diff --git a/developer.ipynb b/developer.ipynb index 151741d..f765ae7 100644 --- a/developer.ipynb +++ b/developer.ipynb @@ -82,9 +82,7 @@ "name": "stderr", "output_type": "stream", "text": [ - "\u001b[32m\u001b[1m Activating\u001b[22m\u001b[39m project at `C:\\Users\\user\\Desktop\\GitHub\\Antiq.jl`\n", - "\u001b[32m\u001b[1m No Changes\u001b[22m\u001b[39m to `C:\\Users\\user\\Desktop\\GitHub\\Antiq.jl\\Project.toml`\n", - "\u001b[32m\u001b[1m No Changes\u001b[22m\u001b[39m to `C:\\Users\\user\\Desktop\\GitHub\\Antiq.jl\\Manifest.toml`\n" + "\u001b[32m\u001b[1m Activating\u001b[22m\u001b[39m project at `C:\\Users\\user\\Desktop\\GitHub\\Antiq.jl`\n" ] } ], @@ -98,7 +96,7 @@ "using Revise\n", "\n", "Pkg.activate(\"./\")\n", - "Pkg.resolve()\n", + "# Pkg.resolve()\n", "using Antiq" ] }, @@ -240,18 +238,18 @@ "output_type": "stream", "text": [ "\u001b[32m\u001b[1m Testing\u001b[22m\u001b[39m Antiq\n", - "\u001b[32m\u001b[1m Status\u001b[22m\u001b[39m `C:\\Users\\user\\AppData\\Local\\Temp\\jl_Qj7MVW\\Project.toml`\n", + "\u001b[32m\u001b[1m Status\u001b[22m\u001b[39m `C:\\Users\\user\\AppData\\Local\\Temp\\jl_firBPY\\Project.toml`\n", " \u001b[90m [be6e5d0e] \u001b[39mAntiq v0.0.16 `C:\\Users\\user\\Desktop\\GitHub\\Antiq.jl`\n", " \u001b[90m [b964fa9f] \u001b[39mLaTeXStrings v1.3.1\n", " \u001b[90m [23fbe1c1] \u001b[39mLatexify v0.16.1\n", " \u001b[90m [1fd47b50] \u001b[39mQuadGK v2.9.1\n", - "\u001b[32m⌃\u001b[39m\u001b[90m [276daf66] \u001b[39mSpecialFunctions v2.3.0\n", + " \u001b[90m [276daf66] \u001b[39mSpecialFunctions v2.3.1\n", " \u001b[90m [fd094767] \u001b[39mSuppressor v0.2.6\n", " \u001b[90m [0c5d862f] \u001b[39mSymbolics v5.10.0\n", " \u001b[90m [d6f4376e] \u001b[39mMarkdown `@stdlib/Markdown`\n", " \u001b[90m [de0858da] \u001b[39mPrintf `@stdlib/Printf`\n", " \u001b[90m [8dfed614] \u001b[39mTest `@stdlib/Test`\n", - "\u001b[32m\u001b[1m Status\u001b[22m\u001b[39m `C:\\Users\\user\\AppData\\Local\\Temp\\jl_Qj7MVW\\Manifest.toml`\n", + "\u001b[32m\u001b[1m Status\u001b[22m\u001b[39m `C:\\Users\\user\\AppData\\Local\\Temp\\jl_firBPY\\Manifest.toml`\n", " \u001b[90m [47edcb42] \u001b[39mADTypes v0.2.5\n", "\u001b[33m⌅\u001b[39m\u001b[90m [c3fe647b] \u001b[39mAbstractAlgebra v0.32.5\n", " \u001b[90m [1520ce14] \u001b[39mAbstractTrees v0.4.4\n", @@ -296,13 +294,13 @@ " \u001b[90m [3587e190] \u001b[39mInverseFunctions v0.1.12\n", " \u001b[90m [92d709cd] \u001b[39mIrrationalConstants v0.2.2\n", " \u001b[90m [82899510] \u001b[39mIteratorInterfaceExtensions v1.0.0\n", - "\u001b[32m⌃\u001b[39m\u001b[90m [692b3bcd] \u001b[39mJLLWrappers v1.4.1\n", + " \u001b[90m [692b3bcd] \u001b[39mJLLWrappers v1.5.0\n", " \u001b[90m [b964fa9f] \u001b[39mLaTeXStrings v1.3.1\n", " \u001b[90m [2ee39098] \u001b[39mLabelledArrays v1.14.0\n", " \u001b[90m [984bce1d] \u001b[39mLambertW v0.4.6\n", " \u001b[90m [23fbe1c1] \u001b[39mLatexify v0.16.1\n", " \u001b[90m [50d2b5c4] \u001b[39mLazy v0.15.1\n", - "\u001b[32m⌃\u001b[39m\u001b[90m [2ab3a3ac] \u001b[39mLogExpFunctions v0.3.24\n", + " \u001b[90m [2ab3a3ac] \u001b[39mLogExpFunctions v0.3.26\n", " \u001b[90m [1914dd2f] \u001b[39mMacroTools v0.5.11\n", " \u001b[90m [e1d29d7a] \u001b[39mMissings v1.1.0\n", " \u001b[90m [102ac46a] \u001b[39mMultivariatePolynomials v0.5.3\n", @@ -312,7 +310,7 @@ " \u001b[90m [90014a1f] \u001b[39mPDMats v0.11.30\n", " \u001b[90m [d236fae5] \u001b[39mPreallocationTools v0.4.12\n", " \u001b[90m [aea7be01] \u001b[39mPrecompileTools v1.2.0\n", - "\u001b[32m⌃\u001b[39m\u001b[90m [21216c6a] \u001b[39mPreferences v1.4.0\n", + " \u001b[90m [21216c6a] \u001b[39mPreferences v1.4.1\n", " \u001b[90m [27ebfcd6] \u001b[39mPrimes v0.5.5\n", " \u001b[90m [1fd47b50] \u001b[39mQuadGK v2.9.1\n", " \u001b[90m [fb686558] \u001b[39mRandomExtensions v0.4.4\n", @@ -328,7 +326,7 @@ " \u001b[90m [efcf1570] \u001b[39mSetfield v1.1.1\n", " \u001b[90m [66db9d55] \u001b[39mSnoopPrecompile v1.0.3\n", " \u001b[90m [a2af1166] \u001b[39mSortingAlgorithms v1.2.0\n", - "\u001b[32m⌃\u001b[39m\u001b[90m [276daf66] \u001b[39mSpecialFunctions v2.3.0\n", + " \u001b[90m [276daf66] \u001b[39mSpecialFunctions v2.3.1\n", " \u001b[90m [90137ffa] \u001b[39mStaticArrays v1.7.0\n", " \u001b[90m [1e83bf80] \u001b[39mStaticArraysCore v1.4.2\n", " \u001b[90m [82ae8749] \u001b[39mStatsAPI v1.7.0\n", @@ -390,11 +388,11 @@ " \u001b[90m [8e850b90] \u001b[39mlibblastrampoline_jll v5.1.1+0 `@stdlib/libblastrampoline_jll`\n", " \u001b[90m [8e850ede] \u001b[39mnghttp2_jll v1.48.0+0 `@stdlib/nghttp2_jll`\n", " \u001b[90m [3f19e933] \u001b[39mp7zip_jll v17.4.0+0 `@stdlib/p7zip_jll`\n", - "\u001b[36m\u001b[1m Info\u001b[22m\u001b[39m Packages marked with \u001b[32m⌃\u001b[39m and \u001b[33m⌅\u001b[39m have new versions available, but those with \u001b[33m⌅\u001b[39m are restricted by compatibility constraints from upgrading.\n", + "\u001b[36m\u001b[1m Info\u001b[22m\u001b[39m Packages marked with \u001b[33m⌅\u001b[39m have new versions available but compatibility constraints restrict them from upgrading.\n", "\u001b[32m\u001b[1m Testing\u001b[22m\u001b[39m Running tests...\n", "WARNING: replacing module MorsePotential1669411154543485985.\n", "\u001b[0m\u001b[1mTest Summary: | \u001b[22m\u001b[32m\u001b[1mPass \u001b[22m\u001b[39m\u001b[36m\u001b[1mTotal \u001b[22m\u001b[39m\u001b[0m\u001b[1m Time\u001b[22m\n", - "Antiq.jl | \u001b[32m2052 \u001b[39m\u001b[36m 2052 \u001b[39m\u001b[0m1m33.8s\n", + "Antiq.jl | \u001b[32m2052 \u001b[39m\u001b[36m 2052 \u001b[39m\u001b[0m1m20.7s\n", "\u001b[32m\u001b[1m Testing\u001b[22m\u001b[39m Antiq tests passed \n" ] } @@ -426,6 +424,7 @@ " juliaup update\n", "\n", "to install Julia 1.9.4+0.x64.w64.mingw32 and update the `release` channel to that version.\n", + " Activating project at `C:\\Users\\user\\Desktop\\GitHub\\Antiq.jl`\n", "┌ Info: Weaving chunk 1 from line 55\n", "└ progress = 0.0\n", "┌ Info: Weaving chunk 2 from line 62\n", @@ -440,21 +439,7 @@ "└ progress = 0.8333333333333334\n", "WARNING: replacing module InfinitePotentialWell8472295553246349096.\n", "┌ Info: Weaved all chunks\n", - "└ progress = 1\n" - ] - }, - { - "name": "stdout", - "output_type": "stream", - "text": [ - "m.match = \".jl\\\")\\n```\\n\\n```\"\n", - "m.offset = " - ] - }, - { - "name": "stderr", - "output_type": "stream", - "text": [ + "└ progress = 1\n", "[ Info: Weaved to C:\\Users\\user\\Desktop\\GitHub\\Antiq.jl\\docs\\src\\InfinitePotentialWell.md\n" ] }, @@ -462,9 +447,10 @@ "name": "stdout", "output_type": "stream", "text": [ - "3032\n", + "m.match = \".jl\\\")\\n```\\n\\n```\"\n", + "m.offset = 3032\n", "m.match = \"```\\n```\\n\\n\\n\"\n", - "m.offset = 25354\n" + "m.offset = 25373\n" ] }, { @@ -486,8 +472,11 @@ "┌ Info: Weaving chunk 7 from line 152\n", "└ progress = 0.8571428571428571\n", "WARNING: replacing module HarmonicOscillator3094684212771689889.\n", + "┌ Warning: ERROR: LoadError occurred, including output in Weaved document\n", + "└ @ Weave C:\\Users\\user\\.julia\\packages\\Weave\\f7Ly3\\src\\run.jl:224\n", "┌ Info: Weaved all chunks\n", - "└ progress = 1\n" + "└ progress = 1\n", + "[ Info: Weaved to C:\\Users\\user\\Desktop\\GitHub\\Antiq.jl\\docs\\src\\HarmonicOscillator.md\n" ] }, { @@ -495,16 +484,13 @@ "output_type": "stream", "text": [ "m.match = \".jl\\\")\\n```\\n\\n```\"\n", - "m.offset = 6290\n", - "m.match = \"```\\n```\\n\\n\\n\"\n", - "m.offset = 32717\n" + "m.offset = 6290\n" ] }, { "name": "stderr", "output_type": "stream", "text": [ - "[ Info: Weaved to C:\\Users\\user\\Desktop\\GitHub\\Antiq.jl\\docs\\src\\HarmonicOscillator.md\n", "┌ Info: Weaving chunk 1 from line 85\n", "└ progress = 0.0\n", "┌ Info: Weaving chunk 2 from line 92\n", @@ -521,8 +507,11 @@ "┌ Info: Weaving chunk 7 from line 152\n", "└ progress = 0.8571428571428571\n", "WARNING: replacing module MorsePotential1669411154543485985.\n", + "┌ Warning: ERROR: LoadError occurred, including output in Weaved document\n", + "└ @ Weave C:\\Users\\user\\.julia\\packages\\Weave\\f7Ly3\\src\\run.jl:224\n", "┌ Info: Weaved all chunks\n", - "└ progress = 1\n" + "└ progress = 1\n", + "[ Info: Weaved to C:\\Users\\user\\Desktop\\GitHub\\Antiq.jl\\docs\\src\\MorsePotential.md\n" ] }, { @@ -530,16 +519,13 @@ "output_type": "stream", "text": [ "m.match = \".jl\\\")\\n```\\n\\n```\"\n", - "m.offset = 5890\n", - "m.match = \"```\\n```\\n\\n\\n\"\n", - "m.offset = 53552\n" + "m.offset = 5890\n" ] }, { "name": "stderr", "output_type": "stream", "text": [ - "[ Info: Weaved to C:\\Users\\user\\Desktop\\GitHub\\Antiq.jl\\docs\\src\\MorsePotential.md\n", "┌ Info: Weaving chunk 1 from line 156\n", "└ progress = 0.0\n", "┌ Info: Weaving chunk 2 from line 163\n", @@ -567,9 +553,9 @@ "output_type": "stream", "text": [ "m.match = \".jl\\\")\\n```\\n\\n```\"\n", - "m.offset = 13690\n", + "m.offset = 13672\n", "m.match = \"```\\n```\\n\\n\\n\"\n", - "m.offset = 104725\n" + "m.offset = 104714\n" ] }, { @@ -584,7 +570,7 @@ { "data": { "text/plain": [ - "Process(`\u001b[4mjulia\u001b[24m \u001b[4m--project=docs/\u001b[24m \u001b[4m-e\u001b[24m \u001b[4m'cd(\"docs\"); include(\"jmd2md.jl\")'\u001b[24m`, ProcessExited(0))" + "Process(`\u001b[4mjulia\u001b[24m \u001b[4m--project=docs/\u001b[24m \u001b[4m-e\u001b[24m \u001b[4m'using Pkg; Pkg.activate(\"./\"); cd(\"docs\"); include(\"jmd2md.jl\")'\u001b[24m`, ProcessExited(0))" ] }, "execution_count": 4, @@ -593,7 +579,7 @@ } ], "source": [ - "run(`julia --project=docs/ -e 'cd(\"docs\"); include(\"jmd2md.jl\")'`)" + "run(`julia --project=docs/ -e 'using Pkg; Pkg.activate(\"./\"); cd(\"docs\"); include(\"jmd2md.jl\")'`)" ] }, { @@ -617,22 +603,46 @@ " juliaup update\n", "\n", "to install Julia 1.9.4+0.x64.w64.mingw32 and update the `release` channel to that version.\n", + " Activating project at `C:\\Users\\user\\Desktop\\GitHub\\Antiq.jl`\n", "[ Info: SetupBuildDirectory: setting up build directory.\n", "[ Info: Doctest: running doctests.\n", "[ Info: ExpandTemplates: expanding markdown templates.\n", + "┌ Warning: In src\\HarmonicOscillator.md:1-3: `@meta` block has an unsupported keyword argument: CurrentModule\n", + "└ @ Documenter C:\\Users\\user\\.julia\\packages\\Documenter\\ifiMq\\src\\expander_pipeline.jl:298\n", + "┌ Warning: In src\\HydrogenAtom.md:1-3: `@meta` block has an unsupported keyword argument: CurrentModule\n", + "└ @ Documenter C:\\Users\\user\\.julia\\packages\\Documenter\\ifiMq\\src\\expander_pipeline.jl:298\n", + "┌ Warning: In src\\InfinitePotentialWell.md:1-3: `@meta` block has an unsupported keyword argument: CurrentModule\n", + "└ @ Documenter C:\\Users\\user\\.julia\\packages\\Documenter\\ifiMq\\src\\expander_pipeline.jl:298\n", + "┌ Warning: In src\\MorsePotential.md:1-3: `@meta` block has an unsupported keyword argument: CurrentModule\n", + "└ @ Documenter C:\\Users\\user\\.julia\\packages\\Documenter\\ifiMq\\src\\expander_pipeline.jl:298\n", + "┌ Warning: In src\\index.md:1-3: `@meta` block has an unsupported keyword argument: CurrentModule\n", + "└ @ Documenter C:\\Users\\user\\.julia\\packages\\Documenter\\ifiMq\\src\\expander_pipeline.jl:298\n", "[ Info: CrossReferences: building cross-references.\n", "[ Info: CheckDocument: running document checks.\n", "[ Info: Populate: populating indices.\n", "[ Info: RenderDocument: rendering document.\n", "[ Info: HTMLWriter: rendering HTML pages.\n", + "┌ Warning: Unable to determine the repository root URL for the navbar link.\n", + "│ This can happen when a string is passed to the `repo` keyword of `makedocs`.\n", + "│ \n", + "│ To remove this warning, either pass a Remotes.Remote object to `repo` to completely\n", + "│ specify the remote repository, or explicitly set the remote URL by setting `repolink`\n", + "│ via `makedocs(format = HTML(repolink = \"...\"), ...)`.\n", + "└ @ Documenter.HTMLWriter C:\\Users\\user\\.julia\\packages\\Documenter\\ifiMq\\src\\html\\HTMLWriter.jl:707\n", + "┌ Warning: Generated HTML over size_threshold_warn limit: HydrogenAtom.md\n", + "│ Generated file size: 118.26 (KiB)\n", + "│ size_threshold_warn: 100.0 (KiB)\n", + "│ size_threshold: 200.0 (KiB)\n", + "│ HTML file: HydrogenAtom.html\n", + "└ @ Documenter.HTMLWriter C:\\Users\\user\\.julia\\packages\\Documenter\\ifiMq\\src\\html\\HTMLWriter.jl:1793\n", "┌ Warning: Documenter could not auto-detect the building environment Skipping deployment.\n", - "└ @ Documenter C:\\Users\\user\\.julia\\packages\\Documenter\\bYYzK\\src\\deployconfig.jl:75\n" + "└ @ Documenter C:\\Users\\user\\.julia\\packages\\Documenter\\ifiMq\\src\\deployconfig.jl:76\n" ] }, { "data": { "text/plain": [ - "Process(`\u001b[4mjulia\u001b[24m \u001b[4m--project=docs/\u001b[24m \u001b[4m-e\u001b[24m \u001b[4m'cd(\"docs\"); include(\"make.jl\")'\u001b[24m`, ProcessExited(0))" + "Process(`\u001b[4mjulia\u001b[24m \u001b[4m--project=docs/\u001b[24m \u001b[4m-e\u001b[24m \u001b[4m'using Pkg; Pkg.activate(\"./\"); cd(\"docs\"); include(\"make.jl\")'\u001b[24m`, ProcessExited(0))" ] }, "execution_count": 5, @@ -643,7 +653,7 @@ "source": [ "# run(`cmd /C cd \\& julia ./docs/make.jl`)\n", "# include(\"./docs/make.jl\")\n", - "run(`julia --project=docs/ -e 'cd(\"docs\"); include(\"make.jl\")'`)" + "run(`julia --project=docs/ -e 'using Pkg; Pkg.activate(\"./\"); cd(\"docs\"); include(\"make.jl\")'`)" ] } ], diff --git a/docs/src/HarmonicOscillator.md b/docs/src/HarmonicOscillator.md index c77f31a..a725c19 100644 --- a/docs/src/HarmonicOscillator.md +++ b/docs/src/HarmonicOscillator.md @@ -156,7 +156,7 @@ 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="") @@ -193,7 +193,7 @@ include("../../../test/HarmonicOscillator.jl") ```math \begin{aligned} H_{0}(x) - = e^{ - x^{2}} e^{x^{2}} + = e^{x^{2}} e^{ - x^{2}} &= 1 \\ &= 1 \end{aligned} @@ -219,12 +219,24 @@ include("../../../test/HarmonicOscillator.jl") \end{aligned} ``` -``n=3:`` ✔ +``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 ```math \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} ``` @@ -243,7 +255,7 @@ include("../../../test/HarmonicOscillator.jl") ```math \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} @@ -254,8 +266,8 @@ include("../../../test/HarmonicOscillator.jl") \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} ``` @@ -264,8 +276,8 @@ include("../../../test/HarmonicOscillator.jl") \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} ``` @@ -274,8 +286,8 @@ include("../../../test/HarmonicOscillator.jl") \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} ``` @@ -284,399 +296,16 @@ include("../../../test/HarmonicOscillator.jl") \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} + &= 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 Total Time -Hₙ(x) = (-1)ⁿ exp(x²) dⁿ/dxⁿ exp(-x²) = ... | 10 10 35.0s -``` - -#### Normalization & Orthogonality of $H_n(x)$ - -```math -\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% ✔ - 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8 8 18295126.8527386672794819 18295126.8527386635541916 0.0000000000000204% ✔ - 8 9 0.0000000016298145 0.0000000000000000 0.0000000000000000% ✔ - 9 0 0.0000000000001137 0.0000000000000000 0.0000000000000000% ✔ - 9 1 0.0000000000143249 0.0000000000000000 0.0000000000000000% ✔ - 9 2 -0.0000000000021032 0.0000000000000000 0.0000000000000000% ✔ - 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 $\psi_n(x)$ - -```math -\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% ✔ - 0 5 -0.0000000000000000 0.0000000000000000 0.0000000000000000% ✔ - 0 6 0.0000000000000001 0.0000000000000000 0.0000000000000000% ✔ - 0 7 0.0000000000000000 0.0000000000000000 0.0000000000000000% ✔ - 0 8 -0.0000000000000002 0.0000000000000000 0.0000000000000000% ✔ - 0 9 0.0000000000000000 0.0000000000000000 0.0000000000000000% ✔ - 1 0 0.0000000000000000 0.0000000000000000 0.0000000000000000% ✔ - 1 1 0.9999999999999435 1.0000000000000000 0.0000000000056510% ✔ - 1 2 0.0000000000000000 0.0000000000000000 0.0000000000000000% ✔ - 1 3 -0.0000000000000000 0.0000000000000000 0.0000000000000000% ✔ - 1 4 -0.0000000000000000 0.0000000000000000 0.0000000000000000% ✔ - 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6 3 0.0000000000000000 0.0000000000000000 0.0000000000000000% ✔ - 6 4 0.0000000000000003 0.0000000000000000 0.0000000000000000% ✔ - 6 5 0.0000000000000000 0.0000000000000000 0.0000000000000000% ✔ - 6 6 0.9999999999999993 1.0000000000000000 0.0000000000000666% ✔ - 6 7 -0.0000000000000000 0.0000000000000000 0.0000000000000000% ✔ - 6 8 0.0000000000000003 0.0000000000000000 0.0000000000000000% ✔ - 6 9 0.0000000000000000 0.0000000000000000 0.0000000000000000% ✔ - 7 0 0.0000000000000000 0.0000000000000000 0.0000000000000000% ✔ - 7 1 -0.0000000000000000 0.0000000000000000 0.0000000000000000% ✔ - 7 2 0.0000000000000000 0.0000000000000000 0.0000000000000000% ✔ - 7 3 0.0000000000000001 0.0000000000000000 0.0000000000000000% ✔ - 7 4 0.0000000000000000 0.0000000000000000 0.0000000000000000% ✔ - 7 5 0.0000000000000002 0.0000000000000000 0.0000000000000000% ✔ - 7 6 -0.0000000000000000 0.0000000000000000 0.0000000000000000% ✔ - 7 7 1.0000000000000004 1.0000000000000000 0.0000000000000444% ✔ - 7 8 0.0000000000000000 0.0000000000000000 0.0000000000000000% ✔ - 7 9 0.0000000000000006 0.0000000000000000 0.0000000000000000% ✔ - 8 0 -0.0000000000000002 0.0000000000000000 0.0000000000000000% ✔ - 8 1 0.0000000000000000 0.0000000000000000 0.0000000000000000% ✔ - 8 2 -0.0000000000000007 0.0000000000000000 0.0000000000000000% ✔ - 8 3 -0.0000000000000000 0.0000000000000000 0.0000000000000000% ✔ - 8 4 -0.0000000000000005 0.0000000000000000 0.0000000000000000% ✔ - 8 5 0.0000000000000000 0.0000000000000000 0.0000000000000000% ✔ - 8 6 0.0000000000000003 0.0000000000000000 0.0000000000000000% ✔ - 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 $\langle H \rangle = 2 \langle V \rangle = 2 \langle T \rangle$. - -```math -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 - -```math - \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: - -```math -\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: - -```math -\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} +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 ``` -``` - 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 -``` diff --git a/docs/src/HydrogenAtom.md b/docs/src/HydrogenAtom.md index fd9289d..156da57 100644 --- a/docs/src/HydrogenAtom.md +++ b/docs/src/HydrogenAtom.md @@ -253,7 +253,7 @@ 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 ``` @@ -265,7 +265,7 @@ 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 @@ -280,7 +280,7 @@ 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\$") @@ -386,8 +386,8 @@ include("../../../test/HydrogenAtom.jl") \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} ``` @@ -426,8 +426,8 @@ include("../../../test/HydrogenAtom.jl") \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} ``` @@ -463,7 +463,7 @@ include("../../../test/HydrogenAtom.jl") ``` 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 +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 $P_n^m(x)$ @@ -829,7 +829,7 @@ Pₙᵐ(x) = √(1-x²)ᵐ dᵐ/dxᵐ Pₙ(x); Pₙ(x) = 1/(2ⁿn!) dⁿ/dxⁿ ( 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 +∫Pᵢᵐ(x)Pⱼᵐ(x)dx = 2(j+m)!/(2j+1)(j-m)! δᵢⱼ | 355 355 1.2s ``` #### Normalization & Orthogonality of $Y_{lm}(\theta,\varphi)$ @@ -925,7 +925,7 @@ Y_{lm}(\theta,\varphi)^* Y_{l'm'}(\theta,\varphi) \sin(\theta) 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 +∫Yₗ₁ₘ₁(θ,φ)Yₗ₂ₘ₂(θ,φ)sinθdθdφ = δₗ₁ₗ₂δₘ₁ₘ₂ | 81 81 2.7s ``` #### Associated Laguerre Polynomials $L_n^{k}(x)$ @@ -944,7 +944,7 @@ Test Summary: | Pass Total Time ```math \begin{aligned} L_{0}^{0}(x) - = e^{ - x} e^{x} + = e^{x} e^{ - x} &= 1 \\ &= 1 \\ &= 1 @@ -955,7 +955,7 @@ Test Summary: | Pass Total Time ```math \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 @@ -966,7 +966,7 @@ Test Summary: | Pass Total Time ```math \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 @@ -977,7 +977,7 @@ Test Summary: | Pass Total Time ```math \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} @@ -988,7 +988,7 @@ Test Summary: | Pass Total Time ```math \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 @@ -999,7 +999,7 @@ Test Summary: | Pass Total Time ```math \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 @@ -1010,10 +1010,10 @@ Test Summary: | Pass Total Time ```math \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} ``` @@ -1021,7 +1021,7 @@ Test Summary: | Pass Total Time ```math \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} @@ -1032,7 +1032,7 @@ Test Summary: | Pass Total Time ```math \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 @@ -1043,7 +1043,7 @@ Test Summary: | Pass Total Time ```math \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 @@ -1054,10 +1054,10 @@ Test Summary: | Pass Total Time ```math \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} ``` @@ -1065,10 +1065,10 @@ Test Summary: | Pass Total Time ```math \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} ``` @@ -1076,7 +1076,7 @@ Test Summary: | Pass Total Time ```math \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} @@ -1087,7 +1087,7 @@ Test Summary: | Pass Total Time ```math \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 @@ -1098,7 +1098,7 @@ Test Summary: | Pass Total Time ```math \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 @@ -1324,7 +1324,7 @@ Replace $n+k$ with $n$ for [the definition of Wolfram MathWorld](https://mathwor 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 +∫exp(-x)xᵏLᵢᵏ(x)Lⱼᵏ(x)dx = (2i+k)!/(i+k)! δᵢⱼ | 204 204 1.0s ``` #### Normalization of $R_{nl}(r)$ @@ -1380,7 +1380,7 @@ Test Summary: | Pass Total Time 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 +∫|Rₙₗ(r)|²r²dr = δₙ₁ₙ₂δₗ₁ₗ₂ | 45 45 0.9s ``` #### Expected Value of $r$ @@ -1444,7 +1444,7 @@ Reference: 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 +∫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 $r^2$ @@ -1508,7 +1508,7 @@ Reference: 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 +∫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 @@ -1531,7 +1531,7 @@ The virial theorem $2\langle T \rangle + \langle V \rangle = 0$ and the definiti 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 +<ψₙ|V|ψₙ> / 2 = Eₙ | 10 10 0.8s ``` #### Normalization & Orthogonality of $\psi_n(r,\theta,\varphi)$ @@ -1737,7 +1737,7 @@ Test Summary: | Pass Total Time 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/docs/src/InfinitePotentialWell.md b/docs/src/InfinitePotentialWell.md index f611e80..71977bd 100644 --- a/docs/src/InfinitePotentialWell.md +++ b/docs/src/InfinitePotentialWell.md @@ -108,7 +108,7 @@ 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="") @@ -239,7 +239,7 @@ include("../../../test/InfinitePotentialWell.jl") 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 +<ψᵢ|ψⱼ> = ∫ψₙ*ψₙdx = δᵢⱼ | 100 100 2.3s ``` #### Eigen Values @@ -386,7 +386,7 @@ are given by the sum of 2 Taylor series: 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 +<ψₙ|H|ψₙ> = ∫ψₙ*Tψₙdx = Eₙ | 80 80 0.9s ``` #### Expected Value of $x$ @@ -406,7 +406,7 @@ Reference: 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 +<ψₙ|x|ψₙ> = L/2 | 4 4 0.5s ``` #### Expected Value of $x^2$ @@ -503,7 +503,7 @@ are given by the sum of 2 Taylor series: 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 +<ψₙ|p|ψₙ> = ∫ψₙ*(-iℏd/dx)ψₙdx = 0 | 4 4 0.7s ``` #### Expected Value of $p^2$ @@ -583,6 +583,6 @@ are given by the sum of 2 Taylor series: 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/docs/src/MorsePotential.md b/docs/src/MorsePotential.md index 3752f84..52f05e7 100644 --- a/docs/src/MorsePotential.md +++ b/docs/src/MorsePotential.md @@ -157,7 +157,7 @@ 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) @@ -197,7 +197,7 @@ include("../../../test/MorsePotential.jl") ```math \begin{aligned} L_{0}^{(0)}(x) - = e^{ - x} e^{x} + = e^{x} e^{ - x} &= 1 \\ &= 1 \end{aligned} @@ -207,7 +207,7 @@ include("../../../test/MorsePotential.jl") ```math \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} @@ -227,7 +227,7 @@ include("../../../test/MorsePotential.jl") ```math \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} @@ -237,7 +237,7 @@ include("../../../test/MorsePotential.jl") ```math \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} @@ -247,7 +247,7 @@ include("../../../test/MorsePotential.jl") ```math \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} @@ -257,39 +257,75 @@ include("../../../test/MorsePotential.jl") ```math \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} + = \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:`` ✔ +``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 ```math \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} + &= 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:`` ✔ +``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 ```math \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} + = \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:`` ✔ +``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 ```math \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} ``` @@ -297,686 +333,105 @@ include("../../../test/MorsePotential.jl") ```math \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} + = \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:`` ✔ +``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 ```math \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} + &= 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:`` ✔ +``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 ```math \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} + = \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:`` ✔ +``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 ```math \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} + &= 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:`` ✔ +``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 ```math \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} + &= 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 Total Time -Lₙ⁽ᵅ⁾(x) = x⁻ᵅeˣ/n! dⁿ/dxⁿ xⁿ⁺ᵅe⁻ˣ | 15 15 20.0s -``` - -#### Normalization & Orthogonality of $L_n^{(\alpha)}(x)$ - -```math -\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} -``` - -``` - α n m numerical analytical |error| -0.1 0 0 1.7724538509055159 1.7724538509055159 0.0000000000000000% ✔ -0.1 0 1 0.0000000000000000 0.0000000000000000 0.0000000000000000% ✔ -0.1 0 2 0.0000000000000000 0.0000000000000000 0.0000000000000000% ✔ -0.1 0 3 0.0000000000000000 0.0000000000000000 0.0000000000000000% ✔ -0.1 0 4 0.0000000000000000 0.0000000000000000 0.0000000000000000% ✔ -0.1 0 5 0.0000000000000000 0.0000000000000000 0.0000000000000000% ✔ -0.1 0 6 0.0000000000000000 0.0000000000000000 0.0000000000000000% ✔ -0.1 0 7 0.0000000000000000 0.0000000000000000 0.0000000000000000% ✔ -0.1 0 8 0.0000000000000000 0.0000000000000000 0.0000000000000000% ✔ -0.1 0 9 0.0000000000000000 0.0000000000000000 0.0000000000000000% ✔ -0.1 1 0 0.0000000000000000 0.0000000000000000 0.0000000000000000% ✔ -0.1 1 1 3.5449077018110318 3.5449077018110318 0.0000000000000000% ✔ -0.1 1 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✔ -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 $\psi_n(r)$ - -```math -\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 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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 - -```math - \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: - -```math -\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: - -```math -\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} +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 ``` -``` - 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 -``` diff --git a/docs/src/assets/fig/HarmonicOscillator_6_1.png b/docs/src/assets/fig/HarmonicOscillator_6_1.png index 0bed97a..89d7af5 100644 Binary files a/docs/src/assets/fig/HarmonicOscillator_6_1.png and b/docs/src/assets/fig/HarmonicOscillator_6_1.png differ diff --git a/docs/src/assets/fig/HydrogenAtom_6_1.png b/docs/src/assets/fig/HydrogenAtom_6_1.png index c8390cc..b5c94f6 100644 Binary files a/docs/src/assets/fig/HydrogenAtom_6_1.png and b/docs/src/assets/fig/HydrogenAtom_6_1.png differ diff --git a/docs/src/assets/fig/HydrogenAtom_7_1.png b/docs/src/assets/fig/HydrogenAtom_7_1.png index 32ffcf8..fc7e833 100644 Binary files a/docs/src/assets/fig/HydrogenAtom_7_1.png and b/docs/src/assets/fig/HydrogenAtom_7_1.png differ diff --git a/docs/src/assets/fig/HydrogenAtom_8_1.png b/docs/src/assets/fig/HydrogenAtom_8_1.png index acc279e..1c767fc 100644 Binary files a/docs/src/assets/fig/HydrogenAtom_8_1.png and b/docs/src/assets/fig/HydrogenAtom_8_1.png differ diff --git a/docs/src/assets/fig/InfinitePotentialWell_5_1.png b/docs/src/assets/fig/InfinitePotentialWell_5_1.png index ca2d25d..f64791d 100644 Binary files a/docs/src/assets/fig/InfinitePotentialWell_5_1.png and b/docs/src/assets/fig/InfinitePotentialWell_5_1.png differ diff --git a/docs/src/assets/fig/MorsePotential_6_1.png b/docs/src/assets/fig/MorsePotential_6_1.png index 776f9d6..b724c27 100644 Binary files a/docs/src/assets/fig/MorsePotential_6_1.png and b/docs/src/assets/fig/MorsePotential_6_1.png differ diff --git a/docs/src/jmd/HarmonicOscillator.jmd b/docs/src/jmd/HarmonicOscillator.jmd index ff1052b..1b6f3a4 100644 --- a/docs/src/jmd/HarmonicOscillator.jmd +++ b/docs/src/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/docs/src/jmd/HydrogenAtom.jmd b/docs/src/jmd/HydrogenAtom.jmd index fe0792a..d1d7336 100644 --- a/docs/src/jmd/HydrogenAtom.jmd +++ b/docs/src/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/docs/src/jmd/InfinitePotentialWell.jmd b/docs/src/jmd/InfinitePotentialWell.jmd index a71cba3..3c0b509 100644 --- a/docs/src/jmd/InfinitePotentialWell.jmd +++ b/docs/src/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/docs/src/jmd/MorsePotential.jmd b/docs/src/jmd/MorsePotential.jmd index e03765c..d7005ca 100644 --- a/docs/src/jmd/MorsePotential.jmd +++ b/docs/src/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)