From 914329ac52a8424826115d3fd73e7cc4d912d7a3 Mon Sep 17 00:00:00 2001 From: ajarifi <53268007+ajarifi@users.noreply.github.com> Date: Fri, 24 May 2024 10:11:59 +0900 Subject: [PATCH] Add eigenvalue test for rigid rotor and delta potential --- test/DeltaPotential.jl | 41 +++++ test/RigidRotor.jl | 51 ++++++ test/result/DeltaPotential.log | 43 +++++ test/result/RigidRotor.log | 311 +++++++++++++++++++-------------- 4 files changed, 311 insertions(+), 135 deletions(-) diff --git a/test/DeltaPotential.jl b/test/DeltaPotential.jl index 17f5388..94e2c00 100644 --- a/test/DeltaPotential.jl +++ b/test/DeltaPotential.jl @@ -31,3 +31,44 @@ end println("""``` """) +# <ψₙ|H|ψₙ> = ∫ψₙ*Tψₙdx = Eₙ + +println(raw""" +#### Eigen Values + +```math + \begin{aligned} + E_n + &= \int_{-\infty}^{\infty} \psi^\ast(x) \hat{H} \psi(x) ~\mathrm{d}x \\ + &= \int_{-\infty}^{\infty} \psi^\ast(x) \left[ \hat{V} + \hat{T} \right] \psi(x) ~\mathrm{d}x \\ + &= \int_{-\infty}^{\infty} \psi^\ast(x) \left[ -\alpha\delta(x) - \frac{\hbar^2}{2m} \frac{\mathrm{d}^{2}}{\mathrm{d} x^{2}} \right] \psi(x) ~\mathrm{d}x \\ + &= \int_{-\infty}^{\infty} \psi^\ast(x) \left[ -\alpha\delta(x) - \frac{\hbar^2}{2m} (\kappa^2 -2\kappa \delta(x))\right]\psi(x) ~\mathrm{d}x \\ + \end{aligned} +``` +where the $\kappa=m\alpha/\hbar^2$ and the integration with the delta function yeild a term proportional to the wave function at the origin $|\psi(0)|^2$. +```""") + +@testset "<ψₙ|H|ψₙ> = ∫ψₙ*Hψₙdx = Eₙ" begin + function ψHψ(DP) + κ = DP.m * DP.α/ DP.ℏ^2 + T = -DP.ℏ^2 / (2 * DP.m) *( κ^2 - 2κ * ψ(DP,0)^2 ) + V = -DP.α * ψ(DP, 0)^2 + return T + V + end + + println(" α | m | ℏ | analytical | numerical ") + println("--- | --- | --- | ----------------- | ----------------- ") + for α in [0.1, 0.5, 2.0] + for m in [0.1, 0.5, 2.0] + for ℏ in [0.1, 0.5, 2.0] + DP = DeltaPotential(α=α, m=m, ℏ=ℏ) + analytical = E(DP) + numerical = ψHψ(DP) + acceptance = iszero(analytical) ? isapprox(analytical, numerical, atol=1e-4) : isapprox(analytical, numerical, rtol=1e-4) + @printf("%.1f | %.1f | %.1f | %17.12f | %17.12f %s\n", α, m, ℏ,analytical, numerical, acceptance ? "✔" : "✗") + end + end + end + end + + println("""```""") \ No newline at end of file diff --git a/test/RigidRotor.jl b/test/RigidRotor.jl index 1b9bafd..2a2b8ea 100644 --- a/test/RigidRotor.jl +++ b/test/RigidRotor.jl @@ -123,5 +123,56 @@ Y_{lm}(\theta,\varphi)^* Y_{l'm'}(\theta,\varphi) \sin(\theta) end end +println("""``` +""") + + +# <ψₙ|H|ψₙ> = ∫ψₙ*Tψₙdx = Eₙ + +println(raw""" +#### Eigen Values + +```math + \begin{aligned} + E_n + &= \int_{0}^{2\pi}~\mathrm{d}\varphi \int_{0}^{\pi}~\mathrm{d}\theta ~ \psi^\ast_{lm}(\theta,\varphi) \hat{H} \psi_{lm}(\theta,\varphi) \\ + &= \int_{0}^{2\pi}~\mathrm{d}\varphi \int_{0}^{\pi}~\mathrm{d}\theta ~ \psi^\ast_{lm}(\theta,\varphi) \left[ \hat{V} + \hat{T} \right] \psi_{lm}(\theta,\varphi) \\ + &= \int_{0}^{2\pi}~\mathrm{d}\varphi \int_{0}^{\pi}~\mathrm{d}\theta ~ \psi^\ast_{lm}(\theta,\varphi) \left[ 0 + \frac{l(l+1)}{2I} \right] \psi_{lm}(\theta,\varphi) \\ + &= \int_{0}^{2\pi}~\mathrm{d}\varphi \int_{0}^{\pi}~\mathrm{d}\theta ~ \frac{l(l+1)}{2I} |\psi_{lm}(\theta,\varphi)|^2 + \end{aligned} +``` +where $l$ is angular momentum quantum number and $I$ is the moment of intertia. +```""") + +@testset "<ψₙ|H|ψₙ> = ∫ψₙ*Hψₙdx = Eₙ" begin + function ψHψ(RR;l) + μ = (1/RR.m₁ + 1/RR.m₂)^(-1) + I = μ * RR.R^2 + YY = real(quadgk(φ -> quadgk(θ -> conj(Y(RR,θ,φ,l=l,m=0)) * Y(RR,θ,φ,l=l,m=0) * sin(θ), 0, π, maxevals=10)[1], 0, 2π, maxevals=10)[1]) + T = l*(l+1)/2/I * YY + return T + end + + ℏ=1 + println(" m₁ | m₂ | R | l | analytical | numerical ") + println("--- | --- | --- | --- | ----------------- | ----------------- ") + for m₁ in [0.5, 2.0] + for m₂ in [0.5, 2.0] + for R in [0.5, 2.0] + for l in [1, 2, 3] + RR = RigidRotor(m₁=m₁, m₂=m₂, R=R, ℏ=ℏ) + analytical = E(RR,l=l) + numerical = ψHψ(RR,l=l) + acceptance = iszero(analytical) ? isapprox(analytical, numerical, atol=1e-4) : isapprox(analytical, numerical, rtol=1e-4) + @printf("%.1f | %.1f | %.1f | %.1f | %17.12f | %17.12f %s\n", m₁, m₂, R, l,analytical, numerical, acceptance ? "✔" : "✗") + end + end + end + end +end + + + + println("""``` """) diff --git a/test/result/DeltaPotential.log b/test/result/DeltaPotential.log index 38dee7e..1c92ec4 100644 --- a/test/result/DeltaPotential.log +++ b/test/result/DeltaPotential.log @@ -36,4 +36,47 @@ 7.0 | 7.0 | 7.0 | 1.000000000000 | 1.000000000000 ✔ ``` +#### Eigen Values + +```math + \begin{aligned} + E_n + &= \int_{-\infty}^{\infty} \psi^\ast(x) \hat{H} \psi(x) ~\mathrm{d}x \\ + &= \int_{-\infty}^{\infty} \psi^\ast(x) \left[ \hat{V} + \hat{T} \right] \psi(x) ~\mathrm{d}x \\ + &= \int_{-\infty}^{\infty} \psi^\ast(x) \left[ -\alpha\delta(x) - \frac{\hbar^2}{2m} \frac{\mathrm{d}^{2}}{\mathrm{d} x^{2}} \right] \psi(x) ~\mathrm{d}x \\ + &= \int_{-\infty}^{\infty} \psi^\ast(x) \left[ -\alpha\delta(x) - \frac{\hbar^2}{2m} (\kappa^2 -2\kappa \delta(x))\right]\psi(x) ~\mathrm{d}x \\ + \end{aligned} +``` +where the $\kappa=m\alpha/\hbar^2$ and the integration with the delta function yeild a term proportional to the wave function at the origin $|\psi(0)|^2$. +``` + α | m | ℏ | analytical | numerical +--- | --- | --- | ----------------- | ----------------- +0.1 | 0.1 | 0.1 | -0.050000000000 | -0.050000000000 ✔ +0.1 | 0.1 | 0.5 | -0.002000000000 | -0.002000000000 ✔ +0.1 | 0.1 | 2.0 | -0.000125000000 | -0.000125000000 ✔ +0.1 | 0.5 | 0.1 | -0.250000000000 | -0.250000000000 ✔ +0.1 | 0.5 | 0.5 | -0.010000000000 | -0.010000000000 ✔ +0.1 | 0.5 | 2.0 | -0.000625000000 | -0.000625000000 ✔ +0.1 | 2.0 | 0.1 | -1.000000000000 | -1.000000000000 ✔ +0.1 | 2.0 | 0.5 | -0.040000000000 | -0.040000000000 ✔ +0.1 | 2.0 | 2.0 | -0.002500000000 | -0.002500000000 ✔ +0.5 | 0.1 | 0.1 | -1.250000000000 | -1.250000000000 ✔ +0.5 | 0.1 | 0.5 | -0.050000000000 | -0.050000000000 ✔ +0.5 | 0.1 | 2.0 | -0.003125000000 | -0.003125000000 ✔ +0.5 | 0.5 | 0.1 | -6.250000000000 | -6.250000000000 ✔ +0.5 | 0.5 | 0.5 | -0.250000000000 | -0.250000000000 ✔ +0.5 | 0.5 | 2.0 | -0.015625000000 | -0.015625000000 ✔ +0.5 | 2.0 | 0.1 | -25.000000000000 | -25.000000000000 ✔ +0.5 | 2.0 | 0.5 | -1.000000000000 | -1.000000000000 ✔ +0.5 | 2.0 | 2.0 | -0.062500000000 | -0.062500000000 ✔ +2.0 | 0.1 | 0.1 | -20.000000000000 | -20.000000000000 ✔ +2.0 | 0.1 | 0.5 | -0.800000000000 | -0.800000000000 ✔ +2.0 | 0.1 | 2.0 | -0.050000000000 | -0.050000000000 ✔ +2.0 | 0.5 | 0.1 | -100.000000000000 | -100.000000000000 ✔ +2.0 | 0.5 | 0.5 | -4.000000000000 | -4.000000000000 ✔ +2.0 | 0.5 | 2.0 | -0.250000000000 | -0.250000000000 ✔ +2.0 | 2.0 | 0.1 | -400.000000000000 | -400.000000000000 ✔ +2.0 | 2.0 | 0.5 | -16.000000000000 | -16.000000000000 ✔ +2.0 | 2.0 | 2.0 | -1.000000000000 | -1.000000000000 ✔ +``` diff --git a/test/result/RigidRotor.log b/test/result/RigidRotor.log index 9917ba7..6d17c0e 100644 --- a/test/result/RigidRotor.log +++ b/test/result/RigidRotor.log @@ -172,20 +172,20 @@ 0 | 0 | 2 | 0.000000000000 | 0.000000000000 ✔ 0 | 0 | 3 | 0.000000000000 | -0.000000000000 ✔ 0 | 0 | 4 | 0.000000000000 | 0.000000000000 ✔ - 0 | 0 | 5 | 0.000000000000 | 0.000000000000 ✔ - 0 | 0 | 6 | 0.000000000000 | 0.000000000000 ✔ + 0 | 0 | 5 | 0.000000000000 | -0.000000000000 ✔ + 0 | 0 | 6 | 0.000000000000 | -0.000000000000 ✔ 0 | 0 | 7 | 0.000000000000 | 0.000000000000 ✔ - 0 | 0 | 8 | 0.000000000000 | 0.000000000000 ✔ + 0 | 0 | 8 | 0.000000000000 | -0.000000000000 ✔ 0 | 0 | 9 | 0.000000000000 | -0.000000000000 ✔ 0 | 1 | 0 | 0.000000000000 | 0.000000000000 ✔ 0 | 1 | 1 | 0.666666666667 | 0.666666666667 ✔ 0 | 1 | 2 | 0.000000000000 | 0.000000000000 ✔ 0 | 1 | 3 | 0.000000000000 | -0.000000000000 ✔ 0 | 1 | 4 | 0.000000000000 | 0.000000000000 ✔ - 0 | 1 | 5 | 0.000000000000 | 0.000000000000 ✔ + 0 | 1 | 5 | 0.000000000000 | -0.000000000000 ✔ 0 | 1 | 6 | 0.000000000000 | 0.000000000000 ✔ - 0 | 1 | 7 | 0.000000000000 | 0.000000000000 ✔ - 0 | 1 | 8 | 0.000000000000 | 0.000000000000 ✔ + 0 | 1 | 7 | 0.000000000000 | -0.000000000000 ✔ + 0 | 1 | 8 | 0.000000000000 | -0.000000000000 ✔ 0 | 1 | 9 | 0.000000000000 | -0.000000000000 ✔ 0 | 2 | 0 | 0.000000000000 | 0.000000000000 ✔ 0 | 2 | 1 | 0.000000000000 | 0.000000000000 ✔ @@ -195,7 +195,7 @@ 0 | 2 | 5 | 0.000000000000 | 0.000000000000 ✔ 0 | 2 | 6 | 0.000000000000 | 0.000000000000 ✔ 0 | 2 | 7 | 0.000000000000 | 0.000000000000 ✔ - 0 | 2 | 8 | 0.000000000000 | 0.000000000000 ✔ + 0 | 2 | 8 | 0.000000000000 | -0.000000000000 ✔ 0 | 2 | 9 | 0.000000000000 | -0.000000000000 ✔ 0 | 3 | 0 | 0.000000000000 | -0.000000000000 ✔ 0 | 3 | 1 | 0.000000000000 | -0.000000000000 ✔ @@ -203,7 +203,7 @@ 0 | 3 | 3 | 0.285714285714 | 0.285714285714 ✔ 0 | 3 | 4 | 0.000000000000 | 0.000000000000 ✔ 0 | 3 | 5 | 0.000000000000 | -0.000000000000 ✔ - 0 | 3 | 6 | 0.000000000000 | 0.000000000000 ✔ + 0 | 3 | 6 | 0.000000000000 | -0.000000000000 ✔ 0 | 3 | 7 | 0.000000000000 | 0.000000000000 ✔ 0 | 3 | 8 | 0.000000000000 | -0.000000000000 ✔ 0 | 3 | 9 | 0.000000000000 | -0.000000000000 ✔ @@ -212,51 +212,51 @@ 0 | 4 | 2 | 0.000000000000 | 0.000000000000 ✔ 0 | 4 | 3 | 0.000000000000 | 0.000000000000 ✔ 0 | 4 | 4 | 0.222222222222 | 0.222222222222 ✔ - 0 | 4 | 5 | 0.000000000000 | -0.000000000000 ✔ + 0 | 4 | 5 | 0.000000000000 | 0.000000000000 ✔ 0 | 4 | 6 | 0.000000000000 | -0.000000000000 ✔ - 0 | 4 | 7 | 0.000000000000 | -0.000000000000 ✔ - 0 | 4 | 8 | 0.000000000000 | 0.000000000000 ✔ + 0 | 4 | 7 | 0.000000000000 | 0.000000000000 ✔ + 0 | 4 | 8 | 0.000000000000 | -0.000000000000 ✔ 0 | 4 | 9 | 0.000000000000 | 0.000000000000 ✔ - 0 | 5 | 0 | 0.000000000000 | 0.000000000000 ✔ - 0 | 5 | 1 | 0.000000000000 | 0.000000000000 ✔ + 0 | 5 | 0 | 0.000000000000 | -0.000000000000 ✔ + 0 | 5 | 1 | 0.000000000000 | -0.000000000000 ✔ 0 | 5 | 2 | 0.000000000000 | 0.000000000000 ✔ 0 | 5 | 3 | 0.000000000000 | -0.000000000000 ✔ - 0 | 5 | 4 | 0.000000000000 | -0.000000000000 ✔ + 0 | 5 | 4 | 0.000000000000 | 0.000000000000 ✔ 0 | 5 | 5 | 0.181818181818 | 0.181818181818 ✔ - 0 | 5 | 6 | 0.000000000000 | 0.000000000000 ✔ - 0 | 5 | 7 | 0.000000000000 | 0.000000000000 ✔ - 0 | 5 | 8 | 0.000000000000 | -0.000000000000 ✔ + 0 | 5 | 6 | 0.000000000000 | -0.000000000000 ✔ + 0 | 5 | 7 | 0.000000000000 | -0.000000000000 ✔ + 0 | 5 | 8 | 0.000000000000 | 0.000000000000 ✔ 0 | 5 | 9 | 0.000000000000 | -0.000000000000 ✔ - 0 | 6 | 0 | 0.000000000000 | 0.000000000000 ✔ + 0 | 6 | 0 | 0.000000000000 | -0.000000000000 ✔ 0 | 6 | 1 | 0.000000000000 | 0.000000000000 ✔ 0 | 6 | 2 | 0.000000000000 | 0.000000000000 ✔ - 0 | 6 | 3 | 0.000000000000 | 0.000000000000 ✔ + 0 | 6 | 3 | 0.000000000000 | -0.000000000000 ✔ 0 | 6 | 4 | 0.000000000000 | -0.000000000000 ✔ - 0 | 6 | 5 | 0.000000000000 | 0.000000000000 ✔ + 0 | 6 | 5 | 0.000000000000 | -0.000000000000 ✔ 0 | 6 | 6 | 0.153846153846 | 0.153846153846 ✔ 0 | 6 | 7 | 0.000000000000 | -0.000000000000 ✔ - 0 | 6 | 8 | 0.000000000000 | 0.000000000000 ✔ + 0 | 6 | 8 | 0.000000000000 | -0.000000000000 ✔ 0 | 6 | 9 | 0.000000000000 | 0.000000000000 ✔ 0 | 7 | 0 | 0.000000000000 | 0.000000000000 ✔ - 0 | 7 | 1 | 0.000000000000 | 0.000000000000 ✔ + 0 | 7 | 1 | 0.000000000000 | -0.000000000000 ✔ 0 | 7 | 2 | 0.000000000000 | 0.000000000000 ✔ 0 | 7 | 3 | 0.000000000000 | 0.000000000000 ✔ - 0 | 7 | 4 | 0.000000000000 | -0.000000000000 ✔ - 0 | 7 | 5 | 0.000000000000 | 0.000000000000 ✔ + 0 | 7 | 4 | 0.000000000000 | 0.000000000000 ✔ + 0 | 7 | 5 | 0.000000000000 | -0.000000000000 ✔ 0 | 7 | 6 | 0.000000000000 | -0.000000000000 ✔ 0 | 7 | 7 | 0.133333333333 | 0.133333333333 ✔ - 0 | 7 | 8 | 0.000000000000 | 0.000000000000 ✔ + 0 | 7 | 8 | 0.000000000000 | -0.000000000000 ✔ 0 | 7 | 9 | 0.000000000000 | -0.000000000000 ✔ - 0 | 8 | 0 | 0.000000000000 | 0.000000000000 ✔ - 0 | 8 | 1 | 0.000000000000 | 0.000000000000 ✔ - 0 | 8 | 2 | 0.000000000000 | 0.000000000000 ✔ + 0 | 8 | 0 | 0.000000000000 | -0.000000000000 ✔ + 0 | 8 | 1 | 0.000000000000 | -0.000000000000 ✔ + 0 | 8 | 2 | 0.000000000000 | -0.000000000000 ✔ 0 | 8 | 3 | 0.000000000000 | -0.000000000000 ✔ - 0 | 8 | 4 | 0.000000000000 | 0.000000000000 ✔ - 0 | 8 | 5 | 0.000000000000 | -0.000000000000 ✔ - 0 | 8 | 6 | 0.000000000000 | 0.000000000000 ✔ - 0 | 8 | 7 | 0.000000000000 | 0.000000000000 ✔ + 0 | 8 | 4 | 0.000000000000 | -0.000000000000 ✔ + 0 | 8 | 5 | 0.000000000000 | 0.000000000000 ✔ + 0 | 8 | 6 | 0.000000000000 | -0.000000000000 ✔ + 0 | 8 | 7 | 0.000000000000 | -0.000000000000 ✔ 0 | 8 | 8 | 0.117647058824 | 0.117647058824 ✔ - 0 | 8 | 9 | 0.000000000000 | 0.000000000000 ✔ + 0 | 8 | 9 | 0.000000000000 | -0.000000000000 ✔ 0 | 9 | 0 | 0.000000000000 | -0.000000000000 ✔ 0 | 9 | 1 | 0.000000000000 | -0.000000000000 ✔ 0 | 9 | 2 | 0.000000000000 | -0.000000000000 ✔ @@ -265,14 +265,14 @@ 0 | 9 | 5 | 0.000000000000 | -0.000000000000 ✔ 0 | 9 | 6 | 0.000000000000 | 0.000000000000 ✔ 0 | 9 | 7 | 0.000000000000 | -0.000000000000 ✔ - 0 | 9 | 8 | 0.000000000000 | 0.000000000000 ✔ + 0 | 9 | 8 | 0.000000000000 | -0.000000000000 ✔ 0 | 9 | 9 | 0.105263157895 | 0.105263157895 ✔ 1 | 1 | 1 | 1.333333333333 | 1.333333333333 ✔ 1 | 1 | 2 | 0.000000000000 | 0.000000000000 ✔ 1 | 1 | 3 | 0.000000000000 | 0.000000000000 ✔ 1 | 1 | 4 | 0.000000000000 | 0.000000000000 ✔ 1 | 1 | 5 | 0.000000000000 | 0.000000000000 ✔ - 1 | 1 | 6 | 0.000000000000 | -0.000000000000 ✔ + 1 | 1 | 6 | 0.000000000000 | 0.000000000000 ✔ 1 | 1 | 7 | 0.000000000000 | 0.000000000000 ✔ 1 | 1 | 8 | 0.000000000000 | 0.000000000000 ✔ 1 | 1 | 9 | 0.000000000000 | 0.000000000000 ✔ @@ -283,17 +283,17 @@ 1 | 2 | 5 | 0.000000000000 | -0.000000000000 ✔ 1 | 2 | 6 | 0.000000000000 | 0.000000000000 ✔ 1 | 2 | 7 | 0.000000000000 | 0.000000000000 ✔ - 1 | 2 | 8 | 0.000000000000 | -0.000000000000 ✔ - 1 | 2 | 9 | 0.000000000000 | 0.000000000000 ✔ + 1 | 2 | 8 | 0.000000000000 | 0.000000000000 ✔ + 1 | 2 | 9 | 0.000000000000 | -0.000000000000 ✔ 1 | 3 | 1 | 0.000000000000 | 0.000000000000 ✔ 1 | 3 | 2 | 0.000000000000 | 0.000000000000 ✔ 1 | 3 | 3 | 3.428571428571 | 3.428571428571 ✔ 1 | 3 | 4 | 0.000000000000 | 0.000000000000 ✔ 1 | 3 | 5 | 0.000000000000 | -0.000000000000 ✔ - 1 | 3 | 6 | 0.000000000000 | -0.000000000000 ✔ - 1 | 3 | 7 | 0.000000000000 | 0.000000000000 ✔ - 1 | 3 | 8 | 0.000000000000 | -0.000000000000 ✔ - 1 | 3 | 9 | 0.000000000000 | -0.000000000000 ✔ + 1 | 3 | 6 | 0.000000000000 | 0.000000000000 ✔ + 1 | 3 | 7 | 0.000000000000 | -0.000000000000 ✔ + 1 | 3 | 8 | 0.000000000000 | 0.000000000000 ✔ + 1 | 3 | 9 | 0.000000000000 | 0.000000000000 ✔ 1 | 4 | 1 | 0.000000000000 | 0.000000000000 ✔ 1 | 4 | 2 | 0.000000000000 | 0.000000000000 ✔ 1 | 4 | 3 | 0.000000000000 | 0.000000000000 ✔ @@ -301,7 +301,7 @@ 1 | 4 | 5 | 0.000000000000 | 0.000000000000 ✔ 1 | 4 | 6 | 0.000000000000 | 0.000000000000 ✔ 1 | 4 | 7 | 0.000000000000 | 0.000000000000 ✔ - 1 | 4 | 8 | 0.000000000000 | -0.000000000000 ✔ + 1 | 4 | 8 | 0.000000000000 | 0.000000000000 ✔ 1 | 4 | 9 | 0.000000000000 | 0.000000000000 ✔ 1 | 5 | 1 | 0.000000000000 | 0.000000000000 ✔ 1 | 5 | 2 | 0.000000000000 | -0.000000000000 ✔ @@ -310,41 +310,41 @@ 1 | 5 | 5 | 5.454545454545 | 5.454545454545 ✔ 1 | 5 | 6 | 0.000000000000 | 0.000000000000 ✔ 1 | 5 | 7 | 0.000000000000 | -0.000000000000 ✔ - 1 | 5 | 8 | 0.000000000000 | -0.000000000000 ✔ - 1 | 5 | 9 | 0.000000000000 | -0.000000000000 ✔ - 1 | 6 | 1 | 0.000000000000 | -0.000000000000 ✔ + 1 | 5 | 8 | 0.000000000000 | 0.000000000000 ✔ + 1 | 5 | 9 | 0.000000000000 | 0.000000000000 ✔ + 1 | 6 | 1 | 0.000000000000 | 0.000000000000 ✔ 1 | 6 | 2 | 0.000000000000 | 0.000000000000 ✔ - 1 | 6 | 3 | 0.000000000000 | -0.000000000000 ✔ + 1 | 6 | 3 | 0.000000000000 | 0.000000000000 ✔ 1 | 6 | 4 | 0.000000000000 | 0.000000000000 ✔ 1 | 6 | 5 | 0.000000000000 | 0.000000000000 ✔ 1 | 6 | 6 | 6.461538461538 | 6.461538461538 ✔ 1 | 6 | 7 | 0.000000000000 | 0.000000000000 ✔ - 1 | 6 | 8 | 0.000000000000 | 0.000000000000 ✔ - 1 | 6 | 9 | 0.000000000000 | 0.000000000000 ✔ + 1 | 6 | 8 | 0.000000000000 | -0.000000000000 ✔ + 1 | 6 | 9 | 0.000000000000 | -0.000000000000 ✔ 1 | 7 | 1 | 0.000000000000 | 0.000000000000 ✔ 1 | 7 | 2 | 0.000000000000 | 0.000000000000 ✔ - 1 | 7 | 3 | 0.000000000000 | 0.000000000000 ✔ + 1 | 7 | 3 | 0.000000000000 | -0.000000000000 ✔ 1 | 7 | 4 | 0.000000000000 | 0.000000000000 ✔ 1 | 7 | 5 | 0.000000000000 | -0.000000000000 ✔ 1 | 7 | 6 | 0.000000000000 | 0.000000000000 ✔ 1 | 7 | 7 | 7.466666666667 | 7.466666666667 ✔ - 1 | 7 | 8 | 0.000000000000 | 0.000000000000 ✔ + 1 | 7 | 8 | 0.000000000000 | -0.000000000000 ✔ 1 | 7 | 9 | 0.000000000000 | 0.000000000000 ✔ 1 | 8 | 1 | 0.000000000000 | 0.000000000000 ✔ - 1 | 8 | 2 | 0.000000000000 | -0.000000000000 ✔ - 1 | 8 | 3 | 0.000000000000 | -0.000000000000 ✔ - 1 | 8 | 4 | 0.000000000000 | -0.000000000000 ✔ - 1 | 8 | 5 | 0.000000000000 | -0.000000000000 ✔ - 1 | 8 | 6 | 0.000000000000 | 0.000000000000 ✔ - 1 | 8 | 7 | 0.000000000000 | 0.000000000000 ✔ + 1 | 8 | 2 | 0.000000000000 | 0.000000000000 ✔ + 1 | 8 | 3 | 0.000000000000 | 0.000000000000 ✔ + 1 | 8 | 4 | 0.000000000000 | 0.000000000000 ✔ + 1 | 8 | 5 | 0.000000000000 | 0.000000000000 ✔ + 1 | 8 | 6 | 0.000000000000 | -0.000000000000 ✔ + 1 | 8 | 7 | 0.000000000000 | -0.000000000000 ✔ 1 | 8 | 8 | 8.470588235294 | 8.470588235294 ✔ 1 | 8 | 9 | 0.000000000000 | -0.000000000000 ✔ 1 | 9 | 1 | 0.000000000000 | 0.000000000000 ✔ - 1 | 9 | 2 | 0.000000000000 | 0.000000000000 ✔ - 1 | 9 | 3 | 0.000000000000 | -0.000000000000 ✔ + 1 | 9 | 2 | 0.000000000000 | -0.000000000000 ✔ + 1 | 9 | 3 | 0.000000000000 | 0.000000000000 ✔ 1 | 9 | 4 | 0.000000000000 | 0.000000000000 ✔ - 1 | 9 | 5 | 0.000000000000 | -0.000000000000 ✔ - 1 | 9 | 6 | 0.000000000000 | 0.000000000000 ✔ + 1 | 9 | 5 | 0.000000000000 | 0.000000000000 ✔ + 1 | 9 | 6 | 0.000000000000 | -0.000000000000 ✔ 1 | 9 | 7 | 0.000000000000 | 0.000000000000 ✔ 1 | 9 | 8 | 0.000000000000 | -0.000000000000 ✔ 1 | 9 | 9 | 9.473684210526 | 9.473684210526 ✔ @@ -355,7 +355,7 @@ 2 | 2 | 6 | 0.000000000000 | -0.000000000000 ✔ 2 | 2 | 7 | 0.000000000000 | 0.000000000000 ✔ 2 | 2 | 8 | 0.000000000000 | 0.000000000000 ✔ - 2 | 2 | 9 | 0.000000000000 | 0.000000000000 ✔ + 2 | 2 | 9 | 0.000000000000 | -0.000000000000 ✔ 2 | 3 | 2 | 0.000000000000 | 0.000000000000 ✔ 2 | 3 | 3 | 34.285714285714 | 34.285714285714 ✔ 2 | 3 | 4 | 0.000000000000 | 0.000000000000 ✔ @@ -371,27 +371,27 @@ 2 | 4 | 6 | 0.000000000000 | -0.000000000000 ✔ 2 | 4 | 7 | 0.000000000000 | -0.000000000000 ✔ 2 | 4 | 8 | 0.000000000000 | 0.000000000000 ✔ - 2 | 4 | 9 | 0.000000000000 | -0.000000000000 ✔ + 2 | 4 | 9 | 0.000000000000 | 0.000000000000 ✔ 2 | 5 | 2 | 0.000000000000 | 0.000000000000 ✔ 2 | 5 | 3 | 0.000000000000 | 0.000000000000 ✔ 2 | 5 | 4 | 0.000000000000 | 0.000000000000 ✔ 2 | 5 | 5 | 152.727272727273 | 152.727272727273 ✔ 2 | 5 | 6 | 0.000000000000 | -0.000000000000 ✔ - 2 | 5 | 7 | 0.000000000000 | -0.000000000000 ✔ + 2 | 5 | 7 | 0.000000000000 | 0.000000000000 ✔ 2 | 5 | 8 | 0.000000000000 | 0.000000000000 ✔ - 2 | 5 | 9 | 0.000000000000 | -0.000000000000 ✔ + 2 | 5 | 9 | 0.000000000000 | 0.000000000000 ✔ 2 | 6 | 2 | 0.000000000000 | -0.000000000000 ✔ 2 | 6 | 3 | 0.000000000000 | 0.000000000000 ✔ 2 | 6 | 4 | 0.000000000000 | -0.000000000000 ✔ 2 | 6 | 5 | 0.000000000000 | -0.000000000000 ✔ 2 | 6 | 6 | 258.461538461538 | 258.461538461538 ✔ 2 | 6 | 7 | 0.000000000000 | 0.000000000000 ✔ - 2 | 6 | 8 | 0.000000000000 | 0.000000000000 ✔ - 2 | 6 | 9 | 0.000000000000 | -0.000000000000 ✔ + 2 | 6 | 8 | 0.000000000000 | -0.000000000000 ✔ + 2 | 6 | 9 | 0.000000000000 | 0.000000000000 ✔ 2 | 7 | 2 | 0.000000000000 | 0.000000000000 ✔ 2 | 7 | 3 | 0.000000000000 | -0.000000000000 ✔ 2 | 7 | 4 | 0.000000000000 | -0.000000000000 ✔ - 2 | 7 | 5 | 0.000000000000 | -0.000000000000 ✔ + 2 | 7 | 5 | 0.000000000000 | 0.000000000000 ✔ 2 | 7 | 6 | 0.000000000000 | 0.000000000000 ✔ 2 | 7 | 7 | 403.200000000000 | 403.200000000000 ✔ 2 | 7 | 8 | 0.000000000000 | -0.000000000000 ✔ @@ -400,80 +400,80 @@ 2 | 8 | 3 | 0.000000000000 | 0.000000000000 ✔ 2 | 8 | 4 | 0.000000000000 | 0.000000000000 ✔ 2 | 8 | 5 | 0.000000000000 | 0.000000000000 ✔ - 2 | 8 | 6 | 0.000000000000 | 0.000000000000 ✔ + 2 | 8 | 6 | 0.000000000000 | -0.000000000000 ✔ 2 | 8 | 7 | 0.000000000000 | -0.000000000000 ✔ 2 | 8 | 8 | 592.941176470588 | 592.941176470588 ✔ - 2 | 8 | 9 | 0.000000000000 | 0.000000000000 ✔ - 2 | 9 | 2 | 0.000000000000 | 0.000000000000 ✔ + 2 | 8 | 9 | 0.000000000000 | -0.000000000000 ✔ + 2 | 9 | 2 | 0.000000000000 | -0.000000000000 ✔ 2 | 9 | 3 | 0.000000000000 | -0.000000000000 ✔ - 2 | 9 | 4 | 0.000000000000 | -0.000000000000 ✔ - 2 | 9 | 5 | 0.000000000000 | -0.000000000000 ✔ - 2 | 9 | 6 | 0.000000000000 | -0.000000000000 ✔ + 2 | 9 | 4 | 0.000000000000 | 0.000000000000 ✔ + 2 | 9 | 5 | 0.000000000000 | 0.000000000000 ✔ + 2 | 9 | 6 | 0.000000000000 | 0.000000000000 ✔ 2 | 9 | 7 | 0.000000000000 | -0.000000000000 ✔ - 2 | 9 | 8 | 0.000000000000 | 0.000000000000 ✔ - 2 | 9 | 9 | 833.684210526316 | 833.684210526316 ✔ + 2 | 9 | 8 | 0.000000000000 | -0.000000000000 ✔ + 2 | 9 | 9 | 833.684210526316 | 833.684210526315 ✔ 3 | 3 | 3 | 205.714285714286 | 205.714285714286 ✔ 3 | 3 | 4 | 0.000000000000 | -0.000000000000 ✔ - 3 | 3 | 5 | 0.000000000000 | -0.000000000000 ✔ + 3 | 3 | 5 | 0.000000000000 | 0.000000000000 ✔ 3 | 3 | 6 | 0.000000000000 | 0.000000000000 ✔ - 3 | 3 | 7 | 0.000000000000 | -0.000000000000 ✔ + 3 | 3 | 7 | 0.000000000000 | 0.000000000000 ✔ 3 | 3 | 8 | 0.000000000000 | 0.000000000000 ✔ - 3 | 3 | 9 | 0.000000000000 | 0.000000000000 ✔ + 3 | 3 | 9 | 0.000000000000 | -0.000000000000 ✔ 3 | 4 | 3 | 0.000000000000 | -0.000000000000 ✔ 3 | 4 | 4 | 1120.000000000000 | 1120.000000000000 ✔ 3 | 4 | 5 | 0.000000000000 | 0.000000000000 ✔ 3 | 4 | 6 | 0.000000000000 | 0.000000000000 ✔ 3 | 4 | 7 | 0.000000000000 | 0.000000000000 ✔ - 3 | 4 | 8 | 0.000000000000 | -0.000000000000 ✔ - 3 | 4 | 9 | 0.000000000000 | -0.000000000000 ✔ - 3 | 5 | 3 | 0.000000000000 | -0.000000000000 ✔ + 3 | 4 | 8 | 0.000000000000 | 0.000000000000 ✔ + 3 | 4 | 9 | 0.000000000000 | 0.000000000000 ✔ + 3 | 5 | 3 | 0.000000000000 | 0.000000000000 ✔ 3 | 5 | 4 | 0.000000000000 | 0.000000000000 ✔ 3 | 5 | 5 | 3665.454545454545 | 3665.454545454545 ✔ - 3 | 5 | 6 | 0.000000000000 | 0.000000000000 ✔ - 3 | 5 | 7 | 0.000000000000 | -0.000000000000 ✔ - 3 | 5 | 8 | 0.000000000000 | -0.000000000000 ✔ - 3 | 5 | 9 | 0.000000000000 | -0.000000000000 ✔ + 3 | 5 | 6 | 0.000000000000 | 0.000000000001 ✔ + 3 | 5 | 7 | 0.000000000000 | 0.000000000000 ✔ + 3 | 5 | 8 | 0.000000000000 | -0.000000000001 ✔ + 3 | 5 | 9 | 0.000000000000 | -0.000000000002 ✔ 3 | 6 | 3 | 0.000000000000 | 0.000000000000 ✔ 3 | 6 | 4 | 0.000000000000 | 0.000000000000 ✔ - 3 | 6 | 5 | 0.000000000000 | 0.000000000000 ✔ + 3 | 6 | 5 | 0.000000000000 | 0.000000000001 ✔ 3 | 6 | 6 | 9304.615384615385 | 9304.615384615387 ✔ - 3 | 6 | 7 | 0.000000000000 | -0.000000000000 ✔ - 3 | 6 | 8 | 0.000000000000 | -0.000000000000 ✔ - 3 | 6 | 9 | 0.000000000000 | -0.000000000002 ✔ - 3 | 7 | 3 | 0.000000000000 | -0.000000000000 ✔ + 3 | 6 | 7 | 0.000000000000 | 0.000000000000 ✔ + 3 | 6 | 8 | 0.000000000000 | 0.000000000001 ✔ + 3 | 6 | 9 | 0.000000000000 | -0.000000000000 ✔ + 3 | 7 | 3 | 0.000000000000 | 0.000000000000 ✔ 3 | 7 | 4 | 0.000000000000 | 0.000000000000 ✔ - 3 | 7 | 5 | 0.000000000000 | -0.000000000000 ✔ - 3 | 7 | 6 | 0.000000000000 | -0.000000000000 ✔ - 3 | 7 | 7 | 20160.000000000000 | 20160.000000000004 ✔ + 3 | 7 | 5 | 0.000000000000 | 0.000000000000 ✔ + 3 | 7 | 6 | 0.000000000000 | 0.000000000000 ✔ + 3 | 7 | 7 | 20160.000000000000 | 20160.000000000007 ✔ 3 | 7 | 8 | 0.000000000000 | 0.000000000000 ✔ - 3 | 7 | 9 | 0.000000000000 | -0.000000000003 ✔ + 3 | 7 | 9 | 0.000000000000 | 0.000000000001 ✔ 3 | 8 | 3 | 0.000000000000 | 0.000000000000 ✔ - 3 | 8 | 4 | 0.000000000000 | -0.000000000000 ✔ - 3 | 8 | 5 | 0.000000000000 | -0.000000000000 ✔ - 3 | 8 | 6 | 0.000000000000 | -0.000000000000 ✔ + 3 | 8 | 4 | 0.000000000000 | 0.000000000000 ✔ + 3 | 8 | 5 | 0.000000000000 | -0.000000000001 ✔ + 3 | 8 | 6 | 0.000000000000 | 0.000000000001 ✔ 3 | 8 | 7 | 0.000000000000 | 0.000000000000 ✔ - 3 | 8 | 8 | 39134.117647058825 | 39134.117647058825 ✔ + 3 | 8 | 8 | 39134.117647058825 | 39134.117647058833 ✔ 3 | 8 | 9 | 0.000000000000 | 0.000000000000 ✔ - 3 | 9 | 3 | 0.000000000000 | 0.000000000000 ✔ - 3 | 9 | 4 | 0.000000000000 | -0.000000000000 ✔ - 3 | 9 | 5 | 0.000000000000 | -0.000000000000 ✔ - 3 | 9 | 6 | 0.000000000000 | -0.000000000002 ✔ - 3 | 9 | 7 | 0.000000000000 | -0.000000000003 ✔ + 3 | 9 | 3 | 0.000000000000 | -0.000000000000 ✔ + 3 | 9 | 4 | 0.000000000000 | 0.000000000000 ✔ + 3 | 9 | 5 | 0.000000000000 | -0.000000000002 ✔ + 3 | 9 | 6 | 0.000000000000 | -0.000000000000 ✔ + 3 | 9 | 7 | 0.000000000000 | 0.000000000001 ✔ 3 | 9 | 8 | 0.000000000000 | 0.000000000000 ✔ - 3 | 9 | 9 | 70029.473684210534 | 70029.473684210505 ✔ + 3 | 9 | 9 | 70029.473684210534 | 70029.473684210519 ✔ 4 | 4 | 4 | 8960.000000000000 | 8960.000000000002 ✔ 4 | 4 | 5 | 0.000000000000 | -0.000000000002 ✔ - 4 | 4 | 6 | 0.000000000000 | -0.000000000001 ✔ + 4 | 4 | 6 | 0.000000000000 | -0.000000000003 ✔ 4 | 4 | 7 | 0.000000000000 | -0.000000000000 ✔ - 4 | 4 | 8 | 0.000000000000 | 0.000000000007 ✔ - 4 | 4 | 9 | 0.000000000000 | 0.000000000000 ✔ + 4 | 4 | 8 | 0.000000000000 | -0.000000000002 ✔ + 4 | 4 | 9 | 0.000000000000 | 0.000000000002 ✔ 4 | 5 | 4 | 0.000000000000 | -0.000000000002 ✔ 4 | 5 | 5 | 65978.181818181823 | 65978.181818181838 ✔ 4 | 5 | 6 | 0.000000000000 | -0.000000000001 ✔ 4 | 5 | 7 | 0.000000000000 | -0.000000000058 ✔ 4 | 5 | 8 | 0.000000000000 | -0.000000000002 ✔ - 4 | 5 | 9 | 0.000000000000 | -0.000000000007 ✔ - 4 | 6 | 4 | 0.000000000000 | -0.000000000001 ✔ + 4 | 5 | 9 | 0.000000000000 | 0.000000000000 ✔ + 4 | 6 | 4 | 0.000000000000 | -0.000000000003 ✔ 4 | 6 | 5 | 0.000000000000 | -0.000000000001 ✔ 4 | 6 | 6 | 279138.461538461561 | 279138.461538461503 ✔ 4 | 6 | 7 | 0.000000000000 | -0.000000000018 ✔ @@ -484,43 +484,43 @@ 4 | 7 | 6 | 0.000000000000 | -0.000000000018 ✔ 4 | 7 | 7 | 887040.000000000000 | 887040.000000000000 ✔ 4 | 7 | 8 | 0.000000000000 | 0.000000000031 ✔ - 4 | 7 | 9 | 0.000000000000 | 0.000000000104 ✔ - 4 | 8 | 4 | 0.000000000000 | 0.000000000007 ✔ + 4 | 7 | 9 | 0.000000000000 | 0.000000000022 ✔ + 4 | 8 | 4 | 0.000000000000 | -0.000000000002 ✔ 4 | 8 | 5 | 0.000000000000 | -0.000000000002 ✔ 4 | 8 | 6 | 0.000000000000 | 0.000000000055 ✔ 4 | 8 | 7 | 0.000000000000 | 0.000000000031 ✔ 4 | 8 | 8 | 2348047.058823529165 | 2348047.058823529631 ✔ - 4 | 8 | 9 | 0.000000000000 | -0.000000000015 ✔ - 4 | 9 | 4 | 0.000000000000 | 0.000000000000 ✔ - 4 | 9 | 5 | 0.000000000000 | -0.000000000007 ✔ + 4 | 8 | 9 | 0.000000000000 | 0.000000000047 ✔ + 4 | 9 | 4 | 0.000000000000 | 0.000000000002 ✔ + 4 | 9 | 5 | 0.000000000000 | 0.000000000000 ✔ 4 | 9 | 6 | 0.000000000000 | 0.000000000029 ✔ - 4 | 9 | 7 | 0.000000000000 | 0.000000000104 ✔ - 4 | 9 | 8 | 0.000000000000 | -0.000000000015 ✔ - 4 | 9 | 9 | 5462298.947368421592 | 5462298.947368418798 ✔ + 4 | 9 | 7 | 0.000000000000 | 0.000000000022 ✔ + 4 | 9 | 8 | 0.000000000000 | 0.000000000047 ✔ + 4 | 9 | 9 | 5462298.947368421592 | 5462298.947368420660 ✔ 5 | 5 | 5 | 659781.818181818235 | 659781.818181818351 ✔ 5 | 5 | 6 | 0.000000000000 | -0.000000000002 ✔ 5 | 5 | 7 | 0.000000000000 | 0.000000000233 ✔ - 5 | 5 | 8 | 0.000000000000 | 0.000000000567 ✔ - 5 | 5 | 9 | 0.000000000000 | 0.000000000000 ✔ + 5 | 5 | 8 | 0.000000000000 | 0.000000000247 ✔ + 5 | 5 | 9 | 0.000000000000 | 0.000000000233 ✔ 5 | 6 | 5 | 0.000000000000 | -0.000000000002 ✔ 5 | 6 | 6 | 6141046.153846153989 | 6141046.153846156783 ✔ - 5 | 6 | 7 | 0.000000000000 | 0.000000000250 ✔ - 5 | 6 | 8 | 0.000000000000 | 0.000000001630 ✔ - 5 | 6 | 9 | 0.000000000000 | 0.000000000931 ✔ + 5 | 6 | 7 | 0.000000000000 | 0.000000000342 ✔ + 5 | 6 | 8 | 0.000000000000 | 0.000000001048 ✔ + 5 | 6 | 9 | 0.000000000000 | 0.000000000233 ✔ 5 | 7 | 5 | 0.000000000000 | 0.000000000233 ✔ - 5 | 7 | 6 | 0.000000000000 | 0.000000000250 ✔ + 5 | 7 | 6 | 0.000000000000 | 0.000000000342 ✔ 5 | 7 | 7 | 31933440.000000000000 | 31933440.000000000000 ✔ 5 | 7 | 8 | 0.000000000000 | 0.000000002503 ✔ - 5 | 7 | 9 | 0.000000000000 | 0.000000003725 ✔ - 5 | 8 | 5 | 0.000000000000 | 0.000000000567 ✔ - 5 | 8 | 6 | 0.000000000000 | 0.000000001630 ✔ + 5 | 7 | 9 | 0.000000000000 | 0.000000005588 ✔ + 5 | 8 | 5 | 0.000000000000 | 0.000000000247 ✔ + 5 | 8 | 6 | 0.000000000000 | 0.000000001048 ✔ 5 | 8 | 7 | 0.000000000000 | 0.000000002503 ✔ 5 | 8 | 8 | 122098447.058823525906 | 122098447.058823525906 ✔ - 5 | 8 | 9 | 0.000000000000 | -0.000000001397 ✔ - 5 | 9 | 5 | 0.000000000000 | 0.000000000000 ✔ - 5 | 9 | 6 | 0.000000000000 | 0.000000000931 ✔ - 5 | 9 | 7 | 0.000000000000 | 0.000000003725 ✔ - 5 | 9 | 8 | 0.000000000000 | -0.000000001397 ✔ + 5 | 8 | 9 | 0.000000000000 | 0.000000002328 ✔ + 5 | 9 | 5 | 0.000000000000 | 0.000000000233 ✔ + 5 | 9 | 6 | 0.000000000000 | 0.000000000233 ✔ + 5 | 9 | 7 | 0.000000000000 | 0.000000005588 ✔ + 5 | 9 | 8 | 0.000000000000 | 0.000000002328 ✔ 5 | 9 | 9 | 382360926.315789461136 | 382360926.315789461136 ✔ ``` @@ -541,9 +541,9 @@ l₁ | l₂ | m₁ | m₂ | analytical | numerical 0 | 1 | 0 | 0 | 0.000000000000 | -0.000000000000 ✔ 0 | 1 | 0 | 1 | 0.000000000000 | 0.000000000000 ✔ 0 | 2 | 0 | -2 | 0.000000000000 | -0.000000000000 ✔ - 0 | 2 | 0 | -1 | 0.000000000000 | 0.000000000000 ✔ + 0 | 2 | 0 | -1 | 0.000000000000 | -0.000000000000 ✔ 0 | 2 | 0 | 0 | 0.000000000000 | 0.000000000000 ✔ - 0 | 2 | 0 | 1 | 0.000000000000 | -0.000000000000 ✔ + 0 | 2 | 0 | 1 | 0.000000000000 | 0.000000000000 ✔ 0 | 2 | 0 | 2 | 0.000000000000 | -0.000000000000 ✔ 1 | 0 | -1 | 0 | 0.000000000000 | 0.000000000000 ✔ 1 | 0 | 0 | 0 | 0.000000000000 | -0.000000000000 ✔ @@ -573,9 +573,9 @@ l₁ | l₂ | m₁ | m₂ | analytical | numerical 1 | 2 | 1 | 1 | 0.000000000000 | -0.000000000000 ✔ 1 | 2 | 1 | 2 | 0.000000000000 | 0.000000000000 ✔ 2 | 0 | -2 | 0 | 0.000000000000 | -0.000000000000 ✔ - 2 | 0 | -1 | 0 | 0.000000000000 | 0.000000000000 ✔ + 2 | 0 | -1 | 0 | 0.000000000000 | -0.000000000000 ✔ 2 | 0 | 0 | 0 | 0.000000000000 | 0.000000000000 ✔ - 2 | 0 | 1 | 0 | 0.000000000000 | -0.000000000000 ✔ + 2 | 0 | 1 | 0 | 0.000000000000 | 0.000000000000 ✔ 2 | 0 | 2 | 0 | 0.000000000000 | -0.000000000000 ✔ 2 | 1 | -2 | -1 | 0.000000000000 | -0.000000000000 ✔ 2 | 1 | -2 | 0 | 0.000000000000 | -0.000000000000 ✔ @@ -619,4 +619,45 @@ l₁ | l₂ | m₁ | m₂ | analytical | numerical 2 | 2 | 2 | 2 | 1.000000000000 | 1.000000000000 ✔ ``` +#### Eigen Values + +```math + \begin{aligned} + E_n + &= \int_{0}^{2\pi}~\mathrm{d}\varphi \int_{0}^{\pi}~\mathrm{d}\theta ~ \psi^\ast_{lm}(\theta,\varphi) \hat{H} \psi_{lm}(\theta,\varphi) \\ + &= \int_{0}^{2\pi}~\mathrm{d}\varphi \int_{0}^{\pi}~\mathrm{d}\theta ~ \psi^\ast_{lm}(\theta,\varphi) \left[ \hat{V} + \hat{T} \right] \psi_{lm}(\theta,\varphi) \\ + &= \int_{0}^{2\pi}~\mathrm{d}\varphi \int_{0}^{\pi}~\mathrm{d}\theta ~ \psi^\ast_{lm}(\theta,\varphi) \left[ 0 + \frac{l(l+1)}{2I} \right] \psi_{lm}(\theta,\varphi) \\ + &= \int_{0}^{2\pi}~\mathrm{d}\varphi \int_{0}^{\pi}~\mathrm{d}\theta ~ \frac{l(l+1)}{2I} |\psi_{lm}(\theta,\varphi)|^2 + \end{aligned} +``` +where $l$ is angular momentum quantum number and $I$ is the moment of intertia. +``` + m₁ | m₂ | R | l | analytical | numerical +--- | --- | --- | --- | ----------------- | ----------------- +0.5 | 0.5 | 0.5 | 1.0 | 16.000000000000 | 16.000000000000 ✔ +0.5 | 0.5 | 0.5 | 2.0 | 48.000000000000 | 48.000000000377 ✔ +0.5 | 0.5 | 0.5 | 3.0 | 96.000000000000 | 95.999998779374 ✔ +0.5 | 0.5 | 2.0 | 1.0 | 1.000000000000 | 1.000000000000 ✔ +0.5 | 0.5 | 2.0 | 2.0 | 3.000000000000 | 3.000000000024 ✔ +0.5 | 0.5 | 2.0 | 3.0 | 6.000000000000 | 5.999999923711 ✔ +0.5 | 2.0 | 0.5 | 1.0 | 10.000000000000 | 10.000000000000 ✔ +0.5 | 2.0 | 0.5 | 2.0 | 30.000000000000 | 30.000000000236 ✔ +0.5 | 2.0 | 0.5 | 3.0 | 60.000000000000 | 59.999999237109 ✔ +0.5 | 2.0 | 2.0 | 1.0 | 0.625000000000 | 0.625000000000 ✔ +0.5 | 2.0 | 2.0 | 2.0 | 1.875000000000 | 1.875000000015 ✔ +0.5 | 2.0 | 2.0 | 3.0 | 3.750000000000 | 3.749999952319 ✔ +2.0 | 0.5 | 0.5 | 1.0 | 10.000000000000 | 10.000000000000 ✔ +2.0 | 0.5 | 0.5 | 2.0 | 30.000000000000 | 30.000000000236 ✔ +2.0 | 0.5 | 0.5 | 3.0 | 60.000000000000 | 59.999999237109 ✔ +2.0 | 0.5 | 2.0 | 1.0 | 0.625000000000 | 0.625000000000 ✔ +2.0 | 0.5 | 2.0 | 2.0 | 1.875000000000 | 1.875000000015 ✔ +2.0 | 0.5 | 2.0 | 3.0 | 3.750000000000 | 3.749999952319 ✔ +2.0 | 2.0 | 0.5 | 1.0 | 4.000000000000 | 4.000000000000 ✔ +2.0 | 2.0 | 0.5 | 2.0 | 12.000000000000 | 12.000000000094 ✔ +2.0 | 2.0 | 0.5 | 3.0 | 24.000000000000 | 23.999999694843 ✔ +2.0 | 2.0 | 2.0 | 1.0 | 0.250000000000 | 0.250000000000 ✔ +2.0 | 2.0 | 2.0 | 2.0 | 0.750000000000 | 0.750000000006 ✔ +2.0 | 2.0 | 2.0 | 3.0 | 1.500000000000 | 1.499999980928 ✔ +``` +