      int psi^ast_n(x) left V(x) - frachbar^22m fracmathrmd^2mathrmd x^2 right psi(x) mathrmdx \n    simeq int psi^ast_n(x) left V(x)psi(x) -frachbar^22m fracpsi(x+Delta x) - 2psi(x) + psi(x-Delta x)Delta x^2 right mathrmdx\n  endaligned","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"Where, the difference formula for the 2nd-order derivative:","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"beginaligned\n   2psi(x)\n   + fracmathrmd^2 psi(x)mathrmd x^2 Delta x^2\n   + Oleft(Delta x^4right)\n   =\n   psi(x+Delta x)\n   + psi(x-Delta x)\n   \n   fracmathrmd^2 psi(x)mathrmd x^2 Delta x^2\n   =\n   psi(x+Delta x)\n   - 2psi(x)\n   + psi(x-Delta x)\n   - Oleft(Delta x^4right)\n   \n   fracmathrmd^2 psi(x)mathrmd x^2\n   =\n   fracpsi(x+Delta x) - 2psi(x) + psi(x-Delta x)Delta x^2\n   - fracOleft(Delta x^4right)Delta x^2\n   \n  fracmathrmd^2 psi(x)mathrmd x^2\n  =\n  fracpsi(x+Delta x) - 2psi(x) + psi(x-Delta x)Delta x^2\n  + Oleft(Delta x^2right)\nendaligned","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"are given by the sum of 2 Taylor series:","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"beginaligned\npsi(x+Delta x)\n= psi(x)\n+ fracmathrmd psi(x)mathrmd x Delta x\n+ frac12 fracmathrmd^2 psi(x)mathrmd x^2 Delta x^2\n+ frac13 fracmathrmd^3 psi(x)mathrmd x^3 Delta x^3\n+ Oleft(Delta x^4right)\n\npsi(x-Delta x)\n= psi(x)\n- fracmathrmd psi(x)mathrmd x Delta x\n+ frac12 fracmathrmd^2 psi(x)mathrmd x^2 Delta x^2\n- frac13 fracmathrmd^3 psi(x)mathrmd x^3 Delta x^3\n+ Oleft(Delta x^4right)\nendaligned","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"   k\t  n\tnumerical         \tanalytical        \t|error|\n0.1\t  0\t0.1581138798827675\t0.1581138830084190\t0.0000019768355921%\t✔\n0.1\t  1\t0.4743416334097137\t0.4743416490252569\t0.0000032920455591%\t✔\n0.1\t  2\t0.7905693744092002\t0.7905694150420949\t0.0000051396998091%\t✔\n0.1\t  3\t1.1067971029282735\t1.1067971810589328\t0.0000070591668115%\t✔\n0.1\t  4\t1.4230248189868937\t1.4230249470757708\t0.0000090011687654%\t✔\n0.1\t  5\t1.7392525225056779\t1.7392527130926088\t0.0000109579780712%\t✔\n0.1\t  6\t2.0554802135001529\t2.0554804791094465\t0.0000129220051647%\t✔\n0.1\t  7\t2.3717078919497427\t2.3717082451262845\t0.0000148912305093%\t✔\n0.1\t  8\t2.6879355581003628\t2.6879360111431225\t0.0000168546705678%\t✔\n0.1\t  9\t3.0041632114497054\t3.0041637771599605\t0.0000188308726503%\t✔\n0.5\t  0\t0.3535533749437250\t0.3535533905932738\t0.0000044263608382%\t✔\n0.5\t  1\t1.0606600936488757\t1.0606601717798214\t0.0000073662562046%\t✔\n0.5\t  2\t1.7677667498779439\t1.7677669529663689\t0.0000114884161979%\t✔\n0.5\t  3\t2.4748733435564381\t2.4748737341529163\t0.0000157824810568%\t✔\n0.5\t  4\t3.1819798748172214\t3.1819805153394642\t0.0000201296720616%\t✔\n0.5\t  5\t3.8890863434634926\t3.8890872965260117\t0.0000245060716425%\t✔\n0.5\t  6\t4.5961927496648043\t4.5961940777125596\t0.0000288945099547%\t✔\n0.5\t  7\t5.3032990935191169\t5.3033008588991066\t0.0000332883243233%\t✔\n0.5\t  8\t6.0104053741967718\t6.0104076400856545\t0.0000376994210448%\t✔\n0.5\t  9\t6.7175115932661038\t6.7175144212722016\t0.0000420989955569%\t✔\n1.0\t  0\t0.4999999687733652\t0.5000000000000000\t0.0000062453269556%\t✔\n1.0\t  1\t1.4999998437740314\t1.5000000000000000\t0.0000104150645737%\t✔\n1.0\t  2\t2.4999995937642381\t2.5000000000000000\t0.0000162494304767%\t✔\n1.0\t  3\t3.4999992187323192\t3.5000000000000000\t0.0000223219337363%\t✔\n1.0\t  4\t4.4999987187474328\t4.5000000000000000\t0.0000284722792701%\t✔\n1.0\t  5\t5.4999980937550053\t5.5000000000000000\t0.0000346589999044%\t✔\n1.0\t  6\t6.4999973436023444\t6.5000000000000000\t0.0000408676562403%\t✔\n1.0\t  7\t7.4999964688869518\t7.5000000000000000\t0.0000470815073091%\t✔\n1.0\t  8\t8.4999954688433572\t8.5000000000000000\t0.0000533077252092%\t✔\n1.0\t  9\t9.4999943434450671\t9.5000000000000000\t0.0000595426835042%\t✔\n5.0\t  0\t1.1180338325225039\t1.1180339887498949\t0.0000139734026511%\t✔\n5.0\t  1\t3.3541011849691116\t3.3541019662496847\t0.0000232932862801%\t✔\n5.0\t  2\t5.5901679125242794\t5.5901699437494745\t0.0000363356609118%\t✔\n5.0\t  3\t7.8262340149842551\t7.8262379212492643\t0.0000499124234216%\t✔\n5.0\t  4\t10.0622994924938265\t10.0623058987490541\t0.0000636658763120%\t✔\n5.0\t  5\t12.2983643449971325\t12.2983738762488439\t0.0000775000972270%\t✔\n5.0\t  6\t14.5344285723091655\t14.5344418537486337\t0.0000913790815076%\t✔\n5.0\t  7\t16.7704921752222305\t16.7705098312484253\t0.0001052801994246%\t✔\n5.0\t  8\t19.0065551524155296\t19.0065778087482116\t0.0001192025882298%\t✔\n5.0\t  9\t21.2426175047498660\t21.2426457862480049\t0.0001331354786194%\t✔\nTest Summary:              | Pass  Total  Time\n∫ψₙ*Hψₙdx = <ψₙ|H|ψₙ> = Eₙ |   40     40  1.4s\n","category":"page"},{"location":"","page":"Home","title":"Home","text":"CurrentModule = Antiq","category":"page"},{"location":"#Antiq.jl","page":"Home","title":"Antiq.jl","text":"","category":"section"},{"location":"","page":"Home","title":"Home","text":"Self-contained, Well-Tested, Well-Documented Functions for Quantum Mechanical Models","category":"page"},{"location":"#Install","page":"Home","title":"Install","text":"","category":"section"},{"location":"","page":"Home","title":"Home","text":"To install this package, run the following code in your Jupyter Notebook:","category":"page"},{"location":"","page":"Home","title":"Home","text":"using Pkg\nPkg.add(path=\"https://github.com/ohno/Antiq.jl.git\")","category":"page"},{"location":"","page":"Home","title":"Home","text":"or use add upon REPL(Package mode):","category":"page"},{"location":"","page":"Home","title":"Home","text":"]\nadd https://github.com/ohno/Antiq.jl.git","category":"page"},{"location":"#Usage","page":"Home","title":"Usage","text":"","category":"section"},{"location":"","page":"Home","title":"Home","text":"To use this package, run the following code before each use:","category":"page"},{"location":"","page":"Home","title":"Home","text":"using Antiq","category":"page"},{"location":"","page":"Home","title":"Home","text":"The function antiq(model, parameters...) returns a module. Each module has E(), ψ(x) and some other functions.","category":"page"},{"location":"#Examples","page":"Home","title":"Examples","text":"","category":"section"},{"location":"","page":"Home","title":"Home","text":"The energy of 1S state in hydrogen atom:","category":"page"},{"location":"","page":"Home","title":"Home","text":"julia> H = antiq(:HydrogenAtom, Z=1)\njulia> H.E(n=1)\n-0.5","category":"page"},{"location":"","page":"Home","title":"Home","text":"The energy of 1S state in helium atom:","category":"page"},{"location":"","page":"Home","title":"Home","text":"julia> He⁺ = antiq(:HydrogenAtom, Z=2)\njulia> He⁺.E(n=1)\n-2.0","category":"page"},{"location":"#Supported-Models","page":"Home","title":"Supported Models","text":"","category":"section"},{"location":"","page":"Home","title":"Home","text":"
\n  
\n    \n      \"InfinitePotentialWell\"/\n    \n    InfinitePotentialWell\n  
\n  
\n    \n      \"HarmonicOscillator\"/\n    \n    HarmonicOscillator\n  
\n  
\n    \n      \"MorsePotential\"/\n    \n    MorsePotential\n  
\n  
\n    \n      \"HydrogenAtom\"/\n    \n    HydrogenAtom\n  
\n
","category":"page"},{"location":"#Future-Works","page":"Home","title":"Future Works","text":"","category":"section"},{"location":"","page":"Home","title":"Home","text":"List of quantum-mechanical systems with analytical solutions","category":"page"},{"location":"#Acknowledgment","page":"Home","title":"Acknowledgment","text":"","category":"section"},{"location":"","page":"Home","title":"Home","text":"This package was named by @KB-satou and @ultimatile:","category":"page"},{"location":"","page":"Home","title":"Home","text":"Analytical soulutions of Schrödinger equations.","category":"page"}]
+[{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"CurrentModule = Antiq","category":"page"},{"location":"InfinitePotentialWell/#Infinite-Potential-Well-(Particle-in-a-Box)","page":"Infinite Potential Well","title":"Infinite Potential Well (Particle in a Box)","text":"","category":"section"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"The infinite potential well (particle in a box) is the simplest model for quantum mechanical system.","category":"page"},{"location":"InfinitePotentialWell/#Definitions","page":"Infinite Potential Well","title":"Definitions","text":"","category":"section"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"L is the length of the 1D-box, m is the mass of particle.","category":"page"},{"location":"InfinitePotentialWell/#Schrödinger-Equation","page":"Infinite Potential Well","title":"Schrödinger Equation","text":"","category":"section"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"  hatH psi(x) = E psi(x)","category":"page"},{"location":"InfinitePotentialWell/#Hamiltonian","page":"Infinite Potential Well","title":"Hamiltonian","text":"","category":"section"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"  hatH = frachbar^22m fracmathrmd^2mathrmdx ^2 + V(x)","category":"page"},{"location":"InfinitePotentialWell/#Potential","page":"Infinite Potential Well","title":"Potential","text":"","category":"section"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"V(x; L=L)","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"  V(x) =\n  left\n    beginarrayll\n    infty  x lt 0 L lt x \n    0       0 leq x leq L\n    endarray\n  right","category":"page"},{"location":"InfinitePotentialWell/#Eigen-Values","page":"Infinite Potential Well","title":"Eigen Values","text":"","category":"section"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"E(; n=0, L=L, m=m, ℏ=ℏ)","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"  E_n = frachbar^2 n^2 pi^22 m L^2","category":"page"},{"location":"InfinitePotentialWell/#Eigen-Functions","page":"Infinite Potential Well","title":"Eigen Functions","text":"","category":"section"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"ψ(x; n=0, L=L)","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"   psi_n(x) = sqrtfrac2L sin fracnpi xL","category":"page"},{"location":"InfinitePotentialWell/#Proofs","page":"Infinite Potential Well","title":"Proofs","text":"","category":"section"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"Eigen Functions & Eigen Values\nNormalization","category":"page"},{"location":"InfinitePotentialWell/#Usage-and-Examples","page":"Infinite Potential Well","title":"Usage & Examples","text":"","category":"section"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"Install Antiq.jl for the first run and run using Antiq before each use. The function antiq(model, parameters...) returns a module that has E(), ψ(x), V(x) and some other functions. In this system, the model name is specified by :InfinitePotentialWell and several parameters L, m and ℏ are set as optional arguments.","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"using Antiq\nIPW = antiq(:InfinitePotentialWell, L=1.0, m=1.0, ℏ=1.0)","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"Parameters:","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"julia> IPW.L\n1.0\n\njulia> IPW.m\n1.0\n\njulia> IPW.ℏ\n1.0","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"Eigen values:","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"julia> IPW.E(n=1)\n4.934802200544679\n\njulia> IPW.E(n=2)\n19.739208802178716","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"Wave functions:","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"using Plots\nplot(xlim=(0,1), xlabel=\"x\", ylabel=\"ψ(x)\")\nplot!(x -> IPW.ψ(x, n=1), label=\"n=1\", lw=2)\nplot!(x -> IPW.ψ(x, n=2), label=\"n=2\", lw=2)\nplot!(x -> IPW.ψ(x, n=3), label=\"n=3\", lw=2)\nplot!(x -> IPW.ψ(x, n=4), label=\"n=4\", lw=2)\nplot!(x -> IPW.ψ(x, n=5), label=\"n=5\", lw=2)","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"(Image: )","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"Potential energy curve, Energy levels, Wave functions:","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"L = 1\nusing Plots\nplot(xlim=(-0.5,1.5), ylim=(-5,140), xlabel=\"\\$x\\$\", ylabel=\"\\$V(x),~E_n,~\\\\psi_n(x)\\\\times5+E_n\\$\", size=(480,400), dpi=300)\nfor n in 1:5\n  # energy\n  plot!([0,L], fill(IPW.E(n=n,L=L),2), lc=:black, lw=2, label=\"\")\n  # wave function\n  plot!(0:0.01:L, x->IPW.E(n=n,L=L)+5*IPW.ψ(x,n=n,L=L), lc=n, lw=2, label=\"\")\nend\n# potential\nplot!([0,0,L,L], [140,0,0,140], lc=:black, lw=2, label=\"\")","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"(Image: )","category":"page"},{"location":"InfinitePotentialWell/#Testing","page":"Infinite Potential Well","title":"Testing","text":"","category":"section"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"Unit testing and Integration testing were done using numerical integration (QuadGK.jl). The test script is here.","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"include(\"../../../test/InfinitePotentialWell.jl\")","category":"page"},{"location":"InfinitePotentialWell/#Normalization-and-Orthogonality-of-\\psi_n(x)","page":"Infinite Potential Well","title":"Normalization & Orthogonality of psi_n(x)","text":"","category":"section"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"int_0^L psi_i^ast(x) psi_j(x) mathrmdx = delta_ij","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"  i\t  j\tnumerical         \tanalytical        \t|error|\n  1\t  1\t1.0000000000000002\t1.0000000000000000\t0.0000000000000222%\t✔\n  1\t  2\t0.0000000000000000\t0.0000000000000000\t0.0000000000000000%\t✔\n  1\t  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8\t-0.0000000000000001\t0.0000000000000000\t0.0000000000000000%\t✔\n  7\t  9\t-0.0000000000000001\t0.0000000000000000\t0.0000000000000000%\t✔\n  7\t 10\t-0.0000000000000000\t0.0000000000000000\t0.0000000000000000%\t✔\n  8\t  1\t-0.0000000000000001\t0.0000000000000000\t0.0000000000000000%\t✔\n  8\t  2\t0.0000000000000000\t0.0000000000000000\t0.0000000000000000%\t✔\n  8\t  3\t0.0000000000000000\t0.0000000000000000\t0.0000000000000000%\t✔\n  8\t  4\t0.0000000000000000\t0.0000000000000000\t0.0000000000000000%\t✔\n  8\t  5\t0.0000000000000000\t0.0000000000000000\t0.0000000000000000%\t✔\n  8\t  6\t-0.0000000000000000\t0.0000000000000000\t0.0000000000000000%\t✔\n  8\t  7\t-0.0000000000000001\t0.0000000000000000\t0.0000000000000000%\t✔\n  8\t  8\t1.0000000000000007\t1.0000000000000000\t0.0000000000000666%\t✔\n  8\t  9\t-0.0000000000000001\t0.0000000000000000\t0.0000000000000000%\t✔\n  8\t 10\t0.0000000000000000\t0.0000000000000000\t0.0000000000000000%\t✔\n  9\t  1\t-0.0000000000000000\t0.0000000000000000\t0.0000000000000000%\t✔\n  9\t  2\t-0.0000000000000001\t0.0000000000000000\t0.0000000000000000%\t✔\n  9\t  3\t-0.0000000000000000\t0.0000000000000000\t0.0000000000000000%\t✔\n  9\t  4\t-0.0000000000000001\t0.0000000000000000\t0.0000000000000000%\t✔\n  9\t  5\t0.0000000000000000\t0.0000000000000000\t0.0000000000000000%\t✔\n  9\t  6\t0.0000000000000000\t0.0000000000000000\t0.0000000000000000%\t✔\n  9\t  7\t-0.0000000000000001\t0.0000000000000000\t0.0000000000000000%\t✔\n  9\t  8\t-0.0000000000000001\t0.0000000000000000\t0.0000000000000000%\t✔\n  9\t  9\t1.0000000000000000\t1.0000000000000000\t0.0000000000000000%\t✔\n  9\t 10\t0.0000000000000001\t0.0000000000000000\t0.0000000000000000%\t✔\n 10\t  1\t0.0000000000000001\t0.0000000000000000\t0.0000000000000000%\t✔\n 10\t  2\t0.0000000000000000\t0.0000000000000000\t0.0000000000000000%\t✔\n 10\t  3\t0.0000000000000000\t0.0000000000000000\t0.0000000000000000%\t✔\n 10\t  4\t0.0000000000000000\t0.0000000000000000\t0.0000000000000000%\t✔\n 10\t  5\t0.0000000000000000\t0.0000000000000000\t0.0000000000000000%\t✔\n 10\t  6\t-0.0000000000000000\t0.0000000000000000\t0.0000000000000000%\t✔\n 10\t  7\t-0.0000000000000000\t0.0000000000000000\t0.0000000000000000%\t✔\n 10\t  8\t0.0000000000000000\t0.0000000000000000\t0.0000000000000000%\t✔\n 10\t  9\t0.0000000000000001\t0.0000000000000000\t0.0000000000000000%\t✔\n 10\t 10\t1.0000000000000002\t1.0000000000000000\t0.0000000000000222%\t✔\nTest Summary:            | Pass  Total  Time\n<ψᵢ|ψⱼ> = ∫ψₙ*ψₙdx = δᵢⱼ |  100    100  2.3s","category":"page"},{"location":"InfinitePotentialWell/#Eigen-Values-2","page":"Infinite Potential Well","title":"Eigen Values","text":"","category":"section"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"  beginaligned\n    E_n\n    =      int_0^L psi^ast_n(x) hatH psi_n(x) mathrmdx \n    =      int_0^L psi^ast_n(x) left hatV + hatT right psi(x) mathrmdx \n    =      int_0^L psi^ast_n(x) left 0 - frachbar^22m fracmathrmd^2mathrmd x^2 right psi(x) mathrmdx \n    simeq int_0^L psi^ast_n(x) left -frachbar^22m fracpsi(x+Delta x) - 2psi(x) + psi(x-Delta x)Delta x^2 right mathrmdx\n  endaligned","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"Where, the difference formula for the 2nd-order derivative:","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"beginaligned\n   2psi(x)\n   + fracmathrmd^2 psi(x)mathrmd x^2 Delta x^2\n   + Oleft(Delta x^4right)\n   =\n   psi(x+Delta x)\n   + psi(x-Delta x)\n   \n   fracmathrmd^2 psi(x)mathrmd x^2 Delta x^2\n   =\n   psi(x+Delta x)\n   - 2psi(x)\n   + psi(x-Delta x)\n   - Oleft(Delta x^4right)\n   \n   fracmathrmd^2 psi(x)mathrmd x^2\n   =\n   fracpsi(x+Delta x) - 2psi(x) + psi(x-Delta x)Delta x^2\n   - fracOleft(Delta x^4right)Delta x^2\n   \n  fracmathrmd^2 psi(x)mathrmd x^2\n  =\n  fracpsi(x+Delta x) - 2psi(x) + psi(x-Delta x)Delta x^2\n  + Oleft(Delta x^2right)\nendaligned","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"are given by the sum of 2 Taylor series:","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"beginaligned\npsi(x+Delta x)\n= psi(x)\n+ fracmathrmd psi(x)mathrmd x Delta x\n+ frac12 fracmathrmd^2 psi(x)mathrmd x^2 Delta x^2\n+ frac13 fracmathrmd^3 psi(x)mathrmd x^3 Delta x^3\n+ Oleft(Delta x^4right)\n\npsi(x-Delta x)\n= psi(x)\n- fracmathrmd psi(x)mathrmd x Delta x\n+ frac12 fracmathrmd^2 psi(x)mathrmd x^2 Delta x^2\n- frac13 fracmathrmd^3 psi(x)mathrmd x^3 Delta x^3\n+ Oleft(Delta x^4right)\nendaligned","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"  L    m    ℏ   n  numerical     \tanalytical    \t|error|\n0.1  0.1  0.1   1  49.348021579139\t49.348022005447\t0.000000863880%\t✔\n0.1  0.1  0.1   2  197.392081461942\t197.392088021787\t0.000003323256%\t✔\n0.1  0.1  0.1   3  444.132165131018\t444.132198049021\t0.000007411758%\t✔\n0.1  0.1  0.1   4  789.568248175979\t789.568352087149\t0.000013160503%\t✔\n0.1  0.1  0.1   5  1233.700296336187\t1233.700550136170\t0.000020572252%\t✔\n0.1  0.1  0.1   6  1776.528266243334\t1776.528792196084\t0.000029605642%\t✔\n0.1  0.1  0.1   7  2418.052103857080\t2418.053078266893\t0.000040297288%\t✔\n0.1  0.1  0.1   8  3158.271745875927\t3158.273408348594\t0.000052638656%\t✔\n0.1  0.1  0.1   9  3997.187119264267\t3997.189782441189\t0.000066626232%\t✔\n0.1  0.1  0.1  10  4934.798141994514\t4934.802200544678\t0.000082243421%\t✔\n0.1  0.1  1.0   1  4934.802157913905\t4934.802200544678\t0.000000863880%\t✔\n0.1  0.1  1.0   2  19739.208146194214\t19739.208802178713\t0.000003323256%\t✔\n0.1  0.1  1.0   3  44413.216513101754\t44413.219804902103\t0.000007411758%\t✔\n0.1  0.1  1.0   4  78956.824817597895\t78956.835208714852\t0.000013160503%\t✔\n0.1  0.1  1.0   5  123370.029633618717\t123370.055013616948\t0.000020572252%\t✔\n0.1  0.1  1.0   6  177652.826624333364\t177652.879219608410\t0.000029605642%\t✔\n0.1  0.1  1.0   7  241805.210385707964\t241805.307826689212\t0.000040297288%\t✔\n0.1  0.1  1.0   8  315827.174587592541\t315827.340834859409\t0.000052638656%\t✔\n0.1  0.1  1.0   9  399718.711926426622\t399718.978244118916\t0.000066626232%\t✔\n0.1  0.1  1.0  10  493479.814199451241\t493480.220054467791\t0.000082243421%\t✔\n0.1  1.0  0.1   1  4.934802157914\t4.934802200545\t0.000000863880%\t✔\n0.1  1.0  0.1   2  19.739208146194\t19.739208802179\t0.000003323256%\t✔\n0.1  1.0  0.1   3  44.413216513102\t44.413219804902\t0.000007411758%\t✔\n0.1  1.0  0.1   4  78.956824817598\t78.956835208715\t0.000013160503%\t✔\n0.1  1.0  0.1   5  123.370029633619\t123.370055013617\t0.000020572252%\t✔\n0.1  1.0  0.1   6  177.652826624333\t177.652879219608\t0.000029605642%\t✔\n0.1  1.0  0.1   7  241.805210385708\t241.805307826689\t0.000040297288%\t✔\n0.1  1.0  0.1   8  315.827174587593\t315.827340834859\t0.000052638656%\t✔\n0.1  1.0  0.1   9  399.718711926427\t399.718978244119\t0.000066626232%\t✔\n0.1  1.0  0.1  10  493.479814199451\t493.480220054468\t0.000082243421%\t✔\n0.1  1.0  1.0   1  493.480215791390\t493.480220054468\t0.000000863880%\t✔\n0.1  1.0  1.0   2  1973.920814619422\t1973.920880217871\t0.000003323256%\t✔\n0.1  1.0  1.0   3  4441.321651310176\t4441.321980490210\t0.000007411758%\t✔\n0.1  1.0  1.0   4  7895.682481759791\t7895.683520871485\t0.000013160503%\t✔\n0.1  1.0  1.0   5  12337.002963361872\t12337.005501361695\t0.000020572252%\t✔\n0.1  1.0  1.0   6  17765.282662433339\t17765.287921960840\t0.000029605642%\t✔\n0.1  1.0  1.0   7  24180.521038570794\t24180.530782668924\t0.000040297288%\t✔\n0.1  1.0  1.0   8  31582.717458759253\t31582.734083485939\t0.000052638656%\t✔\n0.1  1.0  1.0   9  39971.871192642662\t39971.897824411892\t0.000066626232%\t✔\n0.1  1.0  1.0  10  49347.981419945128\t49348.022005446779\t0.000082243421%\t✔\n1.0  0.1  0.1   1  0.493480215948\t0.493480220054\t0.000000832112%\t✔\n1.0  0.1  0.1   2  1.973920815419\t1.973920880218\t0.000003282767%\t✔\n1.0  0.1  0.1   3  4.441321651944\t4.441321980490\t0.000007397478%\t✔\n1.0  0.1  0.1   4  7.895682481265\t7.895683520871\t0.000013166775%\t✔\n1.0  0.1  0.1   5  12.337002965030\t12.337005501362\t0.000020558729%\t✔\n1.0  0.1  0.1   6  17.765282661715\t17.765287921961\t0.000029609687%\t✔\n1.0  0.1  0.1   7  24.180521036064\t24.180530782669\t0.000040307654%\t✔\n1.0  0.1  0.1   8  31.582717460023\t31.582734083486\t0.000052634655%\t✔\n1.0  0.1  0.1   9  39.971871195191\t39.971897824412\t0.000066619856%\t✔\n1.0  0.1  0.1  10  49.347981417827\t49.348022005447\t0.000082247714%\t✔\n1.0  0.1  1.0   1  49.348021594816\t49.348022005447\t0.000000832112%\t✔\n1.0  0.1  1.0   2  197.392081541864\t197.392088021787\t0.000003282767%\t✔\n1.0  0.1  1.0   3  444.132165194438\t444.132198049021\t0.000007397478%\t✔\n1.0  0.1  1.0   4  789.568248126463\t789.568352087149\t0.000013166775%\t✔\n1.0  0.1  1.0   5  1233.700296503016\t1233.700550136170\t0.000020558729%\t✔\n1.0  0.1  1.0   6  1776.528266171473\t1776.528792196084\t0.000029609687%\t✔\n1.0  0.1  1.0   7  2418.052103606433\t2418.053078266892\t0.000040307654%\t✔\n1.0  0.1  1.0   8  3158.271746002275\t3158.273408348594\t0.000052634655%\t✔\n1.0  0.1  1.0   9  3997.187119519121\t3997.189782441190\t0.000066619856%\t✔\n1.0  0.1  1.0  10  4934.798141782662\t4934.802200544679\t0.000082247714%\t✔\n1.0  1.0  0.1   1  0.049348021595\t0.049348022005\t0.000000832112%\t✔\n1.0  1.0  0.1   2  0.197392081542\t0.197392088022\t0.000003282767%\t✔\n1.0  1.0  0.1   3  0.444132165194\t0.444132198049\t0.000007397478%\t✔\n1.0  1.0  0.1   4  0.789568248126\t0.789568352087\t0.000013166775%\t✔\n1.0  1.0  0.1   5  1.233700296503\t1.233700550136\t0.000020558729%\t✔\n1.0  1.0  0.1   6  1.776528266171\t1.776528792196\t0.000029609687%\t✔\n1.0  1.0  0.1   7  2.418052103606\t2.418053078267\t0.000040307654%\t✔\n1.0  1.0  0.1   8  3.158271746002\t3.158273408349\t0.000052634655%\t✔\n1.0  1.0  0.1   9  3.997187119519\t3.997189782441\t0.000066619856%\t✔\n1.0  1.0  0.1  10  4.934798141783\t4.934802200545\t0.000082247714%\t✔\n1.0  1.0  1.0   1  4.934802159482\t4.934802200545\t0.000000832112%\t✔\n1.0  1.0  1.0   2  19.739208154186\t19.739208802179\t0.000003282767%\t✔\n1.0  1.0  1.0   3  44.413216519444\t44.413219804902\t0.000007397478%\t✔\n1.0  1.0  1.0   4  78.956824812646\t78.956835208715\t0.000013166775%\t✔\n1.0  1.0  1.0   5  123.370029650302\t123.370055013617\t0.000020558729%\t✔\n1.0  1.0  1.0   6  177.652826617147\t177.652879219608\t0.000029609687%\t✔\n1.0  1.0  1.0   7  241.805210360643\t241.805307826689\t0.000040307654%\t✔\n1.0  1.0  1.0   8  315.827174600228\t315.827340834859\t0.000052634655%\t✔\n1.0  1.0  1.0   9  399.718711951912\t399.718978244119\t0.000066619856%\t✔\n1.0  1.0  1.0  10  493.479814178266\t493.480220054468\t0.000082247714%\t✔\nTest Summary:               | Pass  Total  Time\n<ψₙ|H|ψₙ>  = ∫ψₙ*Tψₙdx = Eₙ |   80     80  0.9s","category":"page"},{"location":"InfinitePotentialWell/#Expected-Value-of-x","page":"Infinite Potential Well","title":"Expected Value of x","text":"","category":"section"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"langle x rangle_n=1\n= int_0^L psi_1^ast(x) hatx psi_1(x) mathrmdx\n= frac2(2a)^2pi^3 left( fracpi^36 - fracpi4 right)","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"for only n=1.","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"Reference:","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"LibreTexts PHYSICS, 6.4: Expectation Values, Observables, and Uncertainty","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"   L\t  n\tnumerical         \tanalytical        \t|error|\n0.1\t  1\t0.0500000000000000\t0.0500000000000000\t0.0000000000000278%\t✔\n0.5\t  1\t0.2500000000000000\t0.2500000000000000\t0.0000000000000000%\t✔\n1.0\t  1\t0.5000000000000002\t0.5000000000000000\t0.0000000000000444%\t✔\n7.0\t  1\t3.5000000000000000\t3.5000000000000000\t0.0000000000000000%\t✔\nTest Summary:    | Pass  Total  Time\n<ψₙ|x|ψₙ>  = L/2 |    4      4  0.5s","category":"page"},{"location":"InfinitePotentialWell/#Expected-Value-of-x2","page":"Infinite Potential Well","title":"Expected Value of x^2","text":"","category":"section"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"langle x^2 rangle_n=1\n= int_0^L psi_1^ast(x) hatx^2 psi_1(x) mathrmdx\n= frac2(2a)^2pi^3 left( fracpi^36 - fracpi4 right)","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"Reference:","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"LibreTexts PHYSICS, 6.4: Expectation Values, Observables, and Uncertainty","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"   L\t  n\tnumerical         \tanalytical        \t|error|\n0.1\t  1\t0.0028267274151216\t0.0028267274151216\t0.0000000000000460%\t✔\n0.5\t  1\t0.0706681853780411\t0.0706681853780411\t0.0000000000000000%\t✔\n1.0\t  1\t0.2826727415121645\t0.2826727415121645\t0.0000000000000196%\t✔\n7.0\t  1\t13.8509643340960604\t13.8509643340960569\t0.0000000000000256%\t✔\nTest Summary:                 | Pass  Total  Time\n<ψₙ|x²|ψₙ> = 2L²/π³(π³/6-π/4) |    4      4  0.5s","category":"page"},{"location":"InfinitePotentialWell/#Expected-Value-of-p","page":"Infinite Potential Well","title":"Expected Value of p","text":"","category":"section"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"langle p rangle_n=1\n= int_0^L psi_1^ast(x) hatp psi_1(x) mathrmdx\n= 0","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"Reference:","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"LibreTexts PHYSICS, 6.4: Expectation Values, Observables, and Uncertainty","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"  beginaligned\n    langle p rangle_n=1\n    = int_0^L psi^ast_n(x) hatp psi_n(x) mathrmdx \n    = int_0^L psi^ast_n(x) left -ihbarfracmathrmdmathrmd x right psi(x) mathrmdx \n    simeq int_0^L psi^ast_n(x) left -ihbar fracpsi(x+Delta x) - psi(x-Delta x)2Delta x right mathrmdx\n  endaligned","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"Where, the difference formula for the 2nd-order derivative:","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"beginaligned\n   2fracmathrmd psi(x)mathrmdx Delta x\n   + Oleft(Delta x^3right)\n   = \n   psi(x+Delta x)\n   - psi(x-Delta x)\n   \n   2fracmathrmd psi(x)mathrmdx Delta x\n   = \n   psi(x+Delta x)\n   - psi(x-Delta x)\n   - Oleft(Delta x^3right)\n   \n   fracmathrmd psi(x)mathrmdx\n   = \n   fracpsi(x+Delta x)- psi(x-Delta x)2Delta x\n   - fracOleft(Delta x^3right)2Delta x\n   \n  fracmathrmd psi(x)mathrmdx\n  = \n  fracpsi(x+Delta x)- psi(x-Delta x)2Delta x\n  + Oleft(Delta x^2right)\nendaligned","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"are given by the sum of 2 Taylor series:","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"beginaligned\n  psi(x+Delta x)\n  =\n  psi(x)\n  + fracmathrmd psi(x)mathrmdx Delta x\n  + frac12 fracmathrmd^2 psi(x)mathrmdx^2 Delta x^2\n  + Oleft(Delta x^3right)\n  \n  psi(x-Delta x)\n  =\n  psi(x)\n  - fracmathrmd psi(x)mathrmdx Delta x\n  + frac12 fracmathrmd^2 psi(x)mathrmdx^2 Delta x^2\n  + Oleft(Delta x^3right)\nendaligned","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"   L\t  n\tnumerical         \tanalytical        \t|error|\n0.1\t  1\t0.0000000000033684\t0.0000000000000000\t0.0000000000000000%\t✔\n0.5\t  1\t0.0000000000002308\t0.0000000000000000\t0.0000000000000000%\t✔\n1.0\t  1\t0.0000000000001066\t0.0000000000000000\t0.0000000000000000%\t✔\n7.0\t  1\t0.0000000000000252\t0.0000000000000000\t0.0000000000000000%\t✔\nTest Summary:                      | Pass  Total  Time\n<ψₙ|p|ψₙ>  = ∫ψₙ*(-iℏd/dx)ψₙdx = 0 |    4      4  0.7s","category":"page"},{"location":"InfinitePotentialWell/#Expected-Value-of-p2","page":"Infinite Potential Well","title":"Expected Value of p^2","text":"","category":"section"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"langle p^2 rangle\n= int_0^L psi_1^ast(x) hatp^2 psi_1(x) mathrmdx\n= fracpi^2hbar^2L^2","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"Reference:","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"LibreTexts PHYSICS, 6.4: Expectation Values, Observables, and Uncertainty","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"  beginaligned\n    langle p^2 rangle\n    = int_0^L psi^ast_n(x) hatp psi_n(x) mathrmdx \n    = int_0^L psi^ast_n(x) left -hbar^2fracmathrmd^2mathrmdx^2 right psi(x) mathrmdx \n    simeq int_0^L psi^ast_n(x) left -hbar^2 fracpsi(x+Delta x) - 2psi(x) + psi(x-Delta x)Delta x^2 right mathrmdx\n  endaligned","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"Where, the difference formula for the 2nd-order derivative:","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"beginaligned\n   2psi(x)\n   + fracmathrmd^2 psi(x)mathrmd x^2 Delta x^2\n   + Oleft(Delta x^4right)\n   =\n   psi(x+Delta x)\n   + psi(x-Delta x)\n   \n   fracmathrmd^2 psi(x)mathrmd x^2 Delta x^2\n   =\n   psi(x+Delta x)\n   - 2psi(x)\n   + psi(x-Delta x)\n   - Oleft(Delta x^4right)\n   \n   fracmathrmd^2 psi(x)mathrmd x^2\n   =\n   fracpsi(x+Delta x) - 2psi(x) + psi(x-Delta x)Delta x^2\n   - fracOleft(Delta x^4right)Delta x^2\n   \n  fracmathrmd^2 psi(x)mathrmd x^2\n  =\n  fracpsi(x+Delta x) - 2psi(x) + psi(x-Delta x)Delta x^2\n  + Oleft(Delta x^2right)\nendaligned","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"are given by the sum of 2 Taylor series:","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"beginaligned\npsi(x+Delta x)\n= psi(x)\n+ fracmathrmd psi(x)mathrmd x Delta x\n+ frac12 fracmathrmd^2 psi(x)mathrmd x^2 Delta x^2\n+ frac13 fracmathrmd^3 psi(x)mathrmd x^3 Delta x^3\n+ Oleft(Delta x^4right)\n\npsi(x-Delta x)\n= psi(x)\n- fracmathrmd psi(x)mathrmd x Delta x\n+ frac12 fracmathrmd^2 psi(x)mathrmd x^2 Delta x^2\n- frac13 fracmathrmd^3 psi(x)mathrmd x^3 Delta x^3\n+ Oleft(Delta x^4right)\nendaligned","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"   L\t  n\tnumerical         \tanalytical        \t|error|\n0.1\t  1\t986.9604315827808705\t986.9604401089355861\t0.0000008638800877%\t✔\n0.5\t  1\t39.4784172741947543\t39.4784176043574320\t0.0000008363118323%\t✔\n1.0\t  1\t9.8696043189632228\t9.8696044010893580\t0.0000008321117229%\t✔\n7.0\t  1\t0.2014204963826796\t0.2014204979814155\t0.0000007937304979%\t✔\nTest Summary:                              | Pass  Total  Time\n<ψₙ|p²|ψₙ> = ∫ψₙ*(-ℏ²d²/dx²)ψₙdx = π²ℏ²/L² |    4      4  0.5s\n","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"CurrentModule = Antiq","category":"page"},{"location":"MorsePotential/#Morse-Potential","page":"Morse Potential","title":"Morse Potential","text":"","category":"section"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"The Morse potential is a model for inter-nuclear anharmonic vibration in a diatomic molecule.","category":"page"},{"location":"MorsePotential/#Definitions","page":"Morse Potential","title":"Definitions","text":"","category":"section"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"xi = 2lambdamathrme^-a(r-r_e), omega = sqrtkµ, k = 2D_mathrmea^2, lambda = fracsqrt2mD_mathrmeahbar, chi = frachbaromega4D_mathrme, N_n = sqrtfracn(2lambda-2n-1)aGamma(2lambda-n), L_n^(alpha)(x) = fracx^-alpha mathrme^xn  fracmathrmd^nmathrmd x^nleft(mathrme^-x x^n+alpharight) are used.","category":"page"},{"location":"MorsePotential/#Schrödinger-Equation","page":"Morse Potential","title":"Schrödinger Equation","text":"","category":"section"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"  hatHpsi(r) = E psi(r)","category":"page"},{"location":"MorsePotential/#Hamiltonian","page":"Morse Potential","title":"Hamiltonian","text":"","category":"section"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"  hatH = - frachbar^22mu fracmathrmd^2mathrmdr ^2 + V(r)","category":"page"},{"location":"MorsePotential/#Potential","page":"Morse Potential","title":"Potential","text":"","category":"section"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"V(r; rₑ=rₑ, Dₑ=Dₑ, k=k, a=sqrt(k/(2*Dₑ)))","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"  V(r) = D_mathrme left( mathrme^-2a(r-r_e) - 2mathrme^-a(r-r_e) right)","category":"page"},{"location":"MorsePotential/#Eigen-Values","page":"Morse Potential","title":"Eigen Values","text":"","category":"section"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"E(n; rₑ=rₑ, Dₑ=Dₑ, k=k, a=sqrt(k/(2*Dₑ)), µ=µ, ω=sqrt(k/µ), χ=ℏ*ω/(4*Dₑ), ℏ=ℏ)","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"  E_n = - D_mathrme + hbar omega left( n + frac12 right) - chi hbar omega left( n + frac12 right)^2","category":"page"},{"location":"MorsePotential/#Eigen-Functions","page":"Morse Potential","title":"Eigen Functions","text":"","category":"section"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"ψ(n, r; rₑ=rₑ, Dₑ=Dₑ, k=k, a=sqrt(k/(2*Dₑ)), µ=µ, ω=sqrt(k/µ), χ=ℏ*ω/(4*Dₑ), ℏ=ℏ)","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"  psi_n(r) = N_n z^lambda-n-12 mathrme^-z2 L_n^(2lambda-2n-1)(xi)","category":"page"},{"location":"MorsePotential/#Generalized-Laguerre-Polynomials","page":"Morse Potential","title":"Generalized Laguerre Polynomials","text":"","category":"section"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"L(x; n=0, α=0)","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"Rodrigues' formula & closed-form:","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"  beginaligned\n    L_n^(alpha)(x)\n    = fracx^-alphae^xn fracd^ndx^nleft(x^n+alphae^-xright) \n    = sum_k=0^n(-1)^k left(beginarrayl n+alpha  n-k endarrayright) fracx^kk  \n    = sum_k=0^n(-1)^k fracGamma(alpha+n+1)Gamma(alpha+k+1)Gamma(n-k+1) fracx^kk \n  endaligned","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"Examples:","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"  beginaligned\n    L_0^(0)(x) = 1 \n    L_1^(0)(x) = 1 - x \n    L_1^(1)(x) = 2 - x \n    L_2^(0)(x) = 1 - 2 x + 12 x^2 \n    L_2^(1)(x) = 3 - 3 x + 12 x^2 \n    L_2^(2)(x) = 6 - 4 x + 12 x^2 \n    L_3^(0)(x) = 1 - 3 x + 32 x^2 - 16 x^3 \n    L_3^(1)(x) = 4 - 6 x + 2 x^2 - 16 x^3 \n    L_3^(2)(x) = 10 - 10 x + 52 x^2 - 16 x^3 \n    L_3^(3)(x) = 20 - 15 x + 3 x^2 - 16 x^3 \n    L_4^(0)(x) = 1 - 4 x + 3 x^2 - 23 x^3 + 124 x^4 \n    L_4^(1)(x) = 5 - 10 x + 5 x^2 - 56 x^3 + 124 x^4 \n    L_4^(2)(x) = 15 - 20 x + 152 x^2 - 1 x^3 + 124 x^4 \n    L_4^(3)(x) = 35 - 35 x + 212 x^2 - 76 x^3 + 124 x^4 \n    L_4^(4)(x) = 70 - 56 x + 14 x^2 - 43 x^3 + 124 x^4 \n    vdots\n  endaligned","category":"page"},{"location":"MorsePotential/#References","page":"Morse Potential","title":"References","text":"","category":"section"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"P. M. Morse, Phys. Rev. 34, 57 (1929)\nJ. P. Dahl, M. Springborg, J. Chem. Phys. 88, 4535 (1988). (62), (63)\nW. K. Shao, Y. He, J. Pan, J. Nonlinear Sci. Appl., 9, 5, 3388 (2016). (1.6) \nThe Digital Library of Mathematical Functions (DLMF) 18.3 Table1, 18.5 Table1, 18.5.12","category":"page"},{"location":"MorsePotential/#Usage-and-Examples","page":"Morse Potential","title":"Usage & Examples","text":"","category":"section"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"Install Antiq.jl for the first run and run using Antiq before each use. The function antiq(model, parameters...) returns a module that has E(), ψ(r), V(r) and some other functions. In this system, the model name is specified by :MorsePotential and several parameters rₑ, Dₑ, k, µ and ℏ are set as optional arguments.","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"using Antiq\nMP = antiq(:MorsePotential)","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"Parameters (for H₂⁺):","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"julia> MP.rₑ\n1.997193319969992\n\njulia> MP.Dₑ\n0.10263461910653993\n\njulia> MP.k\n0.1027265041900817\n\njulia> MP.µ\n918.076336715\n\njulia> MP.ℏ\n1.0","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"Eigen values:","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"julia> MP.E(n=0)\n-0.09741377794418261\n\njulia> MP.E(n=1)\n-0.08738092406760907","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"Potential energy curve:","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"using Plots\nplot(0.1:0.01:15, r -> MP.V(r), lw=2, label=\"\", xlims=(0.1,9.1), ylims=(-0.11,0.01), xlabel=\"r\", ylabel=\"V(r)\")","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"(Image: )","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"Wave functions:","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"using Plots\nplot(xlim=(0,5), xlabel=\"x\", ylabel=\"ψ(x)\")\nplot!(x -> MP.ψ(x, n=0), label=\"n=0\", lw=2)\nplot!(x -> MP.ψ(x, n=1), label=\"n=1\", lw=2)\nplot!(x -> MP.ψ(x, n=2), label=\"n=2\", lw=2)\nplot!(x -> MP.ψ(x, n=3), label=\"n=3\", lw=2)\nplot!(x -> MP.ψ(x, n=4), label=\"n=4\", lw=2)\nplot!(x -> MP.ψ(x, n=5), label=\"n=5\", lw=2)","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"(Image: )","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"Potential energy curve, Energy levels, Comparison with harmonic oscillator:","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"MP = antiq(:MorsePotential)\nHO = antiq(:HarmonicOscillator, k=MP.k, m=MP.μ)\nusing Plots\nplot(xlims=(0.1,9.1), ylims=(-0.11,0.01), xlabel=\"\\$r\\$\", ylabel=\"\\$V(r), E_n\\$\", legend=:bottomright, size=(480,400), dpi=300)\nfor n in 0:MP.nₘₐₓ()\n  # energy\n  EM = MP.E(n=n)\n  EH = HO.E(n=n) - MP.Dₑ\n  plot!(0.1:0.01:15, r -> EH > HO.V(r-MP.rₑ) - MP.Dₑ ? EH : NaN, lc=\"#BC1C5F\", lw=1, label=\"\")\n  plot!(0.1:0.01:15, r -> EM > MP.V(r) ? EM : NaN, lc=\"#578FC7\", lw=1, label=\"\")\nend\n# potential\nplot!(0.1:0.01:15, r -> HO.V(r-MP.rₑ) - MP.Dₑ, lc=\"#BC1C5F\", lw=2, label=\"Harmonic Oscillator\")\nplot!(0.1:0.01:15, r -> MP.V(r), lc=\"#578FC7\", lw=2, label=\"Morse Potential\")","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"(Image: )","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"where, the potential of harmonic oscillator is defined as V(r) simeq frac12 k (r - r_mathrme)^2 + V_0.","category":"page"},{"location":"MorsePotential/#Testing","page":"Morse Potential","title":"Testing","text":"","category":"section"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"Unit testing and Integration testing were done using computer algebra system (Symbolics.jl) and numerical integration (QuadGK.jl). The test script is here.","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"include(\"../../../test/MorsePotential.jl\")","category":"page"},{"location":"MorsePotential/#Generalized-Laguerre-Polynomials-L_n{(\\alpha)}(x)","page":"Morse Potential","title":"Generalized Laguerre Polynomials L_n^(alpha)(x)","text":"","category":"section"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"  beginaligned\n    L_n^(alpha)(x)\n    = fracx^-alphae^xn fracd^ndx^nleft(x^n+alphae^-xright) \n    = sum_k=0^n(-1)^k fracGamma(alpha+n+1)Gamma(alpha+k+1)Gamma(n-k+1) fracx^kk \n  endaligned","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"n=0 α=0 ✔","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"beginaligned\n  L_0^(0)(x)\n   = e^x e^ - x\n  = 1 \n  = 1\nendaligned","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"n=1 α=0 ✔","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"beginaligned\n  L_1^(0)(x)\n   = e^x fracmathrmdmathrmdx x e^ - x\n  = 1 - x \n  = 1 - x\nendaligned","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"n=1 α=1 ✔","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"beginaligned\n  L_1^(1)(x)\n   = frace^x fracmathrmdmathrmdx x^2 e^ - xx\n  = 2 - x \n  = 2 - x\nendaligned","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"n=2 α=0 ✔","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"beginaligned\n  L_2^(0)(x)\n   = frac12 e^x fracmathrmdmathrmdx fracmathrmdmathrmdx x^2 e^ - x\n  = 1 - 2 x + frac12 x^2 \n  = 1 - 2 x + frac12 x^2\nendaligned","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"n=2 α=1 ✔","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"beginaligned\n  L_2^(1)(x)\n   = fracfrac12 e^x fracmathrmdmathrmdx fracmathrmdmathrmdx x^3 e^ - xx\n  = 3 - 3 x + frac12 x^2 \n  = 3 - 3 x + frac12 x^2\nendaligned","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"n=2 α=2 ✔","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"beginaligned\n  L_2^(2)(x)\n   = fracfrac12 e^x fracmathrmdmathrmdx fracmathrmdmathrmdx x^4 e^ - xx^2\n  = 6 - 4 x + frac12 x^2 \n  = 6 - 4 x + frac12 x^2\nendaligned","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"n=3 α=0 ✔","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"beginaligned\n  L_3^(0)(x)\n   = frac16 e^x fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx x^3 e^ - x\n  = 1 - frac16 x^3 - 3 x + frac32 x^2 \n  = 1 - frac16 x^3 - 3 x + frac32 x^2\nendaligned","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"n=3 α=1 ✗ Lₙ⁽ᵅ⁾(x) = x⁻ᵅeˣ/n! dⁿ/dxⁿ xⁿ⁺ᵅe⁻ˣ: Test Failed at C:\\Users\\user\\Desktop\\GitHub\\Antiq.jl\\test\\MorsePotential.jl:44   Expression: acceptance","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"Stacktrace:  [1] macro expansion    @ C:\\Users\\user.julia\\juliaup\\julia-1.9.2+0.x64.w64.mingw32\\share\\julia\\stdlib\\v1.9\\Test\\src\\Test.jl:478 [inlined]  [2] macro expansion    @ C:\\Users\\user\\Desktop\\GitHub\\Antiq.jl\\test\\MorsePotential.jl:44 [inlined]  [3] macro expansion    @ C:\\Users\\user.julia\\juliaup\\julia-1.9.2+0.x64.w64.mingw32\\share\\julia\\stdlib\\v1.9\\Test\\src\\Test.jl:1498 [inlined]  [4] top-level scope    @ C:\\Users\\user\\Desktop\\GitHub\\Antiq.jl\\test\\MorsePotential.jl:28","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"beginaligned\n  L_3^(1)(x)\n   = fracfrac16 e^x fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx x^4 e^ - xx\n  = 4 - 6 x - frac16 x^3 + 2 x^2 \n  = 4 - frac16 x^3 - 6 x + 2 x^2\nendaligned","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"n=3 α=2 ✗ Lₙ⁽ᵅ⁾(x) = x⁻ᵅeˣ/n! dⁿ/dxⁿ xⁿ⁺ᵅe⁻ˣ: Test Failed at C:\\Users\\user\\Desktop\\GitHub\\Antiq.jl\\test\\MorsePotential.jl:44   Expression: acceptance","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"Stacktrace:  [1] macro expansion    @ C:\\Users\\user.julia\\juliaup\\julia-1.9.2+0.x64.w64.mingw32\\share\\julia\\stdlib\\v1.9\\Test\\src\\Test.jl:478 [inlined]  [2] macro expansion    @ C:\\Users\\user\\Desktop\\GitHub\\Antiq.jl\\test\\MorsePotential.jl:44 [inlined]  [3] macro expansion    @ C:\\Users\\user.julia\\juliaup\\julia-1.9.2+0.x64.w64.mingw32\\share\\julia\\stdlib\\v1.9\\Test\\src\\Test.jl:1498 [inlined]  [4] top-level scope    @ C:\\Users\\user\\Desktop\\GitHub\\Antiq.jl\\test\\MorsePotential.jl:28","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"beginaligned\n  L_3^(2)(x)\n   = fracfrac16 e^x fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx x^5 e^ - xx^2\n  = 10 - 10 x - frac16 x^3 + frac52 x^2 \n  = 10 - frac16 x^3 - 10 x + frac52 x^2\nendaligned","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"n=3 α=3 ✗ Lₙ⁽ᵅ⁾(x) = x⁻ᵅeˣ/n! dⁿ/dxⁿ xⁿ⁺ᵅe⁻ˣ: Test Failed at C:\\Users\\user\\Desktop\\GitHub\\Antiq.jl\\test\\MorsePotential.jl:44   Expression: acceptance","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"Stacktrace:  [1] macro expansion    @ C:\\Users\\user.julia\\juliaup\\julia-1.9.2+0.x64.w64.mingw32\\share\\julia\\stdlib\\v1.9\\Test\\src\\Test.jl:478 [inlined]  [2] macro expansion    @ C:\\Users\\user\\Desktop\\GitHub\\Antiq.jl\\test\\MorsePotential.jl:44 [inlined]  [3] macro expansion    @ C:\\Users\\user.julia\\juliaup\\julia-1.9.2+0.x64.w64.mingw32\\share\\julia\\stdlib\\v1.9\\Test\\src\\Test.jl:1498 [inlined]  [4] top-level scope    @ C:\\Users\\user\\Desktop\\GitHub\\Antiq.jl\\test\\MorsePotential.jl:28","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"beginaligned\n  L_3^(3)(x)\n   = fracfrac16 e^x fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx x^6 e^ - xx^3\n  = 20 - 15 x - frac16 x^3 + 3 x^2 \n  = 20 - frac16 x^3 - 15 x + 3 x^2\nendaligned","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"n=4 α=0 ✔","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"beginaligned\n  L_4^(0)(x)\n   = frac124 e^x fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx x^4 e^ - x\n  = 1 - frac23 x^3 - 4 x + 3 x^2 + frac124 x^4 \n  = 1 - frac23 x^3 - 4 x + 3 x^2 + frac124 x^4\nendaligned","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"n=4 α=1 ✗ Lₙ⁽ᵅ⁾(x) = x⁻ᵅeˣ/n! dⁿ/dxⁿ xⁿ⁺ᵅe⁻ˣ: Test Failed at C:\\Users\\user\\Desktop\\GitHub\\Antiq.jl\\test\\MorsePotential.jl:44   Expression: acceptance","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"Stacktrace:  [1] macro expansion    @ C:\\Users\\user.julia\\juliaup\\julia-1.9.2+0.x64.w64.mingw32\\share\\julia\\stdlib\\v1.9\\Test\\src\\Test.jl:478 [inlined]  [2] macro expansion    @ C:\\Users\\user\\Desktop\\GitHub\\Antiq.jl\\test\\MorsePotential.jl:44 [inlined]  [3] macro expansion    @ C:\\Users\\user.julia\\juliaup\\julia-1.9.2+0.x64.w64.mingw32\\share\\julia\\stdlib\\v1.9\\Test\\src\\Test.jl:1498 [inlined]  [4] top-level scope    @ C:\\Users\\user\\Desktop\\GitHub\\Antiq.jl\\test\\MorsePotential.jl:28","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"beginaligned\n  L_4^(1)(x)\n   = fracfrac124 e^x fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx x^5 e^ - xx\n  = 5 - 10 x - frac56 x^3 + 5 x^2 + frac124 x^4 \n  = 5 - frac56 x^3 - 10 x + 5 x^2 + frac124 x^4\nendaligned","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"n=4 α=2 ✗ Lₙ⁽ᵅ⁾(x) = x⁻ᵅeˣ/n! dⁿ/dxⁿ xⁿ⁺ᵅe⁻ˣ: Test Failed at C:\\Users\\user\\Desktop\\GitHub\\Antiq.jl\\test\\MorsePotential.jl:44   Expression: acceptance","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"Stacktrace:  [1] macro expansion    @ C:\\Users\\user.julia\\juliaup\\julia-1.9.2+0.x64.w64.mingw32\\share\\julia\\stdlib\\v1.9\\Test\\src\\Test.jl:478 [inlined]  [2] macro expansion    @ C:\\Users\\user\\Desktop\\GitHub\\Antiq.jl\\test\\MorsePotential.jl:44 [inlined]  [3] macro expansion    @ C:\\Users\\user.julia\\juliaup\\julia-1.9.2+0.x64.w64.mingw32\\share\\julia\\stdlib\\v1.9\\Test\\src\\Test.jl:1498 [inlined]  [4] top-level scope    @ C:\\Users\\user\\Desktop\\GitHub\\Antiq.jl\\test\\MorsePotential.jl:28","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"beginaligned\n  L_4^(2)(x)\n   = fracfrac124 e^x fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx x^6 e^ - xx^2\n  = 15 - 20 x - x^3 + frac152 x^2 + frac124 x^4 \n  = 15 - x^3 - 20 x + frac152 x^2 + frac124 x^4\nendaligned","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"n=4 α=3 ✗ Lₙ⁽ᵅ⁾(x) = x⁻ᵅeˣ/n! dⁿ/dxⁿ xⁿ⁺ᵅe⁻ˣ: Test Failed at C:\\Users\\user\\Desktop\\GitHub\\Antiq.jl\\test\\MorsePotential.jl:44   Expression: acceptance","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"Stacktrace:  [1] macro expansion    @ C:\\Users\\user.julia\\juliaup\\julia-1.9.2+0.x64.w64.mingw32\\share\\julia\\stdlib\\v1.9\\Test\\src\\Test.jl:478 [inlined]  [2] macro expansion    @ C:\\Users\\user\\Desktop\\GitHub\\Antiq.jl\\test\\MorsePotential.jl:44 [inlined]  [3] macro expansion    @ C:\\Users\\user.julia\\juliaup\\julia-1.9.2+0.x64.w64.mingw32\\share\\julia\\stdlib\\v1.9\\Test\\src\\Test.jl:1498 [inlined]  [4] top-level scope    @ C:\\Users\\user\\Desktop\\GitHub\\Antiq.jl\\test\\MorsePotential.jl:28","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"beginaligned\n  L_4^(3)(x)\n   = fracfrac124 e^x fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx x^7 e^ - xx^3\n  = 35 - 35 x - frac76 x^3 + frac212 x^2 + frac124 x^4 \n  = 35 - frac76 x^3 - 35 x + frac212 x^2 + frac124 x^4\nendaligned","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"n=4 α=4 ✗ Lₙ⁽ᵅ⁾(x) = x⁻ᵅeˣ/n! dⁿ/dxⁿ xⁿ⁺ᵅe⁻ˣ: Test Failed at C:\\Users\\user\\Desktop\\GitHub\\Antiq.jl\\test\\MorsePotential.jl:44   Expression: acceptance","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"Stacktrace:  [1] macro expansion    @ C:\\Users\\user.julia\\juliaup\\julia-1.9.2+0.x64.w64.mingw32\\share\\julia\\stdlib\\v1.9\\Test\\src\\Test.jl:478 [inlined]  [2] macro expansion    @ C:\\Users\\user\\Desktop\\GitHub\\Antiq.jl\\test\\MorsePotential.jl:44 [inlined]  [3] macro expansion    @ C:\\Users\\user.julia\\juliaup\\julia-1.9.2+0.x64.w64.mingw32\\share\\julia\\stdlib\\v1.9\\Test\\src\\Test.jl:1498 [inlined]  [4] top-level scope    @ C:\\Users\\user\\Desktop\\GitHub\\Antiq.jl\\test\\MorsePotential.jl:28","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"beginaligned\n  L_4^(4)(x)\n   = fracfrac124 e^x fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx x^8 e^ - xx^4\n  = 70 - 56 x - frac43 x^3 + 14 x^2 + frac124 x^4 \n  = 70 - frac43 x^3 - 56 x + 14 x^2 + frac124 x^4\nendaligned","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"Test Summary:                      | Pass  Fail  Total   Time\nLₙ⁽ᵅ⁾(x) = x⁻ᵅeˣ/n! dⁿ/dxⁿ xⁿ⁺ᵅe⁻ˣ |    8     7     15  15.8s\nError: LoadError: Some tests did not pass: 8 passed, 7 failed, 0 errored, 0 broken.\nin expression starting at C:\\Users\\user\\Desktop\\GitHub\\Antiq.jl\\test\\MorsePotential.jl:27","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"CurrentModule = Antiq","category":"page"},{"location":"HydrogenAtom/#Hydrogen-Atom","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"","category":"section"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"The hydrogen atom is the simplest 2-body Coulomb system.","category":"page"},{"location":"HydrogenAtom/#Definitions","page":"Hydrogen Atom","title":"Definitions","text":"","category":"section"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"Z is the atomic number.","category":"page"},{"location":"HydrogenAtom/#Schrödinger-Equation","page":"Hydrogen Atom","title":"Schrödinger Equation","text":"","category":"section"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"  hatH psi(pmbr) = E psi(pmbr)","category":"page"},{"location":"HydrogenAtom/#Hamiltonian","page":"Hydrogen Atom","title":"Hamiltonian","text":"","category":"section"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"  hatH = - frachbar^22mu fracmathrmd^2mathrmdr ^2 + V(r)","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"Where, mu=left(frac1m_mathrme+frac1m_mathrmpright)^-1 is the reduced mass of electron mathrme and proton mathrmp. mu = m_mathrme holds in the limit m_mathrmprightarrowinfty. ","category":"page"},{"location":"HydrogenAtom/#Potential","page":"Hydrogen Atom","title":"Potential","text":"","category":"section"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"V(r; Z=Z, a₀=a₀)","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"  beginaligned\n  V(r)\n  = - fracZe^24pivarepsilon_0 r \n  = - frace^24pivarepsilon_0 a_0 fracZra_0\n  = - fracZra_0 E_mathrmh\n  endaligned","category":"page"},{"location":"HydrogenAtom/#Eigen-Values","page":"Hydrogen Atom","title":"Eigen Values","text":"","category":"section"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"E(; n=1, Z=Z, Eₕ=Eₕ)","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"  E_n = -fracm_mathrme e^4 Z^22n^2(4pivarepsilon_0)^2hbar^2 = -fracZ^22n^2 E_mathrmh","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"Where, E_mathrmh is the Hartree energy, one of atomic unit. About atomic units, see section 3.9.2 of the IUPAC GreenBook. In other units, E_mathrmh = 27211386245988(53)mathrmeV from here.","category":"page"},{"location":"HydrogenAtom/#Eigen-Functions","page":"Hydrogen Atom","title":"Eigen Functions","text":"","category":"section"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"ψ(r, θ, φ; n=1, l=0, m=0, Z=Z, a₀=a₀)","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"  psi_nlm(pmbr) = R_nl(r) Y_lm(thetavarphi)","category":"page"},{"location":"HydrogenAtom/#Radial-Functions","page":"Hydrogen Atom","title":"Radial Functions","text":"","category":"section"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"R(r; n=1, l=0, Z=Z, a₀=a₀)","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"    R_nl(r) = -sqrtfrac(n-l-1)2n(n+l) left(frac2Zn a_0right)^3 left(frac2Zrn a_0right)^l exp left(-fracZrn a_0right) L_n+l^2l+1 left(frac2Zrn a_0right)","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"Where, Laguerre polynomials are defined as L_n(x) = frac1n mathrme^x fracmathrmd^nmathrmdx ^n left( mathrme^-x x^n right), and associated Laguerre polynomials are defined as L_n^k(x) = fracmathrmd^kmathrmdx^k L_n(x). Note that, replace 2n(n+l) with 2n(n+l)^3 if Laguerre polynomials are defined as L_n(x) = mathrme^x fracmathrmd^nmathrmdx ^n left( mathrme^-x x^n right).","category":"page"},{"location":"HydrogenAtom/#Associated-Laguerre-Polynomials","page":"Hydrogen Atom","title":"Associated Laguerre Polynomials","text":"","category":"section"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"L(x; n=0, k=0)","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"Rodrigues' formula & closed-form:","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"  beginaligned\n  L_n^k(x)\n    = fracmathrmd^kmathrmdx^k L_n(x) \n    = fracmathrmd^kmathrmdx^k frac1n mathrme^x fracmathrmd^nmathrmdx ^n left( mathrme^-x x^n right) \n    = sum_m=0^n-k (-1)^m+k fracnm(m+k)(n-m-k) x^m \n    = (-1)^k L_n-k^(k)(x)\n  endaligned","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"Where, Laguerre polynomials are defined as L_n(x)=frac1nmathrme^x fracmathrmd^nmathrmdx ^n left( mathrme^-x x^n right).","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"Examples:","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"  beginaligned\n    L_0^0(x) = 1 \n    L_1^0(x) = 1 - x \n    L_1^1(x) = 1 \n    L_2^0(x) = 1 - 2 x + 12 x^2 \n    L_2^1(x) = 2 - x \n    L_2^2(x) = 1 \n    L_3^0(x) = 1 - 3 x + 32 x^2 - 16 x^3 \n    L_3^1(x) = 3 - 3 x + 12 x^2 \n    L_3^2(x) = 3 - x \n    L_3^3(x) = 1 \n    L_4^0(x) = 1 - 4 x + 3 x^2 - 23 x^3 + 512 x^4 \n    L_4^1(x) = 4 - 6 x + 2 x^2 - 16 x^3 \n    L_4^2(x) = 6 - 4 x + 12 x^2 \n    L_4^3(x) = 4 - x \n    L_4^4(x) = 1 \n    vdots\n  endaligned","category":"page"},{"location":"HydrogenAtom/#Spherical-Harmonics","page":"Hydrogen Atom","title":"Spherical Harmonics","text":"","category":"section"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"Y(θ, φ; l=0, m=0)","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"    Y_lm(thetavarphi) = (-1)^fracm+m2 sqrtfrac2l+14pi frac(l-m)(l+m) P_l^m (costheta) mathrme^imvarphi","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"Note that some variants are connected by ","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"i^m+m sqrtfrac(l-m)(l+m) P_l^m = (-1)^fracm+m2 sqrtfrac(l-m)(l+m) P_l^m = (-1)^m sqrtfrac(l-m)(l+m) P_l^m","category":"page"},{"location":"HydrogenAtom/#Associated-Legendre-Polynomials","page":"Hydrogen Atom","title":"Associated Legendre Polynomials","text":"","category":"section"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"P(x; n=0, m=0)","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"Rodrigues' formula & closed-form:","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"  beginaligned\n    P_n^m(x)\n    = left( 1-x^2 right)^m2 fracmathrmd^mmathrmdx^m P_n(x) \n    = left( 1-x^2 right)^m2 fracmathrmd^mmathrmdx^m frac12^n n fracmathrmd^nmathrmdx ^n left left( x^2-1 right)^n right \n    = frac12^n (1-x^2)^m2 sum_j=0^leftlfloorfracn-m2rightrfloor (-1)^j frac(2n-2j)j (n-j) (n-2j-m) x^(n-2j-m)\n  endaligned","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"Where, Legendre polynomials are defined as P_n(x) = frac12^n n fracmathrmd^nmathrmdx ^n left left( x^2-1 right)^n right. Note that P_l^-m = (-1)^m frac(l-m)(l+m) P_l^m for m0. (It is not compatible with P_k^m(t) = (-1)^mleft( 1-t^2 right)^m2 fracmathrmd^m P_k(t)mathrmdt^m caused by (-1)^m.) The specific formulae are given below.","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"Examples:","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"  beginaligned\n    P_0^0(x) = 1 \n    P_1^0(x) = x \n    P_1^1(x) = left(+1right)sqrt1-x^2 \n    P_2^0(x) = -12 + 32 x^2 \n    P_2^1(x) = left(-3 xright)sqrt1-x^2 \n    P_2^2(x) = 3 - 6 x \n    P_3^0(x) = -32 x + 52 x^3 \n    P_3^1(x) = left(32 - 152 x^2right)sqrt1-x^2 \n    P_3^2(x) = 15 x - 30 x^2 \n    P_3^3(x) = left(15 - 30 xright)sqrt1-x^2 \n    P_4^0(x) = 38 - 154 x^2 + 358 x^4 \n    P_4^1(x) = left(- 152 x + 352 x^3right)sqrt1-x^2 \n    P_4^2(x) = -152 + 15 x + 1052 x^2 - 105 x^3 \n    P_4^3(x) = left(105 x - 210 x^2right)sqrt1-x^2 \n    P_4^4(x) = 105 - 420 x + 420 x^2 \n     vdots\n  endaligned","category":"page"},{"location":"HydrogenAtom/#References","page":"Hydrogen Atom","title":"References","text":"","category":"section"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"cpprefjp, legendre, assoc_legendre, laguerre, assoc_laguerre\nThe Digital Library of Mathematical Functions (DLMF), 18.3 Table1, 18.5 Table1, 18.5.16, 18.3 Table1, 18.5 Table1, 18.5.17, 18.3 Table1, 18.5 Table1, 18.5.12\nL. D. Landau, E. M. Lifshitz, Quantum Mechanics (Pergamon Press, 1965), p.598 (c.1), p.598 (c.4), p.603 (d.13), p.603 (d.13)\nL. I. Schiff, Quantum Mechanics (McGraw-Hill Book Company, 1968), p.79 (14.12), p.93 (16.19)\nA. Messiah, Quanfum Mechanics (Dover Publications, 1999), p.493 (B.72), p.494 Table, p.493 (B.72), p.483 (B.12), p.483 (B.12)\nW. Greiner, Quantum Mechanics: An Introduction Third Edition (Springer, 1994), p.83 (4), p.83 (5), p.149 (21)\nD. J. Griffiths, Introduction to Quantum Mechanics (Prentice Hall, 1995), p.126 (4.28), p.96 Table3.1, p.126 (4.27), p.139 (4.88), p.140 Table4.4, p.139 (4.87), p.140 Table4.5\nD. A. McQuarrie, J. D. Simon, Physical Chemistry: A Molecular Approach (University Science Books, 1997), p.195 Table6.1, p.196 (6.26), p.196 Table6.2, p.207 Table6.4\nP. W. Atkins, J. De Paula, Atkins' Physical Chemistry, 8th edition (W. H. Freeman, 2008), p.234\nJ. J. Sakurai, J. Napolitano, Modern Quantum Mechanics Third Edition (Cambridge University Press, 2021), p.245 Problem 3.30.b, ","category":"page"},{"location":"HydrogenAtom/#Usage-and-Examples","page":"Hydrogen Atom","title":"Usage & Examples","text":"","category":"section"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"Install Antiq.jl for the first run and run using Antiq before each use. The function antiq(model, parameters...) returns a module that has E(), ψ(r), V(r) and some other functions. In this system, the model name is specified by :HydrogenAtom and several parameters Z, Eₕ, mₑ, a₀ and ℏ are set as optional arguments.","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"using Antiq\nH = antiq(:HydrogenAtom, Z=1, Eₕ=1.0, a₀=1.0, mₑ=1.0, ℏ=1.0)","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"Parameters:","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"julia> H.Z\n1\n\njulia> H.Eₕ\n1.0\n\njulia> H.mₑ\n1.0\n\njulia> H.a₀\n1.0\n\njulia> H.ℏ\n1.0","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"Eigen values:","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"julia> H.E(n=1)\n-0.5\n\njulia> H.E(n=2)\n-0.125","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"Wave length (n=2rightarrow1, the first line of the Lyman series):","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"Eₕ2nm⁻¹ = 2.1947463136320e-2 # https://physics.nist.gov/cgi-bin/cuu/CCValue?hrminv\nprintln(\"ΔE = \", H.E(n=2) - H.E(n=1), \" Eₕ\")\nprintln(\"λ  = \", ((H.E(n=2)-H.E(n=1))*Eₕ2nm⁻¹)^-1, \" nm\")","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"ΔE = 0.375 Eₕ\nλ  = 121.50227341098497 nm","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"Hyperfine Splitting:","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"# constants: https://doi.org/10.1103/RevModPhys.93.025010\ne  = 1.602176634e-19    # C      https://physics.nist.gov/cgi-bin/cuu/Value?e\nh  = 6.62607015e-34     # J Hz-1 https://physics.nist.gov/cgi-bin/cuu/Value?h\nc  = 299792458          # m s-1  https://physics.nist.gov/cgi-bin/cuu/Value?c\na0 = 5.29177210903e-11  # m      https://physics.nist.gov/cgi-bin/cuu/Value?bohrrada0\nµ0 = 1.25663706212e-6   # N A-2  https://physics.nist.gov/cgi-bin/cuu/Value?mu0\nµB = 9.2740100783e-24   # J T-1  https://physics.nist.gov/cgi-bin/cuu/Value?mub\nµN = 5.0507837461e-27   # J T-1  https://physics.nist.gov/cgi-bin/cuu/Value?mun\nge = 2.00231930436256   #        https://physics.nist.gov/cgi-bin/cuu/Value?gem\ngp = 5.5856946893       #        https://physics.nist.gov/cgi-bin/cuu/Value?gp\n\n# calculation: https://doi.org/10.1119/1.12733\nδ = abs(H.ψ(0,0,0))^2\nΔE = 2 / 3 * µ0 * µN * µB * gp * ge * δ * a0^(-3)\nprintln(\"1/π    = \", 1/π)\nprintln(\"<δ(r)> = \", δ, \" a₀⁻³\")\nprintln(\"<δ(r)> = \", δ * a0^(-3), \" m⁻³\")\nprintln(\"ΔE = \", ΔE, \" J\")\nprintln(\"ν = ΔE/h = \", ΔE / h * 1e-6, \" MHz\")\nprintln(\"λ = hc/ΔE = \", h*c/ΔE*100, \" cm\")","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"1/π    = 0.3183098861837907\n<δ(r)> = 0.3183098861837908 a₀⁻³\n<δ(r)> = 2.1480615849063944e30 m⁻³\nΔE = 9.427622831641132e-25 J\nν = ΔE/h = 1422.8075794882932 MHz\nλ = hc/ΔE = 21.070485027063118 cm","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"Potential energy curve:","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"using Plots\nplot(xlims=(0.0,15.0), ylims=(-0.6,0.05), xlabel=\"\\$r~/~a_0\\$\", ylabel=\"\\$V(r)/E_\\\\mathrm{h},~E_n/E_\\\\mathrm{h}\\$\", legend=:bottomright, size=(480,400))\nplot!(0.1:0.01:15, r -> H.V(r), lc=:black, lw=2, label=\"\") # potential","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"(Image: )","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"Potential energy curve, Energy levels:","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"using Plots\nplot(xlims=(0.0,15.0), ylims=(-0.6,0.05), xlabel=\"\\$r~/~a_0\\$\", ylabel=\"\\$V(r)/E_\\\\mathrm{h}\\$\", legend=:bottomright, size=(480,400))\nfor n in 0:10\n  plot!(0.0:0.01:15, r -> H.E(n=n) > H.V(r) ? H.E(n=n) : NaN, lc=n, lw=1, label=\"\") # energy level\nend\nplot!(0.1:0.01:15, r -> H.V(r), lc=:black, lw=2, label=\"\") # potential","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"(Image: )","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"Radial functions:","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"using Plots\nplot(xlabel=\"\\$r~/~a_0\\$\", ylabel=\"\\$r^2|R_{nl}(r)|^2~/~a_0^{-1}\\$\", ylims=(-0.01,0.55), xticks=0:1:20, size=(480,400), dpi=300)\nfor n in 1:3\n  for l in 0:n-1\n    plot!(0:0.01:20, r->r^2*H.R(r,n=n,l=l)^2, lc=n, lw=2, ls=[:solid,:dash,:dot,:dashdot,:dashdotdot][l+1], label=\"\\$n = $n, l=$l\\$\")\n  end\nend\nplot!()","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"(Image: )","category":"page"},{"location":"HydrogenAtom/#Testing","page":"Hydrogen Atom","title":"Testing","text":"","category":"section"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"Unit testing and Integration testing were done using computer algebra system (Symbolics.jl) and numerical integration (QuadGK.jl). The test script is here.","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"include(\"../../../test/HydrogenAtom.jl\")","category":"page"},{"location":"HydrogenAtom/#Associated-Legendre-Polynomials-P_nm(x)","page":"Hydrogen Atom","title":"Associated Legendre Polynomials P_n^m(x)","text":"","category":"section"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"  beginaligned\n    P_n^m(x)\n    = left( 1-x^2 right)^m2 fracmathrmd^mmathrmdx^m P_n(x) \n    = left( 1-x^2 right)^m2 fracmathrmd^mmathrmdx^m frac12^n n fracmathrmd^nmathrmdx ^n left left( x^2-1 right)^n right \n    = frac12^n (1-x^2)^m2 sum_j=0^leftlfloorfracn-m2rightrfloor (-1)^j frac(2n-2j)j (n-j) (n-2j-m) x^(n-2j-m)\n  endaligned","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"n=0 m=0 ✔","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"beginaligned\n  P_0^0(x)\n    = 1\n  = 1 \n  = 1\nendaligned","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"n=1 m=0 ✔","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"beginaligned\n  P_1^0(x)\n    = frac12 fracmathrmdmathrmdx left( -1 + x^2 right)\n  = x \n  = x\nendaligned","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"n=1 m=1 ✔","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"beginaligned\n  P_1^1(x)\n    = left( 1 - x^2 right)^frac12 fracmathrmdmathrmdx frac12 fracmathrmdmathrmdx left( -1 + x^2 right)\n  = left( 1 - x^2 right)^frac12 \n  = left( 1 - x^2 right)^frac12\nendaligned","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"n=2 m=0 ✔","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"beginaligned\n  P_2^0(x)\n    = frac18 fracmathrmdmathrmdx fracmathrmdmathrmdx left( -1 + x^2 right)^2\n  = frac-12 + frac32 x^2 \n  = frac-12 + frac32 x^2\nendaligned","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"n=2 m=1 ✔","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"beginaligned\n  P_2^1(x)\n    = left( 1 - x^2 right)^frac12 fracmathrmdmathrmdx frac18 fracmathrmdmathrmdx fracmathrmdmathrmdx left( -1 + x^2 right)^2\n  = 3 left( 1 - x^2 right)^frac12 x \n  = 3 left( 1 - x^2 right)^frac12 x\nendaligned","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"n=2 m=2 ✔","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"beginaligned\n  P_2^2(x)\n    = left( 1 - x^2 right) fracmathrmdmathrmdx fracmathrmdmathrmdx frac18 fracmathrmdmathrmdx fracmathrmdmathrmdx left( -1 + x^2 right)^2\n  = 3 - 3 x^2 \n  = 3 - 3 x^2\nendaligned","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"n=3 m=0 ✔","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"beginaligned\n  P_3^0(x)\n    = frac148 fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx left( -1 + x^2 right)^3\n  =  - frac32 x + frac52 x^3 \n  =  - frac32 x + frac52 x^3\nendaligned","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"n=3 m=1 ✔","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"beginaligned\n  P_3^1(x)\n    = left( 1 - x^2 right)^frac12 fracmathrmdmathrmdx frac148 fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx left( -1 + x^2 right)^3\n  =  - frac32 left( 1 - x^2 right)^frac12 + frac152 left( 1 - x^2 right)^frac12 x^2 \n  =  - frac32 left( 1 - x^2 right)^frac12 + frac152 left( 1 - x^2 right)^frac12 x^2\nendaligned","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"n=3 m=2 ✔","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"beginaligned\n  P_3^2(x)\n    = left( 1 - x^2 right) fracmathrmdmathrmdx fracmathrmdmathrmdx frac148 fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx left( -1 + x^2 right)^3\n  = 15 x - 15 x^3 \n  = 15 x - 15 x^3\nendaligned","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"n=3 m=3 ✔","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"beginaligned\n  P_3^3(x)\n    = left( 1 - x^2 right)^frac32 fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx frac148 fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx left( -1 + x^2 right)^3\n  = 15 left( 1 - x^2 right)^frac32 \n  = 15 left( 1 - x^2 right)^frac32\nendaligned","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"n=4 m=0 ✔","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"beginaligned\n  P_4^0(x)\n    = frac1384 fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx left( -1 + x^2 right)^4\n  = frac38 - frac154 x^2 + frac358 x^4 \n  = frac38 - frac154 x^2 + frac358 x^4\nendaligned","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"n=4 m=1 ✔","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"beginaligned\n  P_4^1(x)\n    = left( 1 - x^2 right)^frac12 fracmathrmdmathrmdx frac1384 fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx left( -1 + x^2 right)^4\n  =  - frac152 left( 1 - x^2 right)^frac12 x + frac352 left( 1 - x^2 right)^frac12 x^3 \n  =  - frac152 left( 1 - x^2 right)^frac12 x + frac352 left( 1 - x^2 right)^frac12 x^3\nendaligned","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"n=4 m=2 ✔","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"beginaligned\n  P_4^2(x)\n    = left( 1 - x^2 right) fracmathrmdmathrmdx fracmathrmdmathrmdx frac1384 fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx left( -1 + x^2 right)^4\n  = frac-152 + 60 x^2 - frac1052 x^4 \n  = frac-152 + 60 x^2 - frac1052 x^4\nendaligned","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"n=4 m=3 ✔","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"beginaligned\n  P_4^3(x)\n    = left( 1 - x^2 right)^frac32 fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx frac1384 fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx left( -1 + x^2 right)^4\n  = 105 left( 1 - x^2 right)^frac32 x \n  = 105 left( 1 - x^2 right)^frac32 x\nendaligned","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"n=4 m=4 ✔","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"beginaligned\n  P_4^4(x)\n    = left( 1 - x^2 right)^2 fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx frac1384 fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx left( -1 + x^2 right)^4\n  = 105 left( 1 - x^2 right)^2 \n  = 105 left( 1 - x^2 right)^2\nendaligned","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"Test Summary:                                                   | Pass  Total  Time\nPₙᵐ(x) = √(1-x²)ᵐ dᵐ/dxᵐ Pₙ(x); Pₙ(x) = 1/(2ⁿn!) dⁿ/dxⁿ (x²-1)ⁿ |   15     15  1.7s","category":"page"},{"location":"HydrogenAtom/#Normalization-and-Orthogonality-of-P_nm(x)","page":"Hydrogen Atom","title":"Normalization & Orthogonality of P_n^m(x)","text":"","category":"section"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"int_-1^1 P_i^m(x) P_j^m(x) mathrmdx = frac2(j+m)(2j+1)(j-m) delta_ij","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"  m\t  i\t  j\tnumerical         \tanalytical        \t|error|\n  0\t  0\t  0\t2.0000000000000000\t2.0000000000000000\t0.0000000000000000%\t✔\n  0\t  0\t  1\t0.0000000000000000\t0.0000000000000000\t0.0000000000000000%\t✔\n  0\t  0\t  2\t0.0000000000000000\t0.0000000000000000\t0.0000000000000000%\t✔\n  0\t  0\t  3\t-0.0000000000000000\t0.0000000000000000\t0.0000000000000000%\t✔\n  0\t  0\t  4\t0.0000000000000000\t0.0000000000000000\t0.0000000000000000%\t✔\n  0\t  0\t  5\t0.0000000000000000\t0.0000000000000000\t0.0000000000000000%\t✔\n  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9\t382360926.3157894611358643\t382360926.3157894611358643\t0.0000000000000000%\t✔\nTest Summary:                              | Pass  Total  Time\n∫Pᵢᵐ(x)Pⱼᵐ(x)dx = 2(j+m)!/(2j+1)(j-m)! δᵢⱼ |  355    355  1.2s","category":"page"},{"location":"HydrogenAtom/#Normalization-and-Orthogonality-of-Y_{lm}(\\theta,\\varphi)","page":"Hydrogen Atom","title":"Normalization & Orthogonality of Y_lm(thetavarphi)","text":"","category":"section"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"int_0^2pi\nint_0^pi\nY_lm(thetavarphi)^* Y_lm(thetavarphi) sin(theta)\nmathrmdtheta mathrmdvarphi\n= delta_ll delta_mm","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":" l1 l2 m1 m2\tnumerical\t        analyrical        \t|error|\n  0  0  0  0\t1.0000000000000004\t1.0000000000000000\t0.0000000000000444%\t✔\n  0  1  0 -1\t0.0000000000000000\t0.0000000000000000\t0.0000000000000000%\t✔\n  0  1  0  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-1\t1.0000000000000002\t1.0000000000000000\t0.0000000000000222%\t✔\n  2  2 -1  0\t-0.0000000000000000\t0.0000000000000000\t0.0000000000000000%\t✔\n  2  2 -1  1\t0.0000000000000000\t0.0000000000000000\t0.0000000000000000%\t✔\n  2  2 -1  2\t-0.0000000000000000\t0.0000000000000000\t0.0000000000000000%\t✔\n  2  2  0 -2\t0.0000000000000000\t0.0000000000000000\t0.0000000000000000%\t✔\n  2  2  0 -1\t-0.0000000000000000\t0.0000000000000000\t0.0000000000000000%\t✔\n  2  2  0  0\t1.0000000000000002\t1.0000000000000000\t0.0000000000000222%\t✔\n  2  2  0  1\t0.0000000000000000\t0.0000000000000000\t0.0000000000000000%\t✔\n  2  2  0  2\t0.0000000000000000\t0.0000000000000000\t0.0000000000000000%\t✔\n  2  2  1 -2\t0.0000000000000000\t0.0000000000000000\t0.0000000000000000%\t✔\n  2  2  1 -1\t0.0000000000000000\t0.0000000000000000\t0.0000000000000000%\t✔\n  2  2  1  0\t0.0000000000000000\t0.0000000000000000\t0.0000000000000000%\t✔\n  2  2  1  1\t1.0000000000000002\t1.0000000000000000\t0.0000000000000222%\t✔\n  2  2  1  2\t0.0000000000000000\t0.0000000000000000\t0.0000000000000000%\t✔\n  2  2  2 -2\t-0.0000000000000001\t0.0000000000000000\t0.0000000000000000%\t✔\n  2  2  2 -1\t-0.0000000000000000\t0.0000000000000000\t0.0000000000000000%\t✔\n  2  2  2  0\t0.0000000000000000\t0.0000000000000000\t0.0000000000000000%\t✔\n  2  2  2  1\t0.0000000000000000\t0.0000000000000000\t0.0000000000000000%\t✔\n  2  2  2  2\t1.0000000000000002\t1.0000000000000000\t0.0000000000000222%\t✔\nTest Summary:                              | Pass  Total  Time\n∫Yₗ₁ₘ₁(θ,φ)Yₗ₂ₘ₂(θ,φ)sinθdθdφ = δₗ₁ₗ₂δₘ₁ₘ₂ |   81     81  2.7s","category":"page"},{"location":"HydrogenAtom/#Associated-Laguerre-Polynomials-L_n{k}(x)","page":"Hydrogen Atom","title":"Associated Laguerre Polynomials L_n^k(x)","text":"","category":"section"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"  beginaligned\n  L_n^k(x)\n    = fracmathrmd^kmathrmdx^k L_n(x) \n    = fracmathrmd^kmathrmdx^k frac1n mathrme^x fracmathrmd^nmathrmdx ^n left( mathrme^-x x^n right) \n    = sum_m=0^n-k (-1)^m+k fracnm(m+k)(n-m-k) x^m \n    = (-1)^k L_n-k^(k)(x)\n  endaligned","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"n=0 k=0 ✔","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"beginaligned\n  L_0^0(x)\n   = e^x e^ - x\n  = 1 \n  = 1 \n  = 1\nendaligned","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"n=1 k=0 ✔","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"beginaligned\n  L_1^0(x)\n   = e^x fracmathrmdmathrmdx x e^ - x\n  = 1 - x \n  = 1 - x \n  = 1 - x\nendaligned","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"n=1 k=1 ✔","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"beginaligned\n  L_1^1(x)\n   = fracmathrmdmathrmdx e^x fracmathrmdmathrmdx x e^ - x\n  = -1 \n  = -1 \n  = -1\nendaligned","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"n=2 k=0 ✔","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"beginaligned\n  L_2^0(x)\n   = frac12 e^x fracmathrmdmathrmdx fracmathrmdmathrmdx x^2 e^ - x\n  = 1 - 2 x + frac12 x^2 \n  = 1 - 2 x + frac12 x^2 \n  = 1 - 2 x + frac12 x^2\nendaligned","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"n=2 k=1 ✔","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"beginaligned\n  L_2^1(x)\n   = fracmathrmdmathrmdx frac12 e^x fracmathrmdmathrmdx fracmathrmdmathrmdx x^2 e^ - x\n  = -2 + x \n  = -2 + x \n  = -2 + x\nendaligned","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"n=2 k=2 ✔","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"beginaligned\n  L_2^2(x)\n   = fracmathrmdmathrmdx fracmathrmdmathrmdx frac12 e^x fracmathrmdmathrmdx fracmathrmdmathrmdx x^2 e^ - x\n  = 1 \n  = 1 \n  = 1\nendaligned","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"n=3 k=0 ✔","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"beginaligned\n  L_3^0(x)\n   = frac16 e^x fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx x^3 e^ - x\n  = 1 - frac16 x^3 - 3 x + frac32 x^2 \n  = 1 - frac16 x^3 - 3 x + frac32 x^2 \n  = 1 - frac16 x^3 - 3 x + frac32 x^2\nendaligned","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"n=3 k=1 ✔","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"beginaligned\n  L_3^1(x)\n   = fracmathrmdmathrmdx frac16 e^x fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx x^3 e^ - x\n  = -3 + 3 x - frac12 x^2 \n  = -3 + 3 x - frac12 x^2 \n  = -3 + 3 x - frac12 x^2\nendaligned","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"n=3 k=2 ✔","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"beginaligned\n  L_3^2(x)\n   = fracmathrmdmathrmdx fracmathrmdmathrmdx frac16 e^x fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx x^3 e^ - x\n  = 3 - x \n  = 3 - x \n  = 3 - x\nendaligned","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"n=3 k=3 ✔","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"beginaligned\n  L_3^3(x)\n   = fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx frac16 e^x fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx x^3 e^ - x\n  = -1 \n  = -1 \n  = -1\nendaligned","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"n=4 k=0 ✔","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"beginaligned\n  L_4^0(x)\n   = frac124 e^x fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx x^4 e^ - x\n  = 1 - frac23 x^3 - 4 x + 3 x^2 + frac124 x^4 \n  = 1 - frac23 x^3 - 4 x + 3 x^2 + frac124 x^4 \n  = 1 - frac23 x^3 - 4 x + 3 x^2 + frac124 x^4\nendaligned","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"n=4 k=1 ✔","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"beginaligned\n  L_4^1(x)\n   = fracmathrmdmathrmdx frac124 e^x fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx x^4 e^ - x\n  = -4 + frac16 x^3 + 6 x - 2 x^2 \n  = -4 + frac16 x^3 + 6 x - 2 x^2 \n  = -4 + frac16 x^3 + 6 x - 2 x^2\nendaligned","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"n=4 k=2 ✔","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"beginaligned\n  L_4^2(x)\n   = fracmathrmdmathrmdx fracmathrmdmathrmdx frac124 e^x fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx x^4 e^ - x\n  = 6 - 4 x + frac12 x^2 \n  = 6 - 4 x + frac12 x^2 \n  = 6 - 4 x + frac12 x^2\nendaligned","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"n=4 k=3 ✔","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"beginaligned\n  L_4^3(x)\n   = fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx frac124 e^x fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx x^4 e^ - x\n  = -4 + x \n  = -4 + x \n  = -4 + x\nendaligned","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"n=4 k=4 ✔","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"beginaligned\n  L_4^4(x)\n   = fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx frac124 e^x fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx x^4 e^ - x\n  = 1 \n  = 1 \n  = 1\nendaligned","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"Test Summary:                                          | Pass  Total  Time\nLₙᵏ(x) = dᵏ/dxᵏ Lₙ(x); Lₙ(x) = 1/(n!) eˣ dⁿ/dxⁿ e⁻ˣ xⁿ |   15     15  0.9s","category":"page"},{"location":"HydrogenAtom/#Normalization-and-Orthogonality-of-L_n{k}(x)","page":"Hydrogen Atom","title":"Normalization & 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0\t0.0000000000000012\t0.0000000000000000\t0.0000000000000000%\t✔\n  6\t  7\t  1\t0.0000000000000001\t0.0000000000000000\t0.0000000000000000%\t✔\n  6\t  7\t  2\t-0.0000000000000003\t0.0000000000000000\t0.0000000000000000%\t✔\n  6\t  7\t  3\t0.0000000000001086\t0.0000000000000000\t0.0000000000000000%\t✔\n  6\t  7\t  4\t0.0000000000001137\t0.0000000000000000\t0.0000000000000000%\t✔\n  6\t  7\t  5\t0.0000000000000036\t0.0000000000000000\t0.0000000000000000%\t✔\n  6\t  7\t  6\t0.0000000000000684\t0.0000000000000000\t0.0000000000000000%\t✔\n  7\t  0\t  0\t0.0000000000000011\t0.0000000000000000\t0.0000000000000000%\t✔\n  7\t  1\t  0\t-0.0000000000000041\t0.0000000000000000\t0.0000000000000000%\t✔\n  7\t  1\t  1\t0.0000000000000056\t0.0000000000000000\t0.0000000000000000%\t✔\n  7\t  2\t  0\t0.0000000000000064\t0.0000000000000000\t0.0000000000000000%\t✔\n  7\t  2\t  1\t-0.0000000000000163\t0.0000000000000000\t0.0000000000000000%\t✔\n  7\t  2\t  2\t0.0000000000000170\t0.0000000000000000\t0.0000000000000000%\t✔\n  7\t  3\t  0\t-0.0000000000000079\t0.0000000000000000\t0.0000000000000000%\t✔\n  7\t  3\t  1\t0.0000000000000176\t0.0000000000000000\t0.0000000000000000%\t✔\n  7\t  3\t  2\t-0.0000000000000652\t0.0000000000000000\t0.0000000000000000%\t✔\n  7\t  3\t  3\t-0.0000000000000847\t0.0000000000000000\t0.0000000000000000%\t✔\n  7\t  4\t  0\t0.0000000000000061\t0.0000000000000000\t0.0000000000000000%\t✔\n  7\t  4\t  1\t-0.0000000000000007\t0.0000000000000000\t0.0000000000000000%\t✔\n  7\t  4\t  2\t0.0000000000000793\t0.0000000000000000\t0.0000000000000000%\t✔\n  7\t  4\t  3\t0.0000000000002245\t0.0000000000000000\t0.0000000000000000%\t✔\n  7\t  4\t  4\t0.0000000000000521\t0.0000000000000000\t0.0000000000000000%\t✔\n  7\t  5\t  0\t-0.0000000000000003\t0.0000000000000000\t0.0000000000000000%\t✔\n  7\t  5\t  1\t-0.0000000000000085\t0.0000000000000000\t0.0000000000000000%\t✔\n  7\t  5\t  2\t-0.0000000000000354\t0.0000000000000000\t0.0000000000000000%\t✔\n  7\t  5\t  3\t-0.0000000000002566\t0.0000000000000000\t0.0000000000000000%\t✔\n  7\t  5\t  4\t-0.0000000000001368\t0.0000000000000000\t0.0000000000000000%\t✔\n  7\t  5\t  5\t-0.0000000000000220\t0.0000000000000000\t0.0000000000000000%\t✔\n  7\t  6\t  0\t0.0000000000000012\t0.0000000000000000\t0.0000000000000000%\t✔\n  7\t  6\t  1\t-0.0000000000000003\t0.0000000000000000\t0.0000000000000000%\t✔\n  7\t  6\t  2\t-0.0000000000000012\t0.0000000000000000\t0.0000000000000000%\t✔\n  7\t  6\t  3\t0.0000000000001121\t0.0000000000000000\t0.0000000000000000%\t✔\n  7\t  6\t  4\t0.0000000000001277\t0.0000000000000000\t0.0000000000000000%\t✔\n  7\t  6\t  5\t-0.0000000000000675\t0.0000000000000000\t0.0000000000000000%\t✔\n  7\t  6\t  6\t0.0000000000000684\t0.0000000000000000\t0.0000000000000000%\t✔\n  7\t  7\t  0\t1.0000000000000067\t1.0000000000000000\t0.0000000000006661%\t✔\n  7\t  7\t  1\t6.9999999999999885\t7.0000000000000000\t0.0000000000001649%\t✔\n  7\t  7\t  2\t42.0000000000000000\t42.0000000000000000\t0.0000000000000000%\t✔\n  7\t  7\t  3\t209.9999999999998863\t210.0000000000000000\t0.0000000000000541%\t✔\n  7\t  7\t  4\t840.0000000000002274\t840.0000000000000000\t0.0000000000000271%\t✔\n  7\t  7\t  5\t2519.9999999997753548\t2520.0000000000000000\t0.0000000000089145%\t✔\n  7\t  7\t  6\t5039.9999999999854481\t5040.0000000000000000\t0.0000000000002887%\t✔\n  7\t  7\t  7\t5040.0000000000000000\t5040.0000000000000000\t0.0000000000000000%\t✔\nTest Summary:                                 | Pass  Total  Time\n∫exp(-x)xᵏLᵢᵏ(x)Lⱼᵏ(x)dx = (2i+k)!/(i+k)! δᵢⱼ |  204    204  1.0s","category":"page"},{"location":"HydrogenAtom/#Normalization-of-R_{nl}(r)","page":"Hydrogen Atom","title":"Normalization of R_nl(r)","text":"","category":"section"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"int R_nl(r)^2 r^2 mathrmdr = 1","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"  n\t  l\tnumerical         \tanalytical        \t|error|\n  1\t  0\t1.0000000000000286\t1.0000000000000000\t0.0000000000028644%\t✔\n  2\t  0\t1.0000000000000002\t1.0000000000000000\t0.0000000000000222%\t✔\n  2\t  1\t1.0000000000000000\t1.0000000000000000\t0.0000000000000000%\t✔\n  3\t  0\t0.9999999999999999\t1.0000000000000000\t0.0000000000000111%\t✔\n  3\t  1\t0.9999999999992634\t1.0000000000000000\t0.0000000000736633%\t✔\n  3\t  2\t0.9999999999998338\t1.0000000000000000\t0.0000000000166200%\t✔\n  4\t  0\t0.9999999999999916\t1.0000000000000000\t0.0000000000008438%\t✔\n  4\t  1\t0.9999999999999950\t1.0000000000000000\t0.0000000000004996%\t✔\n  4\t  2\t0.9999999999999987\t1.0000000000000000\t0.0000000000001332%\t✔\n  4\t  3\t1.0000000000000000\t1.0000000000000000\t0.0000000000000000%\t✔\n  5\t  0\t0.9999999999999996\t1.0000000000000000\t0.0000000000000444%\t✔\n  5\t  1\t0.9999999999999989\t1.0000000000000000\t0.0000000000001110%\t✔\n  5\t  2\t0.9999999999999994\t1.0000000000000000\t0.0000000000000555%\t✔\n  5\t  3\t0.9999999999999999\t1.0000000000000000\t0.0000000000000111%\t✔\n  5\t  4\t1.0000000000000000\t1.0000000000000000\t0.0000000000000000%\t✔\n  6\t  0\t1.0000000000000002\t1.0000000000000000\t0.0000000000000222%\t✔\n  6\t  1\t0.9999999999999999\t1.0000000000000000\t0.0000000000000111%\t✔\n  6\t  2\t0.9999999999999997\t1.0000000000000000\t0.0000000000000333%\t✔\n  6\t  3\t0.9999999999999999\t1.0000000000000000\t0.0000000000000111%\t✔\n  6\t  4\t0.9999999999999998\t1.0000000000000000\t0.0000000000000222%\t✔\n  6\t  5\t0.9999999999999994\t1.0000000000000000\t0.0000000000000555%\t✔\n  7\t  0\t0.9999999999999962\t1.0000000000000000\t0.0000000000003775%\t✔\n  7\t  1\t0.9999999999999997\t1.0000000000000000\t0.0000000000000333%\t✔\n  7\t  2\t0.9999999999999978\t1.0000000000000000\t0.0000000000002220%\t✔\n  7\t  3\t0.9999999999999987\t1.0000000000000000\t0.0000000000001332%\t✔\n  7\t  4\t0.9999999999999992\t1.0000000000000000\t0.0000000000000777%\t✔\n  7\t  5\t1.0000000000000000\t1.0000000000000000\t0.0000000000000000%\t✔\n  7\t  6\t0.9999999999999726\t1.0000000000000000\t0.0000000000027423%\t✔\n  8\t  0\t1.0000000000000082\t1.0000000000000000\t0.0000000000008216%\t✔\n  8\t  1\t0.9999999999999961\t1.0000000000000000\t0.0000000000003886%\t✔\n  8\t  2\t1.0000000000000044\t1.0000000000000000\t0.0000000000004441%\t✔\n  8\t  3\t1.0000000000000002\t1.0000000000000000\t0.0000000000000222%\t✔\n  8\t  4\t1.0000000000000009\t1.0000000000000000\t0.0000000000000888%\t✔\n  8\t  5\t0.9999999999999999\t1.0000000000000000\t0.0000000000000111%\t✔\n  8\t  6\t0.9999999999999996\t1.0000000000000000\t0.0000000000000444%\t✔\n  8\t  7\t0.9999999999999999\t1.0000000000000000\t0.0000000000000111%\t✔\n  9\t  0\t0.9999999999999858\t1.0000000000000000\t0.0000000000014211%\t✔\n  9\t  1\t1.0000000000000036\t1.0000000000000000\t0.0000000000003553%\t✔\n  9\t  2\t1.0000000000000187\t1.0000000000000000\t0.0000000000018652%\t✔\n  9\t  3\t0.9999999999999923\t1.0000000000000000\t0.0000000000007661%\t✔\n  9\t  4\t0.9999999999999903\t1.0000000000000000\t0.0000000000009659%\t✔\n  9\t  5\t0.9999999999999982\t1.0000000000000000\t0.0000000000001776%\t✔\n  9\t  6\t0.9999999999999999\t1.0000000000000000\t0.0000000000000111%\t✔\n  9\t  7\t1.0000000000000007\t1.0000000000000000\t0.0000000000000666%\t✔\n  9\t  8\t0.9999999999999994\t1.0000000000000000\t0.0000000000000555%\t✔\nTest Summary:               | Pass  Total  Time\n∫|Rₙₗ(r)|²r²dr = δₙ₁ₙ₂δₗ₁ₗ₂ |   45     45  0.9s","category":"page"},{"location":"HydrogenAtom/#Expected-Value-of-r","page":"Hydrogen Atom","title":"Expected Value of r","text":"","category":"section"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"langle r rangle\n= int r R_n_1 l_1(r)^2 r^2 mathrmdr\n= fraca_mu2Z left 3n^2 - l(l+1) right \na_mu = a_0 fracm_mathrmemu \nfrac1mu = frac1m_mathrme + frac1m_mathrmp","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"Reference:","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"高柳和夫『朝倉物理学大系 11 原子分子物理学』(2000, 朝倉書店) pp.11-22\n Quan­tum Me­chan­ics for En­gi­neers by Leon van Dom­me­len","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"  n\t  l\tnumerical         \tanalytical        \t|error|\n  1\t  0\t1.4999999999999998\t1.5000000000000000\t0.0000000000000148%\t✔\n  2\t  0\t6.0000000000000000\t6.0000000000000000\t0.0000000000000000%\t✔\n  2\t  1\t4.9999999999999991\t5.0000000000000000\t0.0000000000000178%\t✔\n  3\t  0\t13.4999999999999982\t13.5000000000000000\t0.0000000000000132%\t✔\n  3\t  1\t12.5000000000000018\t12.5000000000000000\t0.0000000000000142%\t✔\n  3\t  2\t10.4999999999999982\t10.5000000000000000\t0.0000000000000169%\t✔\n  4\t  0\t23.9999999999989235\t24.0000000000000000\t0.0000000000044853%\t✔\n  4\t  1\t22.9999999999993037\t23.0000000000000000\t0.0000000000030275%\t✔\n  4\t  2\t20.9999999999997513\t21.0000000000000000\t0.0000000000011842%\t✔\n  4\t  3\t17.9999999999999574\t18.0000000000000000\t0.0000000000002368%\t✔\n  5\t  0\t37.4999999999999147\t37.5000000000000000\t0.0000000000002274%\t✔\n  5\t  1\t36.4999999999999716\t36.5000000000000000\t0.0000000000000779%\t✔\n  5\t  2\t34.4999999999999858\t34.5000000000000000\t0.0000000000000412%\t✔\n  5\t  3\t31.4999999999999929\t31.5000000000000000\t0.0000000000000226%\t✔\n  5\t  4\t27.4999999999432916\t27.5000000000000000\t0.0000000002062124%\t✔\n  6\t  0\t54.0000000000009095\t54.0000000000000000\t0.0000000000016842%\t✔\n  6\t  1\t53.0000000000012008\t53.0000000000000000\t0.0000000000022657%\t✔\n  6\t  2\t51.0000000000003268\t51.0000000000000000\t0.0000000000006409%\t✔\n  6\t  3\t48.0000000000001279\t48.0000000000000000\t0.0000000000002665%\t✔\n  6\t  4\t44.0000000000000142\t44.0000000000000000\t0.0000000000000323%\t✔\n  6\t  5\t38.9999999999999858\t39.0000000000000000\t0.0000000000000364%\t✔\n  7\t  0\t73.4999999999998579\t73.5000000000000000\t0.0000000000001933%\t✔\n  7\t  1\t72.5000000000000284\t72.5000000000000000\t0.0000000000000392%\t✔\n  7\t  2\t70.4999999999998437\t70.5000000000000000\t0.0000000000002217%\t✔\n  7\t  3\t67.4999999999999005\t67.5000000000000000\t0.0000000000001474%\t✔\n  7\t  4\t63.4999999999999645\t63.5000000000000000\t0.0000000000000559%\t✔\n  7\t  5\t58.4999999999999858\t58.5000000000000000\t0.0000000000000243%\t✔\n  7\t  6\t52.4999999999920206\t52.5000000000000000\t0.0000000000151988%\t✔\n  8\t  0\t96.0000000000006537\t96.0000000000000000\t0.0000000000006809%\t✔\n  8\t  1\t94.9999999999993179\t95.0000000000000000\t0.0000000000007180%\t✔\n  8\t  2\t93.0000000000000995\t93.0000000000000000\t0.0000000000001070%\t✔\n  8\t  3\t90.0000000000000711\t90.0000000000000000\t0.0000000000000789%\t✔\n  8\t  4\t86.0000000000000568\t86.0000000000000000\t0.0000000000000661%\t✔\n  8\t  5\t81.0000000000000000\t81.0000000000000000\t0.0000000000000000%\t✔\n  8\t  6\t74.9999999999999574\t75.0000000000000000\t0.0000000000000568%\t✔\n  8\t  7\t68.0000000000000000\t68.0000000000000000\t0.0000000000000000%\t✔\n  9\t  0\t121.5000000000011937\t121.5000000000000000\t0.0000000000009825%\t✔\n  9\t  1\t120.5000000000002274\t120.5000000000000000\t0.0000000000001887%\t✔\n  9\t  2\t118.5000000000011653\t118.5000000000000000\t0.0000000000009834%\t✔\n  9\t  3\t115.5000000000000853\t115.5000000000000000\t0.0000000000000738%\t✔\n  9\t  4\t111.4999999999978400\t111.5000000000000000\t0.0000000000019373%\t✔\n  9\t  5\t106.4999999999992895\t106.5000000000000000\t0.0000000000006672%\t✔\n  9\t  6\t100.4999999999998721\t100.5000000000000000\t0.0000000000001273%\t✔\n  9\t  7\t93.5000000000000000\t93.5000000000000000\t0.0000000000000000%\t✔\n  9\t  8\t85.4999999999999716\t85.5000000000000000\t0.0000000000000332%\t✔\nTest Summary:                                                    | Pass  Total  Time\n∫r|Rₙₗ(r)|²r²dr = (a₀×mₑ/μ)/2Z × [3n²-l(l+1)]; 1/μ = 1/mₑ + 1/mₚ |   45     45  0.8s","category":"page"},{"location":"HydrogenAtom/#Expected-Value-of-r2","page":"Hydrogen Atom","title":"Expected Value of r^2","text":"","category":"section"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"langle r^2 rangle\n= int r^2 R_n_1 l_1(r)^2 r^2 mathrmdr\n= fraca_mu^22Z^2 n^2 left 5n^2 + 1 - 3l(l+1) right \na_mu = a_0 fracm_mathrmemu \nfrac1mu = frac1m_mathrme + frac1m_mathrmp","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"Reference:","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"高柳和夫『朝倉物理学大系 11 原子分子物理学』(2000, 朝倉書店) pp.11-22\n Quan­tum Me­chan­ics for En­gi­neers by Leon van Dom­me­len","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"  n\t  l\tnumerical         \tanalytical        \t|error|\n  1\t  0\t3.0000000000000004\t3.0000000000000000\t0.0000000000000148%\t✔\n  2\t  0\t42.0000000000000000\t42.0000000000000000\t0.0000000000000000%\t✔\n  2\t  1\t30.0000000000000000\t30.0000000000000000\t0.0000000000000000%\t✔\n  3\t  0\t207.0000000000000284\t207.0000000000000000\t0.0000000000000137%\t✔\n  3\t  1\t180.0000000000000000\t180.0000000000000000\t0.0000000000000000%\t✔\n  3\t  2\t125.9999999999999858\t126.0000000000000000\t0.0000000000000113%\t✔\n  4\t  0\t647.9999999999030251\t648.0000000000000000\t0.0000000000149653%\t✔\n  4\t  1\t599.9999999999364491\t600.0000000000000000\t0.0000000000105918%\t✔\n  4\t  2\t503.9999999999748184\t504.0000000000000000\t0.0000000000049964%\t✔\n  4\t  3\t359.9999999999955094\t360.0000000000000000\t0.0000000000012474%\t✔\n  5\t  0\t1574.9999999999988631\t1575.0000000000000000\t0.0000000000000722%\t✔\n  5\t  1\t1499.9999999999977263\t1500.0000000000000000\t0.0000000000001516%\t✔\n  5\t  2\t1349.9999999999995453\t1350.0000000000000000\t0.0000000000000337%\t✔\n  5\t  3\t1125.0000000000029559\t1125.0000000000000000\t0.0000000000002627%\t✔\n  5\t  4\t824.9999999999998863\t825.0000000000000000\t0.0000000000000138%\t✔\n  6\t  0\t3257.9999999999968168\t3258.0000000000000000\t0.0000000000000977%\t✔\n  6\t  1\t3149.9999999999922693\t3150.0000000000000000\t0.0000000000002454%\t✔\n  6\t  2\t2933.9999999999981810\t2934.0000000000000000\t0.0000000000000620%\t✔\n  6\t  3\t2610.0000000000327418\t2610.0000000000000000\t0.0000000000012545%\t✔\n  6\t  4\t2178.0000000000081855\t2178.0000000000000000\t0.0000000000003758%\t✔\n  6\t  5\t1638.0000000000002274\t1638.0000000000000000\t0.0000000000000139%\t✔\n  7\t  0\t6026.9999999999918145\t6027.0000000000000000\t0.0000000000001358%\t✔\n  7\t  1\t5880.0000000000027285\t5880.0000000000000000\t0.0000000000000464%\t✔\n  7\t  2\t5585.9999999999899956\t5586.0000000000000000\t0.0000000000001791%\t✔\n  7\t  3\t5144.9999999999918145\t5145.0000000000000000\t0.0000000000001591%\t✔\n  7\t  4\t4556.9999999999972715\t4557.0000000000000000\t0.0000000000000599%\t✔\n  7\t  5\t3821.9999999999990905\t3822.0000000000000000\t0.0000000000000238%\t✔\n  7\t  6\t2940.0000000000013642\t2940.0000000000000000\t0.0000000000000464%\t✔\n  8\t  0\t10272.0000000000291038\t10272.0000000000000000\t0.0000000000002833%\t✔\n  8\t  1\t10079.9999999999945430\t10080.0000000000000000\t0.0000000000000541%\t✔\n  8\t  2\t9695.9999999999927240\t9696.0000000000000000\t0.0000000000000750%\t✔\n  8\t  3\t9120.0000000000109139\t9120.0000000000000000\t0.0000000000001197%\t✔\n  8\t  4\t8352.0000000000018190\t8352.0000000000000000\t0.0000000000000218%\t✔\n  8\t  5\t7392.0000000000100044\t7392.0000000000000000\t0.0000000000001353%\t✔\n  8\t  6\t6240.0000000000000000\t6240.0000000000000000\t0.0000000000000000%\t✔\n  8\t  7\t4896.0000000000081855\t4896.0000000000000000\t0.0000000000001672%\t✔\n  9\t  0\t16443.0000000001018634\t16443.0000000000000000\t0.0000000000006195%\t✔\n  9\t  1\t16200.0000000000400178\t16200.0000000000000000\t0.0000000000002470%\t✔\n  9\t  2\t15714.0000000001491571\t15714.0000000000000000\t0.0000000000009492%\t✔\n  9\t  3\t14984.9999999999181455\t14985.0000000000000000\t0.0000000000005462%\t✔\n  9\t  4\t14012.9999999995452526\t14013.0000000000000000\t0.0000000000032452%\t✔\n  9\t  5\t12797.9999999998071871\t12798.0000000000000000\t0.0000000000015066%\t✔\n  9\t  6\t11339.9999999999454303\t11340.0000000000000000\t0.0000000000004812%\t✔\n  9\t  7\t9638.9999999999909051\t9639.0000000000000000\t0.0000000000000944%\t✔\n  9\t  8\t7694.9999999999981810\t7695.0000000000000000\t0.0000000000000236%\t✔\nTest Summary:                                                            | Pass  Total  Time\n∫r²|Rₙₗ(r)|²r²dr = (a₀×mₑ/μ)²/2Z² × n²[5n²+1-3l(l+1)]; 1/μ = 1/mₑ + 1/mₚ |   45     45  0.8s","category":"page"},{"location":"HydrogenAtom/#Virial-Theorem","page":"Hydrogen Atom","title":"Virial Theorem","text":"","category":"section"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"The virial theorem 2langle T rangle + langle V rangle = 0 and the definition of Hamiltonian langle H rangle = langle T rangle + langle V rangle derive langle H rangle = frac12 langle V rangle and langle H rangle = -langle T rangle.","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"frac12 int psi_n^ast(x) V(x) psi_n(x) mathrmdx = E_n","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"  n\tnumerical         \tanalytical        \t|error|\n  1\t-0.9999999999999999\t-1.0000000000000000\t0.0000000000000111%\t✔\n  2\t-0.2500000000000001\t-0.2500000000000000\t0.0000000000000444%\t✔\n  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-1\t-0.0000000000000429\t0.0000000000000000\t0.0000000000000000%\t✔\n  3\t  3\t  2\t  2\t  1\t  0\t0.0000000000000000\t0.0000000000000000\t0.0000000000000000%\t✔\n  3\t  3\t  2\t  2\t  1\t  1\t1.0003006285590699\t1.0000000000000000\t0.0300628559069871%\t✔\n  3\t  3\t  2\t  2\t  1\t  2\t0.0000000000000000\t0.0000000000000000\t0.0000000000000000%\t✔\n  3\t  3\t  2\t  2\t  2\t -2\t0.0000001937793434\t0.0000000000000000\t0.0000000000000000%\t✔\n  3\t  3\t  2\t  2\t  2\t -1\t-0.0000000000000000\t0.0000000000000000\t0.0000000000000000%\t✔\n  3\t  3\t  2\t  2\t  2\t  0\t-0.0000000000000175\t0.0000000000000000\t0.0000000000000000%\t✔\n  3\t  3\t  2\t  2\t  2\t  1\t0.0000000000000000\t0.0000000000000000\t0.0000000000000000%\t✔\n  3\t  3\t  2\t  2\t  2\t  2\t1.0003006285656155\t1.0000000000000000\t0.0300628565615524%\t✔\nTest Summary:                       | Pass  Total  Time\n<ψₙ₁ₗ₁ₘ₁|ψₙ₂ₗ₂ₘ₂> = δₙ₁ₙ₂δₗ₁ₗ₂δₘ₁ₘ₂ |  196    196  8.0s\n","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"CurrentModule = Antiq","category":"page"},{"location":"HarmonicOscillator/#Harmonic-Oscillator","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"","category":"section"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"The harmonic oscillator is the most frequently used model in quantum physics.","category":"page"},{"location":"HarmonicOscillator/#Definitions","page":"Harmonic Oscillator","title":"Definitions","text":"","category":"section"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"k is the spring constant. And omega = sqrtkm, xi = sqrtfracmomegahbarx, A_n = sqrtfrac1n 2^n sqrtfracmomegapihbar, H_n(x) = (-1)^n mathrme^x^2 fracmathrmd^nmathrmdx^n mathrme^-x^2 are used.","category":"page"},{"location":"HarmonicOscillator/#Schrödinger-Equation","page":"Harmonic Oscillator","title":"Schrödinger Equation","text":"","category":"section"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"  hatHpsi(x) = E psi(x)","category":"page"},{"location":"HarmonicOscillator/#Hamiltonian","page":"Harmonic Oscillator","title":"Hamiltonian","text":"","category":"section"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"  beginaligned\n    hatH\n    = - frachbar^22m fracmathrmd^2mathrmdx ^2 + V(x) \n    = - frac12 hbaromega fracmathrmd^2mathrmdxi^2 + V(x)\n  endaligned","category":"page"},{"location":"HarmonicOscillator/#Potential","page":"Harmonic Oscillator","title":"Potential","text":"","category":"section"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"V(x; k=k, m=m, ω=sqrt(k/m))","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"  V(x)\n  = frac12 k x^2\n  = frac12 m omega^2 x^2\n  = frac12 hbar omega xi^2","category":"page"},{"location":"HarmonicOscillator/#Eigen-Values","page":"Harmonic Oscillator","title":"Eigen Values","text":"","category":"section"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"E(n; k, m, ω=sqrt(k/m), ℏ=ℏ)","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"  E_n = hbar omega left( n + frac12 right)","category":"page"},{"location":"HarmonicOscillator/#Eigen-Functions","page":"Harmonic Oscillator","title":"Eigen Functions","text":"","category":"section"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"ψ(n, x; k=k, m=m, ω=sqrt(k/m), ℏ=ℏ)","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"  psi_n(x) = A_n H_n(xi) expleft( -fracxi^22 right)","category":"page"},{"location":"HarmonicOscillator/#Hermite-Polynomials","page":"Harmonic Oscillator","title":"Hermite Polynomials","text":"","category":"section"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"H(x; n=0)","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"Rodrigues' formula & closed-form:","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"  beginaligned\n    H_n(x)\n    = (-1)^n mathrme^x^2 fracmathrmd^nmathrmdx^n mathrme^-x^2 \n    = n sum_m=0^lfloor n2 rfloor frac(-1)^mm (n-2m)(2 x)^n-2m\n  endaligned","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"Examples:","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"  beginaligned\n    H_0(x)  = 1 \n    H_1(x)  = 2 x \n    H_2(x)  = -2 + 4 x^2 \n    H_3(x)  = -12 x + 8 x^3 \n    H_4(x)  = 12 - 48 x^2 + 16 x^4 \n    H_5(x)  = 120 x - 160 x^3 + 32 x^5 \n    H_6(x)  = -120 + 720 x^2 - 480 x^4 + 64 x^6 \n    H_7(x)  = -1680 x + 3360 x^3 - 1344 x^5 + 128 x^7 \n    H_8(x)  = 1680 - 13440 x^2 + 13440 x^4 - 3584 x^6 + 256 x^8 \n    H_9(x)  = 30240 x - 80640 x^3 + 48384 x^5 - 9216 x^7 + 512 x^9 \n    vdots\n  endaligned","category":"page"},{"location":"HarmonicOscillator/#Reference","page":"Harmonic Oscillator","title":"Reference","text":"","category":"section"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"DLMF 18.5.18\ncpprefjp\nThe Digital Library of Mathematical Functions (DLMF)                                                    18.3 Table1, 18.5 Table1, 18.5.13, 18.5.18\nL. D. Landau, E. M. Lifshitz, Quantum Mechanics (Pergamon Press, 1965)                                  p.595 (a.4), (a.6)\nL. I. Schiff, Quantum Mechanics (McGraw-Hill Book Company, 1968)                                        p.71 (13.12)\nA. Messiah, Quanfum Mechanics (Dover Publications, 1999)                                                p.491 (B.59)\nW. Greiner, Quantum Mechanics: An Introduction Third Edition (Springer, 1994)                           p.152 (7.22)\nD. J. Griffiths, Introduction to Quantum Mechanics (Prentice Hall, 1995)                                p.41 Table 2.1, p.43 (2.70)\nD. A. McQuarrie, J. D. Simon, Physical Chemistry: A Molecular Approach (University Science Books, 1997) p.170 Table 5.2\nP. W. Atkins, J. De Paula, Atkins' Physical Chemistry, 8th edition (W. H. Freeman, 2008)                p.293 Table 9.1\nJ. J. Sakurai, J. Napolitano, Modern Quantum Mechanics Third Edition (Cambridge University Press, 2021) p.524 (B.29)","category":"page"},{"location":"HarmonicOscillator/#Usage-and-Examples","page":"Harmonic Oscillator","title":"Usage & Examples","text":"","category":"section"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"Install Antiq.jl for the first run and run using Antiq before each use. The function antiq(model, parameters...) returns a module that has E(), ψ(x), V(x) and some other functions. In this system, the model name is specified by :HarmonicOscillator and several parameters k, m and ℏ are set as optional arguments.","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"using Antiq\nHO = antiq(:HarmonicOscillator, k=1.0, m=1.0, ℏ=1.0)","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"Parameters:","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"julia> HO.k\n1.0\n\njulia> HO.m\n1.0\n\njulia> HO.ℏ\n1.0","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"Eigen values:","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"julia> HO.E(n=0)\n0.5\n\njulia> HO.E(n=1)\n1.5","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"Potential energy curve:","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"using Plots\nplot(-5:0.1:5, x -> HO.V(x), lw=2, label=\"\", xlabel=\"x\", ylabel=\"V(x)\")","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"(Image: )","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"Wave functions:","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"using Plots\nplot(xlim=(-5,5), xlabel=\"x\", ylabel=\"ψ(x)\")\nplot!(x -> HO.ψ(x, n=0), label=\"n=0\", lw=2)\nplot!(x -> HO.ψ(x, n=1), label=\"n=1\", lw=2)\nplot!(x -> HO.ψ(x, n=2), label=\"n=2\", lw=2)\nplot!(x -> HO.ψ(x, n=3), label=\"n=3\", lw=2)\nplot!(x -> HO.ψ(x, n=4), label=\"n=4\", lw=2)","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"(Image: )","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"Potential energy curve, Energy levels, Wave functions:","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"using Plots\nplot(xlim=(-5.5,5.5), ylim=(-0.2,5.4), xlabel=\"\\$x\\$\", ylabel=\"\\$V(x),~E_n,~\\\\psi_n(x)\\\\times0.5+E_n\\$\", size=(480,400), dpi=300)\nfor n in 0:4\n  # energy\n  hline!([HO.E(n=n)], lc=:black, ls=:dash, label=\"\")\n  plot!([-sqrt(2*HO.k*HO.E(n=n)),sqrt(2*HO.k*HO.E(n=n))], fill(HO.E(n=n),2), lc=:black, lw=2, label=\"\")\n  # wave function\n  plot!(x -> HO.E(n=n) + 0.5*HO.ψ(x,n=n), lc=n+1, lw=2, label=\"\")\nend\n# potential\nplot!(x -> HO.V(x), lc=:black, lw=2, label=\"\")","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"(Image: )","category":"page"},{"location":"HarmonicOscillator/#Testing","page":"Harmonic Oscillator","title":"Testing","text":"","category":"section"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"Unit testing and Integration testing were done using computer algebra system (Symbolics.jl) and numerical integration (QuadGK.jl). The test script is here.","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"include(\"../../../test/HarmonicOscillator.jl\")","category":"page"},{"location":"HarmonicOscillator/#Hermite-Polynomials-H_n(x)","page":"Harmonic Oscillator","title":"Hermite Polynomials H_n(x)","text":"","category":"section"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"  beginaligned\n    H_n(x)\n    = (-1)^n mathrme^x^2 fracmathrmd^nmathrmdx^n mathrme^-x^2 \n    = n sum_m=0^lfloor n2 rfloor frac(-1)^mm (n-2m)(2 x)^n-2m\n  endaligned","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"n=0 ✔","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"beginaligned\n  H_0(x)\n   = e^x^2 e^ - x^2\n  = 1 \n  = 1\nendaligned","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"n=1 ✔","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"beginaligned\n  H_1(x)\n   =  - e^x^2 fracmathrmd e^ - x^2mathrmdx\n  = 2 x \n  = 2 x\nendaligned","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"n=2 ✔","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"beginaligned\n  H_2(x)\n   = e^x^2 fracmathrmdmathrmdx fracmathrmd e^ - x^2mathrmdx\n  = -2 + 4 x^2 \n  = -2 + 4 x^2\nendaligned","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"n=3 ✗ Hₙ(x) = (-1)ⁿ exp(x²) dⁿ/dxⁿ exp(-x²) = ...: Test Failed at C:\\Users\\user\\Desktop\\GitHub\\Antiq.jl\\test\\HarmonicOscillator.jl:43   Expression: acceptance","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"Stacktrace:  [1] macro expansion    @ C:\\Users\\user.julia\\juliaup\\julia-1.9.2+0.x64.w64.mingw32\\share\\julia\\stdlib\\v1.9\\Test\\src\\Test.jl:478 [inlined]  [2] macro expansion    @ C:\\Users\\user\\Desktop\\GitHub\\Antiq.jl\\test\\HarmonicOscillator.jl:43 [inlined]  [3] macro expansion    @ C:\\Users\\user.julia\\juliaup\\julia-1.9.2+0.x64.w64.mingw32\\share\\julia\\stdlib\\v1.9\\Test\\src\\Test.jl:1498 [inlined]  [4] top-level scope    @ C:\\Users\\user\\Desktop\\GitHub\\Antiq.jl\\test\\HarmonicOscillator.jl:28","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"beginaligned\n  H_3(x)\n   =  - e^x^2 fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmd e^ - x^2mathrmdx\n  = 8 x^3 - 12 x \n  =  - 12 x + 8 x^3\nendaligned","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"n=4 ✔","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"beginaligned\n  H_4(x)\n   = e^x^2 fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmd e^ - x^2mathrmdx\n  = 12 - 48 x^2 + 16 x^4 \n  = 12 - 48 x^2 + 16 x^4\nendaligned","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"n=5 ✔","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"beginaligned\n  H_5(x)\n   =  - e^x^2 fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmd e^ - x^2mathrmdx\n  = 120 x - 160 x^3 + 32 x^5 \n  = 120 x - 160 x^3 + 32 x^5\nendaligned","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"n=6 ✔","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"beginaligned\n  H_6(x)\n   = e^x^2 fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmd e^ - x^2mathrmdx\n  = -120 + 720 x^2 + 64 x^6 - 480 x^4 \n  = -120 + 720 x^2 + 64 x^6 - 480 x^4\nendaligned","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"n=7 ✔","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"beginaligned\n  H_7(x)\n   =  - e^x^2 fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmd e^ - x^2mathrmdx\n  =  - 1680 x + 3360 x^3 + 128 x^7 - 1344 x^5 \n  =  - 1680 x + 3360 x^3 + 128 x^7 - 1344 x^5\nendaligned","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"n=8 ✔","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"beginaligned\n  H_8(x)\n   = e^x^2 fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmd e^ - x^2mathrmdx\n  = 1680 + 256 x^8 - 13440 x^2 - 3584 x^6 + 13440 x^4 \n  = 1680 + 256 x^8 - 13440 x^2 - 3584 x^6 + 13440 x^4\nendaligned","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"n=9 ✔","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"beginaligned\n  H_9(x)\n   =  - e^x^2 fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmd e^ - x^2mathrmdx\n  = 30240 x - 80640 x^3 + 512 x^9 - 9216 x^7 + 48384 x^5 \n  = 30240 x - 80640 x^3 + 512 x^9 - 9216 x^7 + 48384 x^5\nendaligned","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"Test Summary:                               | Pass  Fail  Total   Time\nHₙ(x) = (-1)ⁿ exp(x²) dⁿ/dxⁿ exp(-x²) = ... |    9     1     10  29.5s\nError: LoadError: Some tests did not pass: 9 passed, 1 failed, 0 errored, 0 broken.\nin expression starting at C:\\Users\\user\\Desktop\\GitHub\\Antiq.jl\\test\\HarmonicOscillator.jl:27","category":"page"},{"location":"","page":"Home","title":"Home","text":"CurrentModule = Antiq","category":"page"},{"location":"#Antiq.jl","page":"Home","title":"Antiq.jl","text":"","category":"section"},{"location":"","page":"Home","title":"Home","text":"Self-contained, Well-Tested, Well-Documented Functions for Quantum Mechanical Models","category":"page"},{"location":"#Install","page":"Home","title":"Install","text":"","category":"section"},{"location":"","page":"Home","title":"Home","text":"To install this package, run the following code in your Jupyter Notebook:","category":"page"},{"location":"","page":"Home","title":"Home","text":"using Pkg\nPkg.add(path=\"https://github.com/ohno/Antiq.jl.git\")","category":"page"},{"location":"","page":"Home","title":"Home","text":"or use add upon REPL(Package mode):","category":"page"},{"location":"","page":"Home","title":"Home","text":"]\nadd https://github.com/ohno/Antiq.jl.git","category":"page"},{"location":"#Usage","page":"Home","title":"Usage","text":"","category":"section"},{"location":"","page":"Home","title":"Home","text":"To use this package, run the following code before each use:","category":"page"},{"location":"","page":"Home","title":"Home","text":"using Antiq","category":"page"},{"location":"","page":"Home","title":"Home","text":"The function antiq(model, parameters...) returns a module. Each module has E(), ψ(x) and some other functions.","category":"page"},{"location":"#Examples","page":"Home","title":"Examples","text":"","category":"section"},{"location":"","page":"Home","title":"Home","text":"The energy of 1S state in hydrogen atom:","category":"page"},{"location":"","page":"Home","title":"Home","text":"julia> H = antiq(:HydrogenAtom, Z=1)\njulia> H.E(n=1)\n-0.5","category":"page"},{"location":"","page":"Home","title":"Home","text":"The energy of 1S state in helium atom:","category":"page"},{"location":"","page":"Home","title":"Home","text":"julia> He⁺ = antiq(:HydrogenAtom, Z=2)\njulia> He⁺.E(n=1)\n-2.0","category":"page"},{"location":"#Supported-Models","page":"Home","title":"Supported Models","text":"","category":"section"},{"location":"","page":"Home","title":"Home","text":"
\n  
\n    \n      \"InfinitePotentialWell\"/\n    \n    InfinitePotentialWell\n  
\n  
\n    \n      \"HarmonicOscillator\"/\n    \n    HarmonicOscillator\n  
\n  
\n    \n      \"MorsePotential\"/\n    \n    MorsePotential\n  
\n  
\n    \n      \"HydrogenAtom\"/\n    \n    HydrogenAtom\n  
\n
","category":"page"},{"location":"#Future-Works","page":"Home","title":"Future Works","text":"","category":"section"},{"location":"","page":"Home","title":"Home","text":"List of quantum-mechanical systems with analytical solutions","category":"page"},{"location":"#Acknowledgment","page":"Home","title":"Acknowledgment","text":"","category":"section"},{"location":"","page":"Home","title":"Home","text":"This package was named by @KB-satou and @ultimatile:","category":"page"},{"location":"","page":"Home","title":"Home","text":"Analytical soulutions of Schrödinger equations.","category":"page"}]
 }