Spherical Oscillator
The spherical oscillator (3D isotropic harmonic oscillator) is the most frequently used model in quantum physics. This model uses a spherical coordinate system.
Definitions
Antique.SphericalOscillator — TypeModel
This model is described with the time-independent Schrödinger equation
\[ \hat{H} \psi(\pmb{r}) = E \psi(\pmb{r}),\]
and the Hamiltonian
\[ \hat{H} = - \frac{\hbar^2}{2\mu} \nabla^2 + \frac{1}{2} k r^2.\]
Parameters are specified with the following struct:
SO = SphericalOscillator(k=1.0, μ=1.0, ℏ=1.0)$k$ is the force constant, $μ$ is the mass of particle and $\hbar$ is the reduced Planck constant (Dirac's constant).
References
Potential
Antique.V — MethodV(model::SphericalOscillator, r)
\[V(r) = \frac{1}{2} k r^2 = \frac{1}{2} \mu \omega^2 r^2 = \frac{1}{2} \hbar \omega \xi^2,\]
where $\omega = \sqrt{k/\mu}$ is the angular frequency and $\xi = \sqrt{\frac{\mu\omega}{\hbar}}r$.
Eigenvalues
Antique.E — MethodE(model::SphericalOscillator; n::Int=0, l::Int=0)
\[E_{nl} = \left(2n + l + \frac{3}{2}\right)\hbar \omega,\]
where $\omega = \sqrt{k/\mu}$.
Eigenfunctions
Antique.ψ — Methodψ(model::SphericalOscillator, r, θ, φ; n::Int=0, l::Int=0, m::Int=0)
\[\psi_{nlm}(\pmb{r}) = R_{nl}(r) Y_{lm}(\theta,\varphi)\]
The domain is $0\leq r \lt \infty, 0\leq \theta \lt \pi, 0\leq \varphi \lt 2\pi$.
Radial Functions
Antique.R — MethodR(model::SphericalOscillator, r; n=0, l=0)
\[R_{nl}(r) = \sqrt{ \frac{\gamma^{3/2}}{2\sqrt{\pi}}} \sqrt{\frac{2^{n+l+3} n!}{(2n+2l+1)!!}} \xi^l \exp\left(-\xi^2/2\right)L_{n}^{(l+\frac{1}{2})} \left(\xi^2\right),\]
where $\gamma = \mu\omega/\hbar$ and $\xi = \sqrt{\gamma}r = \sqrt{\mu\omega/\hbar}r$ are defined. The generalized Laguerre polynomials are defined as $L_n^{(\alpha)}(x) = \frac{x^{-\alpha} \mathrm{e}^x}{n !} \frac{\mathrm{d}^n}{\mathrm{d} x^n}\left(\mathrm{e}^{-x} x^{n+\alpha}\right)$. The domain is $0\leq r \lt \infty$.
Generalized Laguerre Polynomials
Antique.L — MethodL(model::SphericalOscillator, x; n=0, α=0)
The generalized Laguerre polynomials $L_n^{(\alpha)}(x)$, not the associated Laguerre polynomials $L_n^{k}(x)$, are used in this model.
Rodrigues' formula & closed-form:
\[\begin{aligned} L_n^{(\alpha)}(x) &= \frac{x^{-\alpha}e^x}{n!} \frac{d^n}{dx^n}\left(x^{n+\alpha}e^{-x}\right) \\ &= \sum_{k=0}^n(-1)^k \left(\begin{array}{l} n+\alpha \\ n-k \end{array}\right) \frac{x^k}{k !} \\ &= \sum_{k=0}^n(-1)^k \frac{\Gamma(\alpha+n+1)}{\Gamma(\alpha+k+1)\Gamma(n-k+1)} \frac{x^k}{k !}. \end{aligned}\]
Examples:
\[\begin{aligned} L_0^{(0)}(x) &= 1, \\ L_1^{(0)}(x) &= 1 - x, \\ L_1^{(1)}(x) &= 2 - x, \\ L_2^{(0)}(x) &= 1 - 2 x + 1/2 x^{2}, \\ L_2^{(1)}(x) &= 3 - 3 x + 1/2 x^{2}, \\ L_2^{(2)}(x) &= 6 - 4 x + 1/2 x^{2}, \\ L_3^{(0)}(x) &= 1 - 3 x + 3/2 x^{2} - 1/6 x^{3}, \\ L_3^{(1)}(x) &= 4 - 6 x + 2 x^{2} - 1/6 x^{3}, \\ L_3^{(2)}(x) &= 10 - 10 x + 5/2 x^{2} - 1/6 x^{3}, \\ L_3^{(3)}(x) &= 20 - 15 x + 3 x^{2} - 1/6 x^{3}, \\ L_4^{(0)}(x) &= 1 - 4 x + 3 x^{2} - 2/3 x^{3} + 1/24 x^{4}, \\ L_4^{(1)}(x) &= 5 - 10 x + 5 x^{2} - 5/6 x^{3} + 1/24 x^{4}, \\ L_4^{(2)}(x) &= 15 - 20 x + 15/2 x^{2} - 1 x^{3} + 1/24 x^{4}, \\ L_4^{(3)}(x) &= 35 - 35 x + 21/2 x^{2} - 7/6 x^{3} + 1/24 x^{4}, \\ L_4^{(4)}(x) &= 70 - 56 x + 14 x^{2} - 4/3 x^{3} + 1/24 x^{4}, \\ \vdots \end{aligned}\]
Spherical Harmonics
Antique.Y — MethodY(model::SphericalOscillator, θ, φ; l=0, m=0)
\[Y_{lm}(\theta,\varphi) = (-1)^{\frac{|m|+m}{2}} \sqrt{\frac{2l+1}{4\pi} \frac{(l-|m|)!}{(l+|m|)!}} P_l^{|m|} (\cos\theta) \mathrm{e}^{im\varphi}.\]
The domain is $0\leq \theta \lt \pi, 0\leq \varphi \lt 2\pi$. Note that some variants are connected by
\[i^{|m|+m} \sqrt{\frac{(l-|m|)!}{(l+|m|)!}} P_l^{|m|} = (-1)^{\frac{|m|+m}{2}} \sqrt{\frac{(l-|m|)!}{(l+|m|)!}} P_l^{|m|} = (-1)^m \sqrt{\frac{(l-m)!}{(l+m)!}} P_l^{m}.\]
Associated Legendre Polynomials
Antique.P — MethodP(model::SphericalOscillator, x; n=0, m=0)
Rodrigues' formula & closed-form:
\[\begin{aligned} P_n^m(x) &= \left( 1-x^2 \right)^{m/2} \frac{\mathrm{d}^m}{\mathrm{d}x^m} P_n(x) \\ &= \left( 1-x^2 \right)^{m/2} \frac{\mathrm{d}^m}{\mathrm{d}x^m} \frac{1}{2^n n!} \frac{\mathrm{d}^n}{\mathrm{d}x ^n} \left[ \left( x^2-1 \right)^n \right] \\ &= \frac{1}{2^n} (1-x^2)^{m/2} \sum_{j=0}^{\left\lfloor\frac{n-m}{2}\right\rfloor} (-1)^j \frac{(2n-2j)!}{j! (n-j)! (n-2j-m)!} x^{(n-2j-m)}. \end{aligned},\]
where Legendre polynomials are defined as $P_n(x) = \frac{1}{2^n n!} \frac{\mathrm{d}^n}{\mathrm{d}x ^n} \left[ \left( x^2-1 \right)^n \right]$. Note that $P_l^{-m} = (-1)^m \frac{(l-m)!}{(l+m)!} P_l^m$ for $m<0$. (It is not compatible with $P_k^m(t) = (-1)^m\left( 1-t^2 \right)^{m/2} \frac{\mathrm{d}^m P_k(t)}{\mathrm{d}t^m}$ caused by $(-1)^m$.) The specific formulae are given below.
Examples:
\[\begin{aligned} P_{0}^{0}(x) &= 1, \\ P_{1}^{0}(x) &= x, \\ P_{1}^{1}(x) &= \left(+1\right)\sqrt{1-x^2}, \\ P_{2}^{0}(x) &= -1/2 + 3/2 x^{2}, \\ P_{2}^{1}(x) &= \left(-3 x\right)\sqrt{1-x^2}, \\ P_{2}^{2}(x) &= 3 - 6 x, \\ P_{3}^{0}(x) &= -3/2 x + 5/2 x^{3}, \\ P_{3}^{1}(x) &= \left(3/2 - 15/2 x^{2}\right)\sqrt{1-x^2}, \\ P_{3}^{2}(x) &= 15 x - 30 x^{2}, \\ P_{3}^{3}(x) &= \left(15 - 30 x\right)\sqrt{1-x^2}, \\ P_{4}^{0}(x) &= 3/8 - 15/4 x^{2} + 35/8 x^{4}, \\ P_{4}^{1}(x) &= \left(- 15/2 x + 35/2 x^{3}\right)\sqrt{1-x^2}, \\ P_{4}^{2}(x) &= -15/2 + 15 x + 105/2 x^{2} - 105 x^{3}, \\ P_{4}^{3}(x) &= \left(105 x - 210 x^{2}\right)\sqrt{1-x^2}, \\ P_{4}^{4}(x) &= 105 - 420 x + 420 x^{2}, \\ & \vdots \end{aligned}\]
Usage & Examples
Install Antique.jl for the first use and run using Antique before each use. The energy E(), wavefunction ψ() and potential V() will be exported. In this system, the model is generated by SphericalOscillator and several parameters k, μ and ℏ are set as optional arguments.
using Antique
SO = SphericalOscillator(k=1.0, μ=1.0, ℏ=1.0)Parameters:
julia> SO.k1.0julia> SO.μ1.0julia> SO.ℏ1.0
Eigenvalues:
julia> E(SO, n=0)1.5julia> E(SO, n=1)3.5julia> E(SO, n=2)5.5julia> E(SO, n=0, l=1)2.5julia> E(SO, n=1, l=1)4.5julia> E(SO, n=2, l=1)6.5
Testing
Unit testing and Integration testing were done using computer algebra system (Symbolics.jl) and numerical integration (QuadGK.jl). The test script is here.
Associated Legendre Polynomials $P_n^m(x)$
\[ \begin{aligned} P_n^m(x) &= \left( 1-x^2 \right)^{m/2} \frac{\mathrm{d}^m}{\mathrm{d}x^m} P_n(x) \\ &= \left( 1-x^2 \right)^{m/2} \frac{\mathrm{d}^m}{\mathrm{d}x^m} \frac{1}{2^n n!} \frac{\mathrm{d}^n}{\mathrm{d}x ^n} \left[ \left( x^2-1 \right)^n \right] \\ &= \frac{1}{2^n} (1-x^2)^{m/2} \sum_{j=0}^{\left\lfloor\frac{n-m}{2}\right\rfloor} (-1)^j \frac{(2n-2j)!}{j! (n-j)! (n-2j-m)!} x^{(n-2j-m)}. \end{aligned}\]
$n=0, m=0:$ ✔
\[\begin{aligned} P_{0}^{0}(x) = 1 &= 1 \\ &= 1 \end{aligned}\]
$n=1, m=0:$ ✔
\[\begin{aligned} P_{1}^{0}(x) = \frac{1}{2} \frac{\mathrm{d}}{\mathrm{d}x} \left( -1 + x^{2} \right) &= x \\ &= x \end{aligned}\]
$n=1, m=1:$ ✔
\[\begin{aligned} P_{1}^{1}(x) = \left( 1 - x^{2} \right)^{\frac{1}{2}} \frac{\mathrm{d}}{\mathrm{d}x} \frac{1}{2} \frac{\mathrm{d}}{\mathrm{d}x} \left( -1 + x^{2} \right) &= \left( 1 - x^{2} \right)^{\frac{1}{2}} \\ &= \left( 1 - x^{2} \right)^{\frac{1}{2}} \end{aligned}\]
$n=2, m=0:$ ✔
\[\begin{aligned} P_{2}^{0}(x) = \frac{1}{8} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \left( -1 + x^{2} \right)^{2} &= \frac{-1}{2} + \frac{3}{2} x^{2} \\ &= \frac{-1}{2} + \frac{3}{2} x^{2} \end{aligned}\]
$n=2, m=1:$ ✔
\[\begin{aligned} P_{2}^{1}(x) = \left( 1 - x^{2} \right)^{\frac{1}{2}} \frac{\mathrm{d}}{\mathrm{d}x} \frac{1}{8} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \left( -1 + x^{2} \right)^{2} &= 3 \left( 1 - x^{2} \right)^{\frac{1}{2}} x \\ &= 3 \left( 1 - x^{2} \right)^{\frac{1}{2}} x \end{aligned}\]
$n=2, m=2:$ ✔
\[\begin{aligned} P_{2}^{2}(x) = \left( 1 - x^{2} \right) \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{1}{8} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \left( -1 + x^{2} \right)^{2} &= 3 - 3 x^{2} \\ &= 3 - 3 x^{2} \end{aligned}\]
$n=3, m=0:$ ✔
\[\begin{aligned} P_{3}^{0}(x) = \frac{1}{48} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \left( -1 + x^{2} \right)^{3} &= - \frac{3}{2} x + \frac{5}{2} x^{3} \\ &= - \frac{3}{2} x + \frac{5}{2} x^{3} \end{aligned}\]
$n=3, m=1:$ ✔
\[\begin{aligned} P_{3}^{1}(x) = \left( 1 - x^{2} \right)^{\frac{1}{2}} \frac{\mathrm{d}}{\mathrm{d}x} \frac{1}{48} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \left( -1 + x^{2} \right)^{3} &= - \frac{3}{2} \left( 1 - x^{2} \right)^{\frac{1}{2}} + \frac{15}{2} x^{2} \left( 1 - x^{2} \right)^{\frac{1}{2}} \\ &= - \frac{3}{2} \left( 1 - x^{2} \right)^{\frac{1}{2}} + \frac{15}{2} x^{2} \left( 1 - x^{2} \right)^{\frac{1}{2}} \end{aligned}\]
$n=3, m=2:$ ✔
\[\begin{aligned} P_{3}^{2}(x) = \left( 1 - x^{2} \right) \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{1}{48} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \left( -1 + x^{2} \right)^{3} &= 15 x - 15 x^{3} \\ &= 15 x - 15 x^{3} \end{aligned}\]
$n=3, m=3:$ ✔
\[\begin{aligned} P_{3}^{3}(x) = \left( 1 - x^{2} \right)^{\frac{3}{2}} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{1}{48} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \left( -1 + x^{2} \right)^{3} &= 15 \left( 1 - x^{2} \right)^{\frac{3}{2}} \\ &= 15 \left( 1 - x^{2} \right)^{\frac{3}{2}} \end{aligned}\]
$n=4, m=0:$ ✔
\[\begin{aligned} P_{4}^{0}(x) = \frac{1}{384} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \left( -1 + x^{2} \right)^{4} &= \frac{3}{8} - \frac{15}{4} x^{2} + \frac{35}{8} x^{4} \\ &= \frac{3}{8} - \frac{15}{4} x^{2} + \frac{35}{8} x^{4} \end{aligned}\]
$n=4, m=1:$ ✔
\[\begin{aligned} P_{4}^{1}(x) = \left( 1 - x^{2} \right)^{\frac{1}{2}} \frac{\mathrm{d}}{\mathrm{d}x} \frac{1}{384} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \left( -1 + x^{2} \right)^{4} &= - \frac{15}{2} \left( 1 - x^{2} \right)^{\frac{1}{2}} x + \frac{35}{2} x^{3} \left( 1 - x^{2} \right)^{\frac{1}{2}} \\ &= - \frac{15}{2} \left( 1 - x^{2} \right)^{\frac{1}{2}} x + \frac{35}{2} x^{3} \left( 1 - x^{2} \right)^{\frac{1}{2}} \end{aligned}\]
$n=4, m=2:$ ✔
\[\begin{aligned} P_{4}^{2}(x) = \left( 1 - x^{2} \right) \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{1}{384} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \left( -1 + x^{2} \right)^{4} &= \frac{-15}{2} + 60 x^{2} - \frac{105}{2} x^{4} \\ &= \frac{-15}{2} + 60 x^{2} - \frac{105}{2} x^{4} \end{aligned}\]
$n=4, m=3:$ ✔
\[\begin{aligned} P_{4}^{3}(x) = \left( 1 - x^{2} \right)^{\frac{3}{2}} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{1}{384} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \left( -1 + x^{2} \right)^{4} &= 105 \left( 1 - x^{2} \right)^{\frac{3}{2}} x \\ &= 105 \left( 1 - x^{2} \right)^{\frac{3}{2}} x \end{aligned}\]
$n=4, m=4:$ ✔
\[\begin{aligned} P_{4}^{4}(x) = \left( 1 - x^{2} \right)^{2} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{1}{384} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \left( -1 + x^{2} \right)^{4} &= 105 \left( 1 - x^{2} \right)^{2} \\ &= 105 \left( 1 - x^{2} \right)^{2} \end{aligned}\]
Normalization & Orthogonality of $P_n^m(x)$
\[\int_{-1}^{1} P_i^m(x) P_j^m(x) \mathrm{d}x = \frac{2(j+m)!}{(2j+1)(j-m)!} \delta_{ij}\]
m | i | j | analytical | numerical
-- | -- | -- | -------------- | --------------
0 | 0 | 0 | 2.000000000 | 2.000000000 ✔
0 | 0 | 1 | 0.000000000 | 0.000000000 ✔
0 | 0 | 2 | 0.000000000 | 0.000000000 ✔
0 | 0 | 3 | 0.000000000 | -0.000000000 ✔
0 | 0 | 4 | 0.000000000 | 0.000000000 ✔
0 | 0 | 5 | 0.000000000 | -0.000000000 ✔
0 | 0 | 6 | 0.000000000 | -0.000000000 ✔
0 | 0 | 7 | 0.000000000 | 0.000000000 ✔
0 | 0 | 8 | 0.000000000 | -0.000000000 ✔
0 | 0 | 9 | 0.000000000 | -0.000000000 ✔
0 | 1 | 0 | 0.000000000 | 0.000000000 ✔
0 | 1 | 1 | 0.666666667 | 0.666666667 ✔
0 | 1 | 2 | 0.000000000 | 0.000000000 ✔
0 | 1 | 3 | 0.000000000 | -0.000000000 ✔
0 | 1 | 4 | 0.000000000 | 0.000000000 ✔
0 | 1 | 5 | 0.000000000 | -0.000000000 ✔
0 | 1 | 6 | 0.000000000 | 0.000000000 ✔
0 | 1 | 7 | 0.000000000 | -0.000000000 ✔
0 | 1 | 8 | 0.000000000 | -0.000000000 ✔
0 | 1 | 9 | 0.000000000 | -0.000000000 ✔
0 | 2 | 0 | 0.000000000 | 0.000000000 ✔
0 | 2 | 1 | 0.000000000 | 0.000000000 ✔
0 | 2 | 2 | 0.400000000 | 0.400000000 ✔
0 | 2 | 3 | 0.000000000 | 0.000000000 ✔
0 | 2 | 4 | 0.000000000 | 0.000000000 ✔
0 | 2 | 5 | 0.000000000 | 0.000000000 ✔
0 | 2 | 6 | 0.000000000 | -0.000000000 ✔
0 | 2 | 7 | 0.000000000 | 0.000000000 ✔
0 | 2 | 8 | 0.000000000 | -0.000000000 ✔
0 | 2 | 9 | 0.000000000 | -0.000000000 ✔
0 | 3 | 0 | 0.000000000 | -0.000000000 ✔
0 | 3 | 1 | 0.000000000 | -0.000000000 ✔
0 | 3 | 2 | 0.000000000 | 0.000000000 ✔
0 | 3 | 3 | 0.285714286 | 0.285714286 ✔
0 | 3 | 4 | 0.000000000 | 0.000000000 ✔
0 | 3 | 5 | 0.000000000 | -0.000000000 ✔
0 | 3 | 6 | 0.000000000 | -0.000000000 ✔
0 | 3 | 7 | 0.000000000 | 0.000000000 ✔
0 | 3 | 8 | 0.000000000 | -0.000000000 ✔
0 | 3 | 9 | 0.000000000 | -0.000000000 ✔
0 | 4 | 0 | 0.000000000 | 0.000000000 ✔
0 | 4 | 1 | 0.000000000 | 0.000000000 ✔
0 | 4 | 2 | 0.000000000 | 0.000000000 ✔
0 | 4 | 3 | 0.000000000 | 0.000000000 ✔
0 | 4 | 4 | 0.222222222 | 0.222222222 ✔
0 | 4 | 5 | 0.000000000 | 0.000000000 ✔
0 | 4 | 6 | 0.000000000 | -0.000000000 ✔
0 | 4 | 7 | 0.000000000 | 0.000000000 ✔
0 | 4 | 8 | 0.000000000 | -0.000000000 ✔
0 | 4 | 9 | 0.000000000 | 0.000000000 ✔
0 | 5 | 0 | 0.000000000 | -0.000000000 ✔
0 | 5 | 1 | 0.000000000 | -0.000000000 ✔
0 | 5 | 2 | 0.000000000 | 0.000000000 ✔
0 | 5 | 3 | 0.000000000 | -0.000000000 ✔
0 | 5 | 4 | 0.000000000 | 0.000000000 ✔
0 | 5 | 5 | 0.181818182 | 0.181818182 ✔
0 | 5 | 6 | 0.000000000 | -0.000000000 ✔
0 | 5 | 7 | 0.000000000 | -0.000000000 ✔
0 | 5 | 8 | 0.000000000 | 0.000000000 ✔
0 | 5 | 9 | 0.000000000 | -0.000000000 ✔
0 | 6 | 0 | 0.000000000 | -0.000000000 ✔
0 | 6 | 1 | 0.000000000 | 0.000000000 ✔
0 | 6 | 2 | 0.000000000 | -0.000000000 ✔
0 | 6 | 3 | 0.000000000 | -0.000000000 ✔
0 | 6 | 4 | 0.000000000 | -0.000000000 ✔
0 | 6 | 5 | 0.000000000 | -0.000000000 ✔
0 | 6 | 6 | 0.153846154 | 0.153846154 ✔
0 | 6 | 7 | 0.000000000 | -0.000000000 ✔
0 | 6 | 8 | 0.000000000 | -0.000000000 ✔
0 | 6 | 9 | 0.000000000 | 0.000000000 ✔
0 | 7 | 0 | 0.000000000 | 0.000000000 ✔
0 | 7 | 1 | 0.000000000 | -0.000000000 ✔
0 | 7 | 2 | 0.000000000 | 0.000000000 ✔
0 | 7 | 3 | 0.000000000 | 0.000000000 ✔
0 | 7 | 4 | 0.000000000 | 0.000000000 ✔
0 | 7 | 5 | 0.000000000 | -0.000000000 ✔
0 | 7 | 6 | 0.000000000 | -0.000000000 ✔
0 | 7 | 7 | 0.133333333 | 0.133333333 ✔
0 | 7 | 8 | 0.000000000 | -0.000000000 ✔
0 | 7 | 9 | 0.000000000 | -0.000000000 ✔
0 | 8 | 0 | 0.000000000 | -0.000000000 ✔
0 | 8 | 1 | 0.000000000 | -0.000000000 ✔
0 | 8 | 2 | 0.000000000 | -0.000000000 ✔
0 | 8 | 3 | 0.000000000 | -0.000000000 ✔
0 | 8 | 4 | 0.000000000 | -0.000000000 ✔
0 | 8 | 5 | 0.000000000 | 0.000000000 ✔
0 | 8 | 6 | 0.000000000 | -0.000000000 ✔
0 | 8 | 7 | 0.000000000 | -0.000000000 ✔
0 | 8 | 8 | 0.117647059 | 0.117647059 ✔
0 | 8 | 9 | 0.000000000 | -0.000000000 ✔
0 | 9 | 0 | 0.000000000 | -0.000000000 ✔
0 | 9 | 1 | 0.000000000 | -0.000000000 ✔
0 | 9 | 2 | 0.000000000 | -0.000000000 ✔
0 | 9 | 3 | 0.000000000 | -0.000000000 ✔
0 | 9 | 4 | 0.000000000 | 0.000000000 ✔
0 | 9 | 5 | 0.000000000 | -0.000000000 ✔
0 | 9 | 6 | 0.000000000 | 0.000000000 ✔
0 | 9 | 7 | 0.000000000 | -0.000000000 ✔
0 | 9 | 8 | 0.000000000 | -0.000000000 ✔
0 | 9 | 9 | 0.105263158 | 0.105263158 ✔
1 | 1 | 1 | 1.333333333 | 1.333333333 ✔
1 | 1 | 2 | 0.000000000 | 0.000000000 ✔
1 | 1 | 3 | 0.000000000 | 0.000000000 ✔
1 | 1 | 4 | 0.000000000 | 0.000000000 ✔
1 | 1 | 5 | 0.000000000 | 0.000000000 ✔
1 | 1 | 6 | 0.000000000 | 0.000000000 ✔
1 | 1 | 7 | 0.000000000 | 0.000000000 ✔
1 | 1 | 8 | 0.000000000 | 0.000000000 ✔
1 | 1 | 9 | 0.000000000 | 0.000000000 ✔
1 | 2 | 1 | 0.000000000 | 0.000000000 ✔
1 | 2 | 2 | 2.400000000 | 2.400000000 ✔
1 | 2 | 3 | 0.000000000 | 0.000000000 ✔
1 | 2 | 4 | 0.000000000 | 0.000000000 ✔
1 | 2 | 5 | 0.000000000 | -0.000000000 ✔
1 | 2 | 6 | 0.000000000 | 0.000000000 ✔
1 | 2 | 7 | 0.000000000 | 0.000000000 ✔
1 | 2 | 8 | 0.000000000 | 0.000000000 ✔
1 | 2 | 9 | 0.000000000 | -0.000000000 ✔
1 | 3 | 1 | 0.000000000 | 0.000000000 ✔
1 | 3 | 2 | 0.000000000 | 0.000000000 ✔
1 | 3 | 3 | 3.428571429 | 3.428571429 ✔
1 | 3 | 4 | 0.000000000 | 0.000000000 ✔
1 | 3 | 5 | 0.000000000 | -0.000000000 ✔
1 | 3 | 6 | 0.000000000 | 0.000000000 ✔
1 | 3 | 7 | 0.000000000 | -0.000000000 ✔
1 | 3 | 8 | 0.000000000 | 0.000000000 ✔
1 | 3 | 9 | 0.000000000 | 0.000000000 ✔
1 | 4 | 1 | 0.000000000 | 0.000000000 ✔
1 | 4 | 2 | 0.000000000 | 0.000000000 ✔
1 | 4 | 3 | 0.000000000 | 0.000000000 ✔
1 | 4 | 4 | 4.444444444 | 4.444444444 ✔
1 | 4 | 5 | 0.000000000 | 0.000000000 ✔
1 | 4 | 6 | 0.000000000 | 0.000000000 ✔
1 | 4 | 7 | 0.000000000 | 0.000000000 ✔
1 | 4 | 8 | 0.000000000 | 0.000000000 ✔
1 | 4 | 9 | 0.000000000 | 0.000000000 ✔
1 | 5 | 1 | 0.000000000 | 0.000000000 ✔
1 | 5 | 2 | 0.000000000 | -0.000000000 ✔
1 | 5 | 3 | 0.000000000 | -0.000000000 ✔
1 | 5 | 4 | 0.000000000 | 0.000000000 ✔
1 | 5 | 5 | 5.454545455 | 5.454545455 ✔
1 | 5 | 6 | 0.000000000 | -0.000000000 ✔
1 | 5 | 7 | 0.000000000 | -0.000000000 ✔
1 | 5 | 8 | 0.000000000 | 0.000000000 ✔
1 | 5 | 9 | 0.000000000 | 0.000000000 ✔
1 | 6 | 1 | 0.000000000 | 0.000000000 ✔
1 | 6 | 2 | 0.000000000 | 0.000000000 ✔
1 | 6 | 3 | 0.000000000 | 0.000000000 ✔
1 | 6 | 4 | 0.000000000 | 0.000000000 ✔
1 | 6 | 5 | 0.000000000 | -0.000000000 ✔
1 | 6 | 6 | 6.461538462 | 6.461538462 ✔
1 | 6 | 7 | 0.000000000 | 0.000000000 ✔
1 | 6 | 8 | 0.000000000 | -0.000000000 ✔
1 | 6 | 9 | 0.000000000 | -0.000000000 ✔
1 | 7 | 1 | 0.000000000 | 0.000000000 ✔
1 | 7 | 2 | 0.000000000 | 0.000000000 ✔
1 | 7 | 3 | 0.000000000 | -0.000000000 ✔
1 | 7 | 4 | 0.000000000 | 0.000000000 ✔
1 | 7 | 5 | 0.000000000 | -0.000000000 ✔
1 | 7 | 6 | 0.000000000 | 0.000000000 ✔
1 | 7 | 7 | 7.466666667 | 7.466666667 ✔
1 | 7 | 8 | 0.000000000 | -0.000000000 ✔
1 | 7 | 9 | 0.000000000 | 0.000000000 ✔
1 | 8 | 1 | 0.000000000 | 0.000000000 ✔
1 | 8 | 2 | 0.000000000 | 0.000000000 ✔
1 | 8 | 3 | 0.000000000 | 0.000000000 ✔
1 | 8 | 4 | 0.000000000 | 0.000000000 ✔
1 | 8 | 5 | 0.000000000 | 0.000000000 ✔
1 | 8 | 6 | 0.000000000 | -0.000000000 ✔
1 | 8 | 7 | 0.000000000 | -0.000000000 ✔
1 | 8 | 8 | 8.470588235 | 8.470588235 ✔
1 | 8 | 9 | 0.000000000 | -0.000000000 ✔
1 | 9 | 1 | 0.000000000 | 0.000000000 ✔
1 | 9 | 2 | 0.000000000 | -0.000000000 ✔
1 | 9 | 3 | 0.000000000 | 0.000000000 ✔
1 | 9 | 4 | 0.000000000 | 0.000000000 ✔
1 | 9 | 5 | 0.000000000 | 0.000000000 ✔
1 | 9 | 6 | 0.000000000 | -0.000000000 ✔
1 | 9 | 7 | 0.000000000 | 0.000000000 ✔
1 | 9 | 8 | 0.000000000 | -0.000000000 ✔
1 | 9 | 9 | 9.473684211 | 9.473684211 ✔
2 | 2 | 2 | 9.600000000 | 9.600000000 ✔
2 | 2 | 3 | 0.000000000 | 0.000000000 ✔
2 | 2 | 4 | 0.000000000 | 0.000000000 ✔
2 | 2 | 5 | 0.000000000 | 0.000000000 ✔
2 | 2 | 6 | 0.000000000 | -0.000000000 ✔
2 | 2 | 7 | 0.000000000 | 0.000000000 ✔
2 | 2 | 8 | 0.000000000 | 0.000000000 ✔
2 | 2 | 9 | 0.000000000 | -0.000000000 ✔
2 | 3 | 2 | 0.000000000 | 0.000000000 ✔
2 | 3 | 3 | 34.285714286 | 34.285714286 ✔
2 | 3 | 4 | 0.000000000 | 0.000000000 ✔
2 | 3 | 5 | 0.000000000 | 0.000000000 ✔
2 | 3 | 6 | 0.000000000 | 0.000000000 ✔
2 | 3 | 7 | 0.000000000 | -0.000000000 ✔
2 | 3 | 8 | 0.000000000 | 0.000000000 ✔
2 | 3 | 9 | 0.000000000 | -0.000000000 ✔
2 | 4 | 2 | 0.000000000 | 0.000000000 ✔
2 | 4 | 3 | 0.000000000 | 0.000000000 ✔
2 | 4 | 4 | 80.000000000 | 80.000000000 ✔
2 | 4 | 5 | 0.000000000 | 0.000000000 ✔
2 | 4 | 6 | 0.000000000 | -0.000000000 ✔
2 | 4 | 7 | 0.000000000 | -0.000000000 ✔
2 | 4 | 8 | 0.000000000 | 0.000000000 ✔
2 | 4 | 9 | 0.000000000 | 0.000000000 ✔
2 | 5 | 2 | 0.000000000 | 0.000000000 ✔
2 | 5 | 3 | 0.000000000 | 0.000000000 ✔
2 | 5 | 4 | 0.000000000 | 0.000000000 ✔
2 | 5 | 5 | 152.727272727 | 152.727272727 ✔
2 | 5 | 6 | 0.000000000 | -0.000000000 ✔
2 | 5 | 7 | 0.000000000 | 0.000000000 ✔
2 | 5 | 8 | 0.000000000 | 0.000000000 ✔
2 | 5 | 9 | 0.000000000 | 0.000000000 ✔
2 | 6 | 2 | 0.000000000 | -0.000000000 ✔
2 | 6 | 3 | 0.000000000 | 0.000000000 ✔
2 | 6 | 4 | 0.000000000 | -0.000000000 ✔
2 | 6 | 5 | 0.000000000 | -0.000000000 ✔
2 | 6 | 6 | 258.461538462 | 258.461538462 ✔
2 | 6 | 7 | 0.000000000 | 0.000000000 ✔
2 | 6 | 8 | 0.000000000 | -0.000000000 ✔
2 | 6 | 9 | 0.000000000 | 0.000000000 ✔
2 | 7 | 2 | 0.000000000 | 0.000000000 ✔
2 | 7 | 3 | 0.000000000 | -0.000000000 ✔
2 | 7 | 4 | 0.000000000 | -0.000000000 ✔
2 | 7 | 5 | 0.000000000 | 0.000000000 ✔
2 | 7 | 6 | 0.000000000 | 0.000000000 ✔
2 | 7 | 7 | 403.200000000 | 403.200000000 ✔
2 | 7 | 8 | 0.000000000 | -0.000000000 ✔
2 | 7 | 9 | 0.000000000 | -0.000000000 ✔
2 | 8 | 2 | 0.000000000 | 0.000000000 ✔
2 | 8 | 3 | 0.000000000 | 0.000000000 ✔
2 | 8 | 4 | 0.000000000 | 0.000000000 ✔
2 | 8 | 5 | 0.000000000 | 0.000000000 ✔
2 | 8 | 6 | 0.000000000 | -0.000000000 ✔
2 | 8 | 7 | 0.000000000 | -0.000000000 ✔
2 | 8 | 8 | 592.941176471 | 592.941176471 ✔
2 | 8 | 9 | 0.000000000 | -0.000000000 ✔
2 | 9 | 2 | 0.000000000 | -0.000000000 ✔
2 | 9 | 3 | 0.000000000 | -0.000000000 ✔
2 | 9 | 4 | 0.000000000 | 0.000000000 ✔
2 | 9 | 5 | 0.000000000 | 0.000000000 ✔
2 | 9 | 6 | 0.000000000 | 0.000000000 ✔
2 | 9 | 7 | 0.000000000 | -0.000000000 ✔
2 | 9 | 8 | 0.000000000 | -0.000000000 ✔
2 | 9 | 9 | 833.684210526 | 833.684210526 ✔
3 | 3 | 3 | 205.714285714 | 205.714285714 ✔
3 | 3 | 4 | 0.000000000 | -0.000000000 ✔
3 | 3 | 5 | 0.000000000 | -0.000000000 ✔
3 | 3 | 6 | 0.000000000 | 0.000000000 ✔
3 | 3 | 7 | 0.000000000 | -0.000000000 ✔
3 | 3 | 8 | 0.000000000 | -0.000000000 ✔
3 | 3 | 9 | 0.000000000 | -0.000000000 ✔
3 | 4 | 3 | 0.000000000 | -0.000000000 ✔
3 | 4 | 4 | 1120.000000000 | 1120.000000000 ✔
3 | 4 | 5 | 0.000000000 | 0.000000000 ✔
3 | 4 | 6 | 0.000000000 | 0.000000000 ✔
3 | 4 | 7 | 0.000000000 | 0.000000000 ✔
3 | 4 | 8 | 0.000000000 | 0.000000000 ✔
3 | 4 | 9 | 0.000000000 | 0.000000000 ✔
3 | 5 | 3 | 0.000000000 | -0.000000000 ✔
3 | 5 | 4 | 0.000000000 | 0.000000000 ✔
3 | 5 | 5 | 3665.454545455 | 3665.454545455 ✔
3 | 5 | 6 | 0.000000000 | 0.000000000 ✔
3 | 5 | 7 | 0.000000000 | -0.000000000 ✔
3 | 5 | 8 | 0.000000000 | -0.000000000 ✔
3 | 5 | 9 | 0.000000000 | -0.000000000 ✔
3 | 6 | 3 | 0.000000000 | 0.000000000 ✔
3 | 6 | 4 | 0.000000000 | 0.000000000 ✔
3 | 6 | 5 | 0.000000000 | 0.000000000 ✔
3 | 6 | 6 | 9304.615384615 | 9304.615384615 ✔
3 | 6 | 7 | 0.000000000 | -0.000000000 ✔
3 | 6 | 8 | 0.000000000 | 0.000000000 ✔
3 | 6 | 9 | 0.000000000 | 0.000000000 ✔
3 | 7 | 3 | 0.000000000 | -0.000000000 ✔
3 | 7 | 4 | 0.000000000 | 0.000000000 ✔
3 | 7 | 5 | 0.000000000 | -0.000000000 ✔
3 | 7 | 6 | 0.000000000 | -0.000000000 ✔
3 | 7 | 7 | 20160.000000000 | 20160.000000000 ✔
3 | 7 | 8 | 0.000000000 | 0.000000000 ✔
3 | 7 | 9 | 0.000000000 | 0.000000000 ✔
3 | 8 | 3 | 0.000000000 | -0.000000000 ✔
3 | 8 | 4 | 0.000000000 | 0.000000000 ✔
3 | 8 | 5 | 0.000000000 | -0.000000000 ✔
3 | 8 | 6 | 0.000000000 | 0.000000000 ✔
3 | 8 | 7 | 0.000000000 | 0.000000000 ✔
3 | 8 | 8 | 39134.117647059 | 39134.117647059 ✔
3 | 8 | 9 | 0.000000000 | -0.000000000 ✔
3 | 9 | 3 | 0.000000000 | -0.000000000 ✔
3 | 9 | 4 | 0.000000000 | 0.000000000 ✔
3 | 9 | 5 | 0.000000000 | -0.000000000 ✔
3 | 9 | 6 | 0.000000000 | 0.000000000 ✔
3 | 9 | 7 | 0.000000000 | 0.000000000 ✔
3 | 9 | 8 | 0.000000000 | -0.000000000 ✔
3 | 9 | 9 | 70029.473684211 | 70029.473684211 ✔
4 | 4 | 4 | 8960.000000000 | 8960.000000000 ✔
4 | 4 | 5 | 0.000000000 | -0.000000000 ✔
4 | 4 | 6 | 0.000000000 | -0.000000000 ✔
4 | 4 | 7 | 0.000000000 | -0.000000000 ✔
4 | 4 | 8 | 0.000000000 | 0.000000000 ✔
4 | 4 | 9 | 0.000000000 | 0.000000000 ✔
4 | 5 | 4 | 0.000000000 | -0.000000000 ✔
4 | 5 | 5 | 65978.181818182 | 65978.181818182 ✔
4 | 5 | 6 | 0.000000000 | -0.000000000 ✔
4 | 5 | 7 | 0.000000000 | -0.000000000 ✔
4 | 5 | 8 | 0.000000000 | -0.000000000 ✔
4 | 5 | 9 | 0.000000000 | 0.000000000 ✔
4 | 6 | 4 | 0.000000000 | -0.000000000 ✔
4 | 6 | 5 | 0.000000000 | -0.000000000 ✔
4 | 6 | 6 | 279138.461538462 | 279138.461538462 ✔
4 | 6 | 7 | 0.000000000 | -0.000000000 ✔
4 | 6 | 8 | 0.000000000 | 0.000000000 ✔
4 | 6 | 9 | 0.000000000 | 0.000000000 ✔
4 | 7 | 4 | 0.000000000 | -0.000000000 ✔
4 | 7 | 5 | 0.000000000 | -0.000000000 ✔
4 | 7 | 6 | 0.000000000 | -0.000000000 ✔
4 | 7 | 7 | 887040.000000000 | 887040.000000000 ✔
4 | 7 | 8 | 0.000000000 | 0.000000000 ✔
4 | 7 | 9 | 0.000000000 | 0.000000000 ✔
4 | 8 | 4 | 0.000000000 | 0.000000000 ✔
4 | 8 | 5 | 0.000000000 | -0.000000000 ✔
4 | 8 | 6 | 0.000000000 | 0.000000000 ✔
4 | 8 | 7 | 0.000000000 | 0.000000000 ✔
4 | 8 | 8 | 2348047.058823529 | 2348047.058823530 ✔
4 | 8 | 9 | 0.000000000 | 0.000000000 ✔
4 | 9 | 4 | 0.000000000 | 0.000000000 ✔
4 | 9 | 5 | 0.000000000 | 0.000000000 ✔
4 | 9 | 6 | 0.000000000 | 0.000000000 ✔
4 | 9 | 7 | 0.000000000 | 0.000000000 ✔
4 | 9 | 8 | 0.000000000 | 0.000000000 ✔
4 | 9 | 9 | 5462298.947368422 | 5462298.947368421 ✔
5 | 5 | 5 | 659781.818181818 | 659781.818181818 ✔
5 | 5 | 6 | 0.000000000 | -0.000000000 ✔
5 | 5 | 7 | 0.000000000 | 0.000000000 ✔
5 | 5 | 8 | 0.000000000 | 0.000000001 ✔
5 | 5 | 9 | 0.000000000 | 0.000000000 ✔
5 | 6 | 5 | 0.000000000 | -0.000000000 ✔
5 | 6 | 6 | 6141046.153846154 | 6141046.153846157 ✔
5 | 6 | 7 | 0.000000000 | 0.000000000 ✔
5 | 6 | 8 | 0.000000000 | 0.000000002 ✔
5 | 6 | 9 | 0.000000000 | 0.000000001 ✔
5 | 7 | 5 | 0.000000000 | 0.000000000 ✔
5 | 7 | 6 | 0.000000000 | 0.000000000 ✔
5 | 7 | 7 | 31933440.000000000 | 31933440.000000000 ✔
5 | 7 | 8 | 0.000000000 | 0.000000003 ✔
5 | 7 | 9 | 0.000000000 | 0.000000004 ✔
5 | 8 | 5 | 0.000000000 | 0.000000001 ✔
5 | 8 | 6 | 0.000000000 | 0.000000002 ✔
5 | 8 | 7 | 0.000000000 | 0.000000003 ✔
5 | 8 | 8 | 122098447.058823526 | 122098447.058823526 ✔
5 | 8 | 9 | 0.000000000 | -0.000000001 ✔
5 | 9 | 5 | 0.000000000 | 0.000000000 ✔
5 | 9 | 6 | 0.000000000 | 0.000000001 ✔
5 | 9 | 7 | 0.000000000 | 0.000000004 ✔
5 | 9 | 8 | 0.000000000 | -0.000000001 ✔
5 | 9 | 9 | 382360926.315789461 | 382360926.315789461 ✔Normalization & Orthogonality of $Y_{lm}(\theta,\varphi)$
\[\int_0^{2\pi} \int_0^\pi Y_{lm}(\theta,\varphi)^* Y_{l'm'}(\theta,\varphi) \sin(\theta) ~\mathrm{d}\theta \mathrm{d}\varphi = \delta_{ll'} \delta_{mm'}\]
l₁ | l₂ | m₁ | m₂ | analytical | numerical
-- | -- | -- | -- | -------------- | --------------
0 | 0 | 0 | 0 | 1.000000000 | 1.000000000 ✔
0 | 1 | 0 | -1 | 0.000000000 | 0.000000000 ✔
0 | 1 | 0 | 0 | 0.000000000 | -0.000000000 ✔
0 | 1 | 0 | 1 | 0.000000000 | 0.000000000 ✔
0 | 2 | 0 | -2 | 0.000000000 | -0.000000000 ✔
0 | 2 | 0 | -1 | 0.000000000 | 0.000000000 ✔
0 | 2 | 0 | 0 | 0.000000000 | 0.000000000 ✔
0 | 2 | 0 | 1 | 0.000000000 | -0.000000000 ✔
0 | 2 | 0 | 2 | 0.000000000 | -0.000000000 ✔
1 | 0 | -1 | 0 | 0.000000000 | 0.000000000 ✔
1 | 0 | 0 | 0 | 0.000000000 | -0.000000000 ✔
1 | 0 | 1 | 0 | 0.000000000 | 0.000000000 ✔
1 | 1 | -1 | -1 | 1.000000000 | 1.000000000 ✔
1 | 1 | -1 | 0 | 0.000000000 | 0.000000000 ✔
1 | 1 | -1 | 1 | 0.000000000 | 0.000000000 ✔
1 | 1 | 0 | -1 | 0.000000000 | 0.000000000 ✔
1 | 1 | 0 | 0 | 1.000000000 | 1.000000000 ✔
1 | 1 | 0 | 1 | 0.000000000 | -0.000000000 ✔
1 | 1 | 1 | -1 | 0.000000000 | 0.000000000 ✔
1 | 1 | 1 | 0 | 0.000000000 | -0.000000000 ✔
1 | 1 | 1 | 1 | 1.000000000 | 1.000000000 ✔
1 | 2 | -1 | -2 | 0.000000000 | -0.000000000 ✔
1 | 2 | -1 | -1 | 0.000000000 | -0.000000000 ✔
1 | 2 | -1 | 0 | 0.000000000 | 0.000000000 ✔
1 | 2 | -1 | 1 | 0.000000000 | -0.000000000 ✔
1 | 2 | -1 | 2 | 0.000000000 | 0.000000000 ✔
1 | 2 | 0 | -2 | 0.000000000 | -0.000000000 ✔
1 | 2 | 0 | -1 | 0.000000000 | -0.000000000 ✔
1 | 2 | 0 | 0 | 0.000000000 | 0.000000000 ✔
1 | 2 | 0 | 1 | 0.000000000 | 0.000000000 ✔
1 | 2 | 0 | 2 | 0.000000000 | -0.000000000 ✔
1 | 2 | 1 | -2 | 0.000000000 | -0.000000000 ✔
1 | 2 | 1 | -1 | 0.000000000 | -0.000000000 ✔
1 | 2 | 1 | 0 | 0.000000000 | -0.000000000 ✔
1 | 2 | 1 | 1 | 0.000000000 | -0.000000000 ✔
1 | 2 | 1 | 2 | 0.000000000 | 0.000000000 ✔
2 | 0 | -2 | 0 | 0.000000000 | -0.000000000 ✔
2 | 0 | -1 | 0 | 0.000000000 | 0.000000000 ✔
2 | 0 | 0 | 0 | 0.000000000 | 0.000000000 ✔
2 | 0 | 1 | 0 | 0.000000000 | -0.000000000 ✔
2 | 0 | 2 | 0 | 0.000000000 | -0.000000000 ✔
2 | 1 | -2 | -1 | 0.000000000 | -0.000000000 ✔
2 | 1 | -2 | 0 | 0.000000000 | -0.000000000 ✔
2 | 1 | -2 | 1 | 0.000000000 | -0.000000000 ✔
2 | 1 | -1 | -1 | 0.000000000 | -0.000000000 ✔
2 | 1 | -1 | 0 | 0.000000000 | -0.000000000 ✔
2 | 1 | -1 | 1 | 0.000000000 | -0.000000000 ✔
2 | 1 | 0 | -1 | 0.000000000 | 0.000000000 ✔
2 | 1 | 0 | 0 | 0.000000000 | 0.000000000 ✔
2 | 1 | 0 | 1 | 0.000000000 | -0.000000000 ✔
2 | 1 | 1 | -1 | 0.000000000 | -0.000000000 ✔
2 | 1 | 1 | 0 | 0.000000000 | 0.000000000 ✔
2 | 1 | 1 | 1 | 0.000000000 | -0.000000000 ✔
2 | 1 | 2 | -1 | 0.000000000 | 0.000000000 ✔
2 | 1 | 2 | 0 | 0.000000000 | -0.000000000 ✔
2 | 1 | 2 | 1 | 0.000000000 | 0.000000000 ✔
2 | 2 | -2 | -2 | 1.000000000 | 1.000000000 ✔
2 | 2 | -2 | -1 | 0.000000000 | -0.000000000 ✔
2 | 2 | -2 | 0 | 0.000000000 | 0.000000000 ✔
2 | 2 | -2 | 1 | 0.000000000 | 0.000000000 ✔
2 | 2 | -2 | 2 | 0.000000000 | -0.000000000 ✔
2 | 2 | -1 | -2 | 0.000000000 | -0.000000000 ✔
2 | 2 | -1 | -1 | 1.000000000 | 1.000000000 ✔
2 | 2 | -1 | 0 | 0.000000000 | -0.000000000 ✔
2 | 2 | -1 | 1 | 0.000000000 | 0.000000000 ✔
2 | 2 | -1 | 2 | 0.000000000 | -0.000000000 ✔
2 | 2 | 0 | -2 | 0.000000000 | 0.000000000 ✔
2 | 2 | 0 | -1 | 0.000000000 | -0.000000000 ✔
2 | 2 | 0 | 0 | 1.000000000 | 1.000000000 ✔
2 | 2 | 0 | 1 | 0.000000000 | 0.000000000 ✔
2 | 2 | 0 | 2 | 0.000000000 | 0.000000000 ✔
2 | 2 | 1 | -2 | 0.000000000 | 0.000000000 ✔
2 | 2 | 1 | -1 | 0.000000000 | 0.000000000 ✔
2 | 2 | 1 | 0 | 0.000000000 | 0.000000000 ✔
2 | 2 | 1 | 1 | 1.000000000 | 1.000000000 ✔
2 | 2 | 1 | 2 | 0.000000000 | 0.000000000 ✔
2 | 2 | 2 | -2 | 0.000000000 | -0.000000000 ✔
2 | 2 | 2 | -1 | 0.000000000 | -0.000000000 ✔
2 | 2 | 2 | 0 | 0.000000000 | 0.000000000 ✔
2 | 2 | 2 | 1 | 0.000000000 | 0.000000000 ✔
2 | 2 | 2 | 2 | 1.000000000 | 1.000000000 ✔Generalized Laguerre Polynomials $L_n^{(\alpha)}(x)$
\[ \begin{aligned} L_n^{(\alpha)}(x) &= \frac{x^{-\alpha}e^x}{n!} \frac{d^n}{dx^n}\left(x^{n+\alpha}e^{-x}\right) \\ &= \sum_{k=0}^n(-1)^k \frac{\Gamma(\alpha+n+1)}{\Gamma(\alpha+k+1)\Gamma(n-k+1)} \frac{x^k}{k !}. \end{aligned}\]
$n=0, α=0:$ ✔
\[\begin{aligned} L_{0}^{(0)}(x) = e^{ - x} e^{x} &= 1 \\ &= 1 \end{aligned}\]
$n=1, α=0:$ ✔
\[\begin{aligned} L_{1}^{(0)}(x) = \frac{\mathrm{d}}{\mathrm{d}x} x e^{ - x} e^{x} &= 1 - x \\ &= 1 - x \end{aligned}\]
$n=1, α=1:$ ✔
\[\begin{aligned} L_{1}^{(1)}(x) = \frac{\frac{\mathrm{d}}{\mathrm{d}x} x^{2} e^{ - x} e^{x}}{x} &= 2 - x \\ &= 2 - x \end{aligned}\]
$n=2, α=0:$ ✔
\[\begin{aligned} L_{2}^{(0)}(x) = \frac{1}{2} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} x^{2} e^{ - x} e^{x} &= 1 - 2 x + \frac{1}{2} x^{2} \\ &= 1 - 2 x + \frac{1}{2} x^{2} \end{aligned}\]
$n=2, α=1:$ ✔
\[\begin{aligned} L_{2}^{(1)}(x) = \frac{\frac{1}{2} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} x^{3} e^{ - x} e^{x}}{x} &= 3 - 3 x + \frac{1}{2} x^{2} \\ &= 3 - 3 x + \frac{1}{2} x^{2} \end{aligned}\]
$n=2, α=2:$ ✔
\[\begin{aligned} L_{2}^{(2)}(x) = \frac{\frac{1}{2} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} x^{4} e^{ - x} e^{x}}{x^{2}} &= 6 - 4 x + \frac{1}{2} x^{2} \\ &= 6 - 4 x + \frac{1}{2} x^{2} \end{aligned}\]
$n=3, α=0:$ ✔
\[\begin{aligned} L_{3}^{(0)}(x) = \frac{1}{6} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} x^{3} e^{ - x} e^{x} &= 1 - 3 x + \frac{3}{2} x^{2} - \frac{1}{6} x^{3} \\ &= 1 - 3 x + \frac{3}{2} x^{2} - \frac{1}{6} x^{3} \end{aligned}\]
$n=3, α=1:$ ✔
\[\begin{aligned} L_{3}^{(1)}(x) = \frac{\frac{1}{6} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} x^{4} e^{ - x} e^{x}}{x} &= 4 - 6 x + 2 x^{2} - \frac{1}{6} x^{3} \\ &= 4 - 6 x + 2 x^{2} - \frac{1}{6} x^{3} \end{aligned}\]
$n=3, α=2:$ ✔
\[\begin{aligned} L_{3}^{(2)}(x) = \frac{\frac{1}{6} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} x^{5} e^{ - x} e^{x}}{x^{2}} &= 10 - 10 x + \frac{5}{2} x^{2} - \frac{1}{6} x^{3} \\ &= 10 - 10 x + \frac{5}{2} x^{2} - \frac{1}{6} x^{3} \end{aligned}\]
$n=3, α=3:$ ✔
\[\begin{aligned} L_{3}^{(3)}(x) = \frac{\frac{1}{6} e^{x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} x^{6} e^{ - x}}{x^{3}} &= 20 - 15 x + 3 x^{2} - \frac{1}{6} x^{3} \\ &= 20 - 15 x + 3 x^{2} - \frac{1}{6} x^{3} \end{aligned}\]
$n=4, α=0:$ ✔
\[\begin{aligned} L_{4}^{(0)}(x) = \frac{1}{24} e^{x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} x^{4} e^{ - x} &= 1 - 4 x + 3 x^{2} - \frac{2}{3} x^{3} + \frac{1}{24} x^{4} \\ &= 1 - 4 x + 3 x^{2} - \frac{2}{3} x^{3} + \frac{1}{24} x^{4} \end{aligned}\]
$n=4, α=1:$ ✔
\[\begin{aligned} L_{4}^{(1)}(x) = \frac{\frac{1}{24} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} x^{5} e^{ - x} e^{x}}{x} &= 5 - 10 x + 5 x^{2} - \frac{5}{6} x^{3} + \frac{1}{24} x^{4} \\ &= 5 - 10 x + 5 x^{2} - \frac{5}{6} x^{3} + \frac{1}{24} x^{4} \end{aligned}\]
$n=4, α=2:$ ✔
\[\begin{aligned} L_{4}^{(2)}(x) = \frac{\frac{1}{24} e^{x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} x^{6} e^{ - x}}{x^{2}} &= 15 - 20 x + \frac{15}{2} x^{2} - x^{3} + \frac{1}{24} x^{4} \\ &= 15 - 20 x + \frac{15}{2} x^{2} - x^{3} + \frac{1}{24} x^{4} \end{aligned}\]
$n=4, α=3:$ ✔
\[\begin{aligned} L_{4}^{(3)}(x) = \frac{\frac{1}{24} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} x^{7} e^{ - x} e^{x}}{x^{3}} &= 35 - 35 x + \frac{21}{2} x^{2} - \frac{7}{6} x^{3} + \frac{1}{24} x^{4} \\ &= 35 - 35 x + \frac{21}{2} x^{2} - \frac{7}{6} x^{3} + \frac{1}{24} x^{4} \end{aligned}\]
$n=4, α=4:$ ✔
\[\begin{aligned} L_{4}^{(4)}(x) = \frac{\frac{1}{24} e^{x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} x^{8} e^{ - x}}{x^{4}} &= 70 - 56 x + 14 x^{2} - \frac{4}{3} x^{3} + \frac{1}{24} x^{4} \\ &= 70 - 56 x + 14 x^{2} - \frac{4}{3} x^{3} + \frac{1}{24} x^{4} \end{aligned}\]
Normalization & Orthogonality of $L_n^{(\alpha)}(x)$
\[\int_0^\infty L_i^{(\alpha)}(x) L_j^{(\alpha)}(x) x^\alpha \mathrm{e}^{-x} \mathrm{d}x = \frac{\Gamma(n+\alpha+1)}{n!} \delta_{ij}\]
α | i | j | analytical | numerical
---- | -- | -- | -------------- | --------------
0.01 | 0 | 0 | 0.994325851 | 0.994325853 ✔
0.01 | 0 | 1 | 0.000000000 | 0.000000000 ✔
0.01 | 0 | 2 | 0.000000000 | 0.000000000 ✔
0.01 | 0 | 3 | 0.000000000 | 0.000000000 ✔
0.01 | 0 | 4 | 0.000000000 | 0.000000000 ✔
0.01 | 0 | 5 | 0.000000000 | 0.000000000 ✔
0.01 | 0 | 6 | 0.000000000 | 0.000000000 ✔
0.01 | 0 | 7 | 0.000000000 | 0.000000000 ✔
0.01 | 0 | 8 | 0.000000000 | 0.000000000 ✔
0.01 | 0 | 9 | 0.000000000 | 0.000000000 ✔
0.01 | 1 | 0 | 0.000000000 | 0.000000000 ✔
0.01 | 1 | 1 | 1.004269110 | 1.004269111 ✔
0.01 | 1 | 2 | 0.000000000 | 0.000000000 ✔
0.01 | 1 | 3 | 0.000000000 | 0.000000000 ✔
0.01 | 1 | 4 | 0.000000000 | 0.000000000 ✔
0.01 | 1 | 5 | 0.000000000 | 0.000000000 ✔
0.01 | 1 | 6 | 0.000000000 | 0.000000000 ✔
0.01 | 1 | 7 | 0.000000000 | 0.000000000 ✔
0.01 | 1 | 8 | 0.000000000 | 0.000000000 ✔
0.01 | 1 | 9 | 0.000000000 | 0.000000000 ✔
0.01 | 2 | 0 | 0.000000000 | 0.000000000 ✔
0.01 | 2 | 1 | 0.000000000 | 0.000000000 ✔
0.01 | 2 | 2 | 1.009290455 | 1.009290456 ✔
0.01 | 2 | 3 | 0.000000000 | 0.000000000 ✔
0.01 | 2 | 4 | 0.000000000 | 0.000000000 ✔
0.01 | 2 | 5 | 0.000000000 | 0.000000000 ✔
0.01 | 2 | 6 | 0.000000000 | 0.000000000 ✔
0.01 | 2 | 7 | 0.000000000 | 0.000000000 ✔
0.01 | 2 | 8 | 0.000000000 | 0.000000000 ✔
0.01 | 2 | 9 | 0.000000000 | 0.000000000 ✔
0.01 | 3 | 0 | 0.000000000 | 0.000000000 ✔
0.01 | 3 | 1 | 0.000000000 | 0.000000000 ✔
0.01 | 3 | 2 | 0.000000000 | 0.000000000 ✔
0.01 | 3 | 3 | 1.012654757 | 1.012654759 ✔
0.01 | 3 | 4 | 0.000000000 | 0.000000000 ✔
0.01 | 3 | 5 | 0.000000000 | 0.000000000 ✔
0.01 | 3 | 6 | 0.000000000 | 0.000000000 ✔
0.01 | 3 | 7 | 0.000000000 | 0.000000000 ✔
0.01 | 3 | 8 | 0.000000000 | 0.000000000 ✔
0.01 | 3 | 9 | 0.000000000 | 0.000000000 ✔
0.01 | 4 | 0 | 0.000000000 | 0.000000000 ✔
0.01 | 4 | 1 | 0.000000000 | 0.000000000 ✔
0.01 | 4 | 2 | 0.000000000 | 0.000000000 ✔
0.01 | 4 | 3 | 0.000000000 | 0.000000000 ✔
0.01 | 4 | 4 | 1.015186394 | 1.015186395 ✔
0.01 | 4 | 5 | 0.000000000 | 0.000000000 ✔
0.01 | 4 | 6 | 0.000000000 | 0.000000000 ✔
0.01 | 4 | 7 | 0.000000000 | 0.000000000 ✔
0.01 | 4 | 8 | 0.000000000 | 0.000000000 ✔
0.01 | 4 | 9 | 0.000000000 | 0.000000000 ✔
0.01 | 5 | 0 | 0.000000000 | 0.000000000 ✔
0.01 | 5 | 1 | 0.000000000 | 0.000000000 ✔
0.01 | 5 | 2 | 0.000000000 | 0.000000000 ✔
0.01 | 5 | 3 | 0.000000000 | 0.000000000 ✔
0.01 | 5 | 4 | 0.000000000 | 0.000000000 ✔
0.01 | 5 | 5 | 1.017216766 | 1.017216768 ✔
0.01 | 5 | 6 | 0.000000000 | 0.000000000 ✔
0.01 | 5 | 7 | 0.000000000 | 0.000000000 ✔
0.01 | 5 | 8 | 0.000000000 | 0.000000000 ✔
0.01 | 5 | 9 | 0.000000000 | 0.000000000 ✔
0.01 | 6 | 0 | 0.000000000 | 0.000000000 ✔
0.01 | 6 | 1 | 0.000000000 | 0.000000000 ✔
0.01 | 6 | 2 | 0.000000000 | 0.000000000 ✔
0.01 | 6 | 3 | 0.000000000 | 0.000000000 ✔
0.01 | 6 | 4 | 0.000000000 | 0.000000000 ✔
0.01 | 6 | 5 | 0.000000000 | 0.000000000 ✔
0.01 | 6 | 6 | 1.018912128 | 1.018912129 ✔
0.01 | 6 | 7 | 0.000000000 | 0.000000000 ✔
0.01 | 6 | 8 | 0.000000000 | 0.000000000 ✔
0.01 | 6 | 9 | 0.000000000 | 0.000000000 ✔
0.01 | 7 | 0 | 0.000000000 | 0.000000000 ✔
0.01 | 7 | 1 | 0.000000000 | 0.000000000 ✔
0.01 | 7 | 2 | 0.000000000 | 0.000000000 ✔
0.01 | 7 | 3 | 0.000000000 | 0.000000000 ✔
0.01 | 7 | 4 | 0.000000000 | 0.000000000 ✔
0.01 | 7 | 5 | 0.000000000 | 0.000000000 ✔
0.01 | 7 | 6 | 0.000000000 | 0.000000000 ✔
0.01 | 7 | 7 | 1.020367716 | 1.020367717 ✔
0.01 | 7 | 8 | 0.000000000 | 0.000000000 ✔
0.01 | 7 | 9 | 0.000000000 | 0.000000000 ✔
0.01 | 8 | 0 | 0.000000000 | 0.000000000 ✔
0.01 | 8 | 1 | 0.000000000 | 0.000000000 ✔
0.01 | 8 | 2 | 0.000000000 | 0.000000000 ✔
0.01 | 8 | 3 | 0.000000000 | 0.000000000 ✔
0.01 | 8 | 4 | 0.000000000 | 0.000000000 ✔
0.01 | 8 | 5 | 0.000000000 | 0.000000000 ✔
0.01 | 8 | 6 | 0.000000000 | 0.000000000 ✔
0.01 | 8 | 7 | 0.000000000 | 0.000000000 ✔
0.01 | 8 | 8 | 1.021643176 | 1.021643178 ✔
0.01 | 8 | 9 | 0.000000000 | 0.000000000 ✔
0.01 | 9 | 0 | 0.000000000 | 0.000000000 ✔
0.01 | 9 | 1 | 0.000000000 | 0.000000000 ✔
0.01 | 9 | 2 | 0.000000000 | 0.000000000 ✔
0.01 | 9 | 3 | 0.000000000 | 0.000000000 ✔
0.01 | 9 | 4 | 0.000000000 | 0.000000000 ✔
0.01 | 9 | 5 | 0.000000000 | 0.000000000 ✔
0.01 | 9 | 6 | 0.000000000 | 0.000000000 ✔
0.01 | 9 | 7 | 0.000000000 | 0.000000000 ✔
0.01 | 9 | 8 | 0.000000000 | 0.000000000 ✔
0.01 | 9 | 9 | 1.022778335 | 1.022778336 ✔
0.10 | 0 | 0 | 0.951350770 | 0.951350772 ✔
0.10 | 0 | 1 | 0.000000000 | 0.000000000 ✔
0.10 | 0 | 2 | 0.000000000 | 0.000000000 ✔
0.10 | 0 | 3 | 0.000000000 | 0.000000000 ✔
0.10 | 0 | 4 | 0.000000000 | 0.000000000 ✔
0.10 | 0 | 5 | 0.000000000 | 0.000000000 ✔
0.10 | 0 | 6 | 0.000000000 | 0.000000000 ✔
0.10 | 0 | 7 | 0.000000000 | 0.000000000 ✔
0.10 | 0 | 8 | 0.000000000 | 0.000000000 ✔
0.10 | 0 | 9 | 0.000000000 | 0.000000000 ✔
0.10 | 1 | 0 | 0.000000000 | 0.000000000 ✔
0.10 | 1 | 1 | 1.046485847 | 1.046485848 ✔
0.10 | 1 | 2 | 0.000000000 | 0.000000000 ✔
0.10 | 1 | 3 | 0.000000000 | 0.000000000 ✔
0.10 | 1 | 4 | 0.000000000 | 0.000000000 ✔
0.10 | 1 | 5 | 0.000000000 | 0.000000000 ✔
0.10 | 1 | 6 | 0.000000000 | 0.000000000 ✔
0.10 | 1 | 7 | 0.000000000 | 0.000000000 ✔
0.10 | 1 | 8 | 0.000000000 | 0.000000000 ✔
0.10 | 1 | 9 | 0.000000000 | 0.000000000 ✔
0.10 | 2 | 0 | 0.000000000 | 0.000000000 ✔
0.10 | 2 | 1 | 0.000000000 | 0.000000000 ✔
0.10 | 2 | 2 | 1.098810139 | 1.098810140 ✔
0.10 | 2 | 3 | 0.000000000 | 0.000000000 ✔
0.10 | 2 | 4 | 0.000000000 | 0.000000000 ✔
0.10 | 2 | 5 | 0.000000000 | 0.000000000 ✔
0.10 | 2 | 6 | 0.000000000 | 0.000000000 ✔
0.10 | 2 | 7 | 0.000000000 | 0.000000000 ✔
0.10 | 2 | 8 | 0.000000000 | 0.000000000 ✔
0.10 | 2 | 9 | 0.000000000 | 0.000000000 ✔
0.10 | 3 | 0 | 0.000000000 | 0.000000000 ✔
0.10 | 3 | 1 | 0.000000000 | 0.000000000 ✔
0.10 | 3 | 2 | 0.000000000 | 0.000000000 ✔
0.10 | 3 | 3 | 1.135437144 | 1.135437145 ✔
0.10 | 3 | 4 | 0.000000000 | 0.000000000 ✔
0.10 | 3 | 5 | 0.000000000 | 0.000000000 ✔
0.10 | 3 | 6 | 0.000000000 | 0.000000000 ✔
0.10 | 3 | 7 | 0.000000000 | 0.000000000 ✔
0.10 | 3 | 8 | 0.000000000 | 0.000000000 ✔
0.10 | 3 | 9 | 0.000000000 | 0.000000000 ✔
0.10 | 4 | 0 | 0.000000000 | 0.000000000 ✔
0.10 | 4 | 1 | 0.000000000 | 0.000000000 ✔
0.10 | 4 | 2 | 0.000000000 | 0.000000000 ✔
0.10 | 4 | 3 | 0.000000000 | 0.000000000 ✔
0.10 | 4 | 4 | 1.163823072 | 1.163823074 ✔
0.10 | 4 | 5 | 0.000000000 | 0.000000000 ✔
0.10 | 4 | 6 | 0.000000000 | 0.000000000 ✔
0.10 | 4 | 7 | 0.000000000 | 0.000000000 ✔
0.10 | 4 | 8 | 0.000000000 | 0.000000000 ✔
0.10 | 4 | 9 | 0.000000000 | 0.000000000 ✔
0.10 | 5 | 0 | 0.000000000 | 0.000000000 ✔
0.10 | 5 | 1 | 0.000000000 | 0.000000000 ✔
0.10 | 5 | 2 | 0.000000000 | 0.000000000 ✔
0.10 | 5 | 3 | 0.000000000 | 0.000000000 ✔
0.10 | 5 | 4 | 0.000000000 | 0.000000000 ✔
0.10 | 5 | 5 | 1.187099534 | 1.187099535 ✔
0.10 | 5 | 6 | 0.000000000 | 0.000000000 ✔
0.10 | 5 | 7 | 0.000000000 | 0.000000000 ✔
0.10 | 5 | 8 | 0.000000000 | 0.000000000 ✔
0.10 | 5 | 9 | 0.000000000 | 0.000000000 ✔
0.10 | 6 | 0 | 0.000000000 | 0.000000000 ✔
0.10 | 6 | 1 | 0.000000000 | 0.000000000 ✔
0.10 | 6 | 2 | 0.000000000 | 0.000000000 ✔
0.10 | 6 | 3 | 0.000000000 | 0.000000000 ✔
0.10 | 6 | 4 | 0.000000000 | 0.000000000 ✔
0.10 | 6 | 5 | 0.000000000 | 0.000000000 ✔
0.10 | 6 | 6 | 1.206884526 | 1.206884527 ✔
0.10 | 6 | 7 | 0.000000000 | 0.000000000 ✔
0.10 | 6 | 8 | 0.000000000 | 0.000000000 ✔
0.10 | 6 | 9 | 0.000000000 | 0.000000000 ✔
0.10 | 7 | 0 | 0.000000000 | 0.000000000 ✔
0.10 | 7 | 1 | 0.000000000 | 0.000000000 ✔
0.10 | 7 | 2 | 0.000000000 | 0.000000000 ✔
0.10 | 7 | 3 | 0.000000000 | 0.000000000 ✔
0.10 | 7 | 4 | 0.000000000 | 0.000000000 ✔
0.10 | 7 | 5 | 0.000000000 | 0.000000000 ✔
0.10 | 7 | 6 | 0.000000000 | 0.000000000 ✔
0.10 | 7 | 7 | 1.224125734 | 1.224125734 ✔
0.10 | 7 | 8 | 0.000000000 | 0.000000000 ✔
0.10 | 7 | 9 | 0.000000000 | 0.000000000 ✔
0.10 | 8 | 0 | 0.000000000 | 0.000000000 ✔
0.10 | 8 | 1 | 0.000000000 | 0.000000000 ✔
0.10 | 8 | 2 | 0.000000000 | 0.000000000 ✔
0.10 | 8 | 3 | 0.000000000 | 0.000000000 ✔
0.10 | 8 | 4 | 0.000000000 | 0.000000000 ✔
0.10 | 8 | 5 | 0.000000000 | 0.000000000 ✔
0.10 | 8 | 6 | 0.000000000 | 0.000000000 ✔
0.10 | 8 | 7 | 0.000000000 | 0.000000000 ✔
0.10 | 8 | 8 | 1.239427305 | 1.239427307 ✔
0.10 | 8 | 9 | 0.000000000 | 0.000000000 ✔
0.10 | 9 | 0 | 0.000000000 | 0.000000000 ✔
0.10 | 9 | 1 | 0.000000000 | 0.000000000 ✔
0.10 | 9 | 2 | 0.000000000 | 0.000000000 ✔
0.10 | 9 | 3 | 0.000000000 | 0.000000000 ✔
0.10 | 9 | 4 | 0.000000000 | 0.000000000 ✔
0.10 | 9 | 5 | 0.000000000 | 0.000000000 ✔
0.10 | 9 | 6 | 0.000000000 | 0.000000000 ✔
0.10 | 9 | 7 | 0.000000000 | 0.000000000 ✔
0.10 | 9 | 8 | 0.000000000 | 0.000000000 ✔
0.10 | 9 | 9 | 1.253198720 | 1.253198721 ✔
1.00 | 0 | 0 | 1.000000000 | 1.000000000 ✔
1.00 | 0 | 1 | 0.000000000 | -0.000000000 ✔
1.00 | 0 | 2 | 0.000000000 | -0.000000000 ✔
1.00 | 0 | 3 | 0.000000000 | -0.000000000 ✔
1.00 | 0 | 4 | 0.000000000 | -0.000000000 ✔
1.00 | 0 | 5 | 0.000000000 | -0.000000000 ✔
1.00 | 0 | 6 | 0.000000000 | 0.000000000 ✔
1.00 | 0 | 7 | 0.000000000 | -0.000000000 ✔
1.00 | 0 | 8 | 0.000000000 | 0.000000000 ✔
1.00 | 0 | 9 | 0.000000000 | -0.000000000 ✔
1.00 | 1 | 0 | 0.000000000 | -0.000000000 ✔
1.00 | 1 | 1 | 2.000000000 | 2.000000000 ✔
1.00 | 1 | 2 | 0.000000000 | 0.000000000 ✔
1.00 | 1 | 3 | 0.000000000 | 0.000000000 ✔
1.00 | 1 | 4 | 0.000000000 | 0.000000000 ✔
1.00 | 1 | 5 | 0.000000000 | 0.000000000 ✔
1.00 | 1 | 6 | 0.000000000 | -0.000000000 ✔
1.00 | 1 | 7 | 0.000000000 | 0.000000000 ✔
1.00 | 1 | 8 | 0.000000000 | -0.000000000 ✔
1.00 | 1 | 9 | 0.000000000 | 0.000000000 ✔
1.00 | 2 | 0 | 0.000000000 | -0.000000000 ✔
1.00 | 2 | 1 | 0.000000000 | 0.000000000 ✔
1.00 | 2 | 2 | 3.000000000 | 3.000000000 ✔
1.00 | 2 | 3 | 0.000000000 | -0.000000000 ✔
1.00 | 2 | 4 | 0.000000000 | 0.000000000 ✔
1.00 | 2 | 5 | 0.000000000 | -0.000000000 ✔
1.00 | 2 | 6 | 0.000000000 | -0.000000000 ✔
1.00 | 2 | 7 | 0.000000000 | -0.000000000 ✔
1.00 | 2 | 8 | 0.000000000 | 0.000000000 ✔
1.00 | 2 | 9 | 0.000000000 | -0.000000000 ✔
1.00 | 3 | 0 | 0.000000000 | -0.000000000 ✔
1.00 | 3 | 1 | 0.000000000 | 0.000000000 ✔
1.00 | 3 | 2 | 0.000000000 | -0.000000000 ✔
1.00 | 3 | 3 | 4.000000000 | 4.000000000 ✔
1.00 | 3 | 4 | 0.000000000 | -0.000000000 ✔
1.00 | 3 | 5 | 0.000000000 | 0.000000000 ✔
1.00 | 3 | 6 | 0.000000000 | 0.000000000 ✔
1.00 | 3 | 7 | 0.000000000 | 0.000000000 ✔
1.00 | 3 | 8 | 0.000000000 | -0.000000000 ✔
1.00 | 3 | 9 | 0.000000000 | -0.000000000 ✔
1.00 | 4 | 0 | 0.000000000 | -0.000000000 ✔
1.00 | 4 | 1 | 0.000000000 | 0.000000000 ✔
1.00 | 4 | 2 | 0.000000000 | 0.000000000 ✔
1.00 | 4 | 3 | 0.000000000 | -0.000000000 ✔
1.00 | 4 | 4 | 5.000000000 | 5.000000000 ✔
1.00 | 4 | 5 | 0.000000000 | -0.000000000 ✔
1.00 | 4 | 6 | 0.000000000 | -0.000000000 ✔
1.00 | 4 | 7 | 0.000000000 | -0.000000000 ✔
1.00 | 4 | 8 | 0.000000000 | 0.000000000 ✔
1.00 | 4 | 9 | 0.000000000 | 0.000000000 ✔
1.00 | 5 | 0 | 0.000000000 | -0.000000000 ✔
1.00 | 5 | 1 | 0.000000000 | 0.000000000 ✔
1.00 | 5 | 2 | 0.000000000 | -0.000000000 ✔
1.00 | 5 | 3 | 0.000000000 | 0.000000000 ✔
1.00 | 5 | 4 | 0.000000000 | -0.000000000 ✔
1.00 | 5 | 5 | 6.000000000 | 6.000000000 ✔
1.00 | 5 | 6 | 0.000000000 | 0.000000000 ✔
1.00 | 5 | 7 | 0.000000000 | -0.000000000 ✔
1.00 | 5 | 8 | 0.000000000 | -0.000000000 ✔
1.00 | 5 | 9 | 0.000000000 | 0.000000000 ✔
1.00 | 6 | 0 | 0.000000000 | 0.000000000 ✔
1.00 | 6 | 1 | 0.000000000 | -0.000000000 ✔
1.00 | 6 | 2 | 0.000000000 | -0.000000000 ✔
1.00 | 6 | 3 | 0.000000000 | 0.000000000 ✔
1.00 | 6 | 4 | 0.000000000 | -0.000000000 ✔
1.00 | 6 | 5 | 0.000000000 | 0.000000000 ✔
1.00 | 6 | 6 | 7.000000000 | 7.000000000 ✔
1.00 | 6 | 7 | 0.000000000 | 0.000000000 ✔
1.00 | 6 | 8 | 0.000000000 | 0.000000000 ✔
1.00 | 6 | 9 | 0.000000000 | 0.000000000 ✔
1.00 | 7 | 0 | 0.000000000 | -0.000000000 ✔
1.00 | 7 | 1 | 0.000000000 | 0.000000000 ✔
1.00 | 7 | 2 | 0.000000000 | -0.000000000 ✔
1.00 | 7 | 3 | 0.000000000 | 0.000000000 ✔
1.00 | 7 | 4 | 0.000000000 | -0.000000000 ✔
1.00 | 7 | 5 | 0.000000000 | -0.000000000 ✔
1.00 | 7 | 6 | 0.000000000 | 0.000000000 ✔
1.00 | 7 | 7 | 8.000000000 | 8.000000000 ✔
1.00 | 7 | 8 | 0.000000000 | 0.000000000 ✔
1.00 | 7 | 9 | 0.000000000 | -0.000000000 ✔
1.00 | 8 | 0 | 0.000000000 | 0.000000000 ✔
1.00 | 8 | 1 | 0.000000000 | -0.000000000 ✔
1.00 | 8 | 2 | 0.000000000 | 0.000000000 ✔
1.00 | 8 | 3 | 0.000000000 | -0.000000000 ✔
1.00 | 8 | 4 | 0.000000000 | 0.000000000 ✔
1.00 | 8 | 5 | 0.000000000 | -0.000000000 ✔
1.00 | 8 | 6 | 0.000000000 | 0.000000000 ✔
1.00 | 8 | 7 | 0.000000000 | 0.000000000 ✔
1.00 | 8 | 8 | 9.000000000 | 9.000000000 ✔
1.00 | 8 | 9 | 0.000000000 | -0.000000000 ✔
1.00 | 9 | 0 | 0.000000000 | -0.000000000 ✔
1.00 | 9 | 1 | 0.000000000 | 0.000000000 ✔
1.00 | 9 | 2 | 0.000000000 | -0.000000000 ✔
1.00 | 9 | 3 | 0.000000000 | -0.000000000 ✔
1.00 | 9 | 4 | 0.000000000 | 0.000000000 ✔
1.00 | 9 | 5 | 0.000000000 | 0.000000000 ✔
1.00 | 9 | 6 | 0.000000000 | 0.000000000 ✔
1.00 | 9 | 7 | 0.000000000 | -0.000000000 ✔
1.00 | 9 | 8 | 0.000000000 | -0.000000000 ✔
1.00 | 9 | 9 | 10.000000000 | 10.000000000 ✔Normalization of $R_{nl}(r)$
\[\int |R_{nl}(r)|^2 r^2 \mathrm{d}r = 1\]
n | l | analytical | numerical
-- | -- | -------------- | --------------
0 | 0 | 1.000000000 | 1.000000000 ✔
1 | 0 | 1.000000000 | 1.000000000 ✔
1 | 1 | 1.000000000 | 1.000000000 ✔
2 | 0 | 1.000000000 | 1.000000000 ✔
2 | 1 | 1.000000000 | 1.000000000 ✔
2 | 2 | 1.000000000 | 1.000000000 ✔
3 | 0 | 1.000000000 | 1.000000000 ✔
3 | 1 | 1.000000000 | 1.000000000 ✔
3 | 2 | 1.000000000 | 1.000000000 ✔
3 | 3 | 1.000000000 | 1.000000000 ✔
4 | 0 | 1.000000000 | 1.000000000 ✔
4 | 1 | 1.000000000 | 1.000000000 ✔
4 | 2 | 1.000000000 | 1.000000000 ✔
4 | 3 | 1.000000000 | 1.000000000 ✔
4 | 4 | 1.000000000 | 1.000000000 ✔
5 | 0 | 1.000000000 | 1.000000000 ✔
5 | 1 | 1.000000000 | 1.000000000 ✔
5 | 2 | 1.000000000 | 1.000000000 ✔
5 | 3 | 1.000000000 | 1.000000000 ✔
5 | 4 | 1.000000000 | 1.000000000 ✔
5 | 5 | 1.000000000 | 1.000000000 ✔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 $\langle H \rangle = 2 \langle V \rangle$.
\[2 \langle V \rangle = 2 \times \int \psi_i^\ast(r,\theta,\varphi) V(r) \psi_j(r,\theta,\varphi) r^2 \sin(\theta) \mathrm{d}r \mathrm{d}\theta \mathrm{d}\varphi = 2 \times \int V(r) |R_{nl}(r)|^2 r^2 \mathrm{d}r = E_n\]
n | l | m | analytical | numerical
-- | -- | -- | -------------- | --------------
0 | 0 | 0 | 1.500000000 | 1.500000000 ✔
1 | 0 | 0 | 3.500000000 | 3.499999870 ✔
1 | 1 | -1 | 4.500000000 | 4.499999499 ✔
1 | 1 | 0 | 4.500000000 | 4.499999499 ✔
1 | 1 | 1 | 4.500000000 | 4.499999499 ✔
2 | 0 | 0 | 5.500000000 | 5.500004237 ✔
2 | 1 | -1 | 6.500000000 | 6.500046126 ✔
2 | 1 | 0 | 6.500000000 | 6.500046126 ✔
2 | 1 | 1 | 6.500000000 | 6.500046126 ✔
2 | 2 | -2 | 7.500000000 | 7.500128111 ✔
2 | 2 | -1 | 7.500000000 | 7.500128111 ✔
2 | 2 | 0 | 7.500000000 | 7.500128112 ✔
2 | 2 | 1 | 7.500000000 | 7.500128111 ✔
2 | 2 | 2 | 7.500000000 | 7.500128111 ✔
3 | 0 | 0 | 7.500000000 | 7.500149607 ✔
3 | 1 | -1 | 8.500000000 | 8.499578116 ✔
3 | 1 | 0 | 8.500000000 | 8.499578116 ✔
3 | 1 | 1 | 8.500000000 | 8.499578116 ✔
3 | 2 | -2 | 9.500000000 | 9.496427080 ✔
3 | 2 | -1 | 9.500000000 | 9.496427080 ✔
3 | 2 | 0 | 9.500000000 | 9.496427080 ✔
3 | 2 | 1 | 9.500000000 | 9.496427080 ✔
3 | 2 | 2 | 9.500000000 | 9.496427080 ✔
3 | 3 | -3 | 10.500000000 | 10.492207413 ✔
3 | 3 | -2 | 10.500000000 | 10.492207367 ✔
3 | 3 | -1 | 10.500000000 | 10.492207507 ✔
3 | 3 | 0 | 10.500000000 | 10.492207273 ✔
3 | 3 | 1 | 10.500000000 | 10.492207507 ✔
3 | 3 | 2 | 10.500000000 | 10.492207367 ✔
3 | 3 | 3 | 10.500000000 | 10.492207413 ✔ n | l | analytical | numerical
-- | -- | -------------- | --------------
0 | 0 | 1.500000000 | 1.500000000 ✔
1 | 0 | 3.500000000 | 3.500000000 ✔
1 | 1 | 4.500000000 | 4.500000000 ✔
2 | 0 | 5.500000000 | 5.500000000 ✔
2 | 1 | 6.500000000 | 6.500000000 ✔
2 | 2 | 7.500000000 | 7.500000000 ✔
3 | 0 | 7.500000000 | 7.500000000 ✔
3 | 1 | 8.500000000 | 8.500000000 ✔
3 | 2 | 9.500000000 | 9.500000000 ✔
3 | 3 | 10.500000000 | 10.500000000 ✔
4 | 0 | 9.500000000 | 9.500000000 ✔
4 | 1 | 10.500000000 | 10.500000000 ✔
4 | 2 | 11.500000000 | 11.500000000 ✔
4 | 3 | 12.500000000 | 12.500000000 ✔
4 | 4 | 13.500000000 | 13.500000000 ✔
5 | 0 | 11.500000000 | 11.500000000 ✔
5 | 1 | 12.500000000 | 12.500000000 ✔
5 | 2 | 13.500000000 | 13.500000000 ✔
5 | 3 | 14.500000000 | 14.500000000 ✔
5 | 4 | 15.500000000 | 15.500000000 ✔
5 | 5 | 16.500000000 | 16.500000000 ✔Normalization & Orthogonality of $\psi_n(r,\theta,\varphi)$
\[\int \psi_i^\ast(r,\theta,\varphi) \psi_j(r,\theta,\varphi) r^2 \sin(\theta) \mathrm{d}r \mathrm{d}\theta \mathrm{d}\varphi = \delta_{ij}\]
n₁ | n₂ | l₁ | l₂ | m₁ | m₂ | analytical | numerical
-- | -- | -- | -- | -- | -- | -------------- | --------------
0 | 0 | 0 | 0 | 0 | 0 | 1.000000000 | 1.000000301 ✔
0 | 1 | 0 | 0 | 0 | 0 | 0.000000000 | -0.000001218 ✔
0 | 1 | 0 | 1 | 0 | -1 | 0.000000000 | 0.000004079 ✔
0 | 1 | 0 | 1 | 0 | 0 | 0.000000000 | 0.000000001 ✔
0 | 1 | 0 | 1 | 0 | 1 | 0.000000000 | -0.000004079 ✔
0 | 2 | 0 | 0 | 0 | 0 | 0.000000000 | 0.000000128 ✔
0 | 2 | 0 | 1 | 0 | -1 | 0.000000000 | -0.000016724 ✔
0 | 2 | 0 | 1 | 0 | 0 | 0.000000000 | 0.000000001 ✔
0 | 2 | 0 | 1 | 0 | 1 | 0.000000000 | 0.000016724 ✔
0 | 2 | 0 | 2 | 0 | -2 | 0.000000000 | 0.007549677 ✔
0 | 2 | 0 | 2 | 0 | -1 | 0.000000000 | 0.000000000 ✔
0 | 2 | 0 | 2 | 0 | 0 | 0.000000000 | -0.000012343 ✔
0 | 2 | 0 | 2 | 0 | 1 | 0.000000000 | -0.000000000 ✔
0 | 2 | 0 | 2 | 0 | 2 | 0.000000000 | 0.007549677 ✔
1 | 0 | 0 | 0 | 0 | 0 | 0.000000000 | -0.000001218 ✔
1 | 0 | 1 | 0 | -1 | 0 | 0.000000000 | 0.000004079 ✔
1 | 0 | 1 | 0 | 0 | 0 | 0.000000000 | 0.000000001 ✔
1 | 0 | 1 | 0 | 1 | 0 | 0.000000000 | -0.000004079 ✔
1 | 1 | 0 | 0 | 0 | 0 | 1.000000000 | 1.000000508 ✔
1 | 1 | 0 | 1 | 0 | -1 | 0.000000000 | -0.000046754 ✔
1 | 1 | 0 | 1 | 0 | 0 | 0.000000000 | -0.000000000 ✔
1 | 1 | 0 | 1 | 0 | 1 | 0.000000000 | 0.000046754 ✔
1 | 1 | 1 | 0 | -1 | 0 | 0.000000000 | -0.000046754 ✔
1 | 1 | 1 | 0 | 0 | 0 | 0.000000000 | -0.000000000 ✔
1 | 1 | 1 | 0 | 1 | 0 | 0.000000000 | 0.000046754 ✔
1 | 1 | 1 | 1 | -1 | -1 | 1.000000000 | 0.999997164 ✔
1 | 1 | 1 | 1 | -1 | 0 | 0.000000000 | 0.000000000 ✔
1 | 1 | 1 | 1 | -1 | 1 | 0.000000000 | 0.000158558 ✔
1 | 1 | 1 | 1 | 0 | -1 | 0.000000000 | 0.000000000 ✔
1 | 1 | 1 | 1 | 0 | 0 | 1.000000000 | 1.000010290 ✔
1 | 1 | 1 | 1 | 0 | 1 | 0.000000000 | -0.000000000 ✔
1 | 1 | 1 | 1 | 1 | -1 | 0.000000000 | 0.000158558 ✔
1 | 1 | 1 | 1 | 1 | 0 | 0.000000000 | -0.000000000 ✔
1 | 1 | 1 | 1 | 1 | 1 | 1.000000000 | 0.999997164 ✔
1 | 2 | 0 | 0 | 0 | 0 | 0.000000000 | 0.000001168 ✔
1 | 2 | 0 | 1 | 0 | -1 | 0.000000000 | 0.000001694 ✔
1 | 2 | 0 | 1 | 0 | 0 | 0.000000000 | -0.000000000 ✔
1 | 2 | 0 | 1 | 0 | 1 | 0.000000000 | -0.000001694 ✔
1 | 2 | 0 | 2 | 0 | -2 | 0.000000000 | -0.000019162 ✔
1 | 2 | 0 | 2 | 0 | -1 | 0.000000000 | 0.000000000 ✔
1 | 2 | 0 | 2 | 0 | 0 | 0.000000000 | 0.000027271 ✔
1 | 2 | 0 | 2 | 0 | 1 | 0.000000000 | -0.000000000 ✔
1 | 2 | 0 | 2 | 0 | 2 | 0.000000000 | -0.000019162 ✔
1 | 2 | 1 | 0 | -1 | 0 | 0.000000000 | 0.000002122 ✔
1 | 2 | 1 | 0 | 0 | 0 | 0.000000000 | 0.000000000 ✔
1 | 2 | 1 | 0 | 1 | 0 | 0.000000000 | -0.000002122 ✔
1 | 2 | 1 | 1 | -1 | -1 | 0.000000000 | -0.000030502 ✔
1 | 2 | 1 | 1 | -1 | 0 | 0.000000000 | 0.000000000 ✔
1 | 2 | 1 | 1 | -1 | 1 | 0.000000000 | -0.000007300 ✔
1 | 2 | 1 | 1 | 0 | -1 | 0.000000000 | 0.000000000 ✔
1 | 2 | 1 | 1 | 0 | 0 | 0.000000000 | 0.000039908 ✔
1 | 2 | 1 | 1 | 0 | 1 | 0.000000000 | -0.000000000 ✔
1 | 2 | 1 | 1 | 1 | -1 | 0.000000000 | -0.000007300 ✔
1 | 2 | 1 | 1 | 1 | 0 | 0.000000000 | -0.000000000 ✔
1 | 2 | 1 | 1 | 1 | 1 | 0.000000000 | -0.000030502 ✔
1 | 2 | 1 | 2 | -1 | -2 | 0.000000000 | 0.000018054 ✔
1 | 2 | 1 | 2 | -1 | -1 | 0.000000000 | 0.000000121 ✔
1 | 2 | 1 | 2 | -1 | 0 | 0.000000000 | 0.000258496 ✔
1 | 2 | 1 | 2 | -1 | 1 | 0.000000000 | -0.000000000 ✔
1 | 2 | 1 | 2 | -1 | 2 | 0.000000000 | 0.002185307 ✔
1 | 2 | 1 | 2 | 0 | -2 | 0.000000000 | 0.000000000 ✔
1 | 2 | 1 | 2 | 0 | -1 | 0.000000000 | 0.000009501 ✔
1 | 2 | 1 | 2 | 0 | 0 | 0.000000000 | -0.000000213 ✔
1 | 2 | 1 | 2 | 0 | 1 | 0.000000000 | -0.000009501 ✔
1 | 2 | 1 | 2 | 0 | 2 | 0.000000000 | 0.000000000 ✔
1 | 2 | 1 | 2 | 1 | -2 | 0.000000000 | -0.002185307 ✔
1 | 2 | 1 | 2 | 1 | -1 | 0.000000000 | -0.000000000 ✔
1 | 2 | 1 | 2 | 1 | 0 | 0.000000000 | -0.000258496 ✔
1 | 2 | 1 | 2 | 1 | 1 | 0.000000000 | 0.000000121 ✔
1 | 2 | 1 | 2 | 1 | 2 | 0.000000000 | -0.000018054 ✔
2 | 0 | 0 | 0 | 0 | 0 | 0.000000000 | 0.000000128 ✔
2 | 0 | 1 | 0 | -1 | 0 | 0.000000000 | -0.000016724 ✔
2 | 0 | 1 | 0 | 0 | 0 | 0.000000000 | 0.000000001 ✔
2 | 0 | 1 | 0 | 1 | 0 | 0.000000000 | 0.000016724 ✔
2 | 0 | 2 | 0 | -2 | 0 | 0.000000000 | 0.007549677 ✔
2 | 0 | 2 | 0 | -1 | 0 | 0.000000000 | 0.000000000 ✔
2 | 0 | 2 | 0 | 0 | 0 | 0.000000000 | -0.000012343 ✔
2 | 0 | 2 | 0 | 1 | 0 | 0.000000000 | -0.000000000 ✔
2 | 0 | 2 | 0 | 2 | 0 | 0.000000000 | 0.007549677 ✔
2 | 1 | 0 | 0 | 0 | 0 | 0.000000000 | 0.000001168 ✔
2 | 1 | 0 | 1 | 0 | -1 | 0.000000000 | 0.000002122 ✔
2 | 1 | 0 | 1 | 0 | 0 | 0.000000000 | 0.000000000 ✔
2 | 1 | 0 | 1 | 0 | 1 | 0.000000000 | -0.000002122 ✔
2 | 1 | 1 | 0 | -1 | 0 | 0.000000000 | 0.000001694 ✔
2 | 1 | 1 | 0 | 0 | 0 | 0.000000000 | -0.000001119 ✔
2 | 1 | 1 | 0 | 1 | 0 | 0.000000000 | -0.000001694 ✔
2 | 1 | 1 | 1 | -1 | -1 | 0.000000000 | -0.000030502 ✔
2 | 1 | 1 | 1 | -1 | 0 | 0.000000000 | 0.000000000 ✔
2 | 1 | 1 | 1 | -1 | 1 | 0.000000000 | -0.000007300 ✔
2 | 1 | 1 | 1 | 0 | -1 | 0.000000000 | 0.000000000 ✔
2 | 1 | 1 | 1 | 0 | 0 | 0.000000000 | 0.000039908 ✔
2 | 1 | 1 | 1 | 0 | 1 | 0.000000000 | -0.000000000 ✔
2 | 1 | 1 | 1 | 1 | -1 | 0.000000000 | -0.000007300 ✔
2 | 1 | 1 | 1 | 1 | 0 | 0.000000000 | -0.000000000 ✔
2 | 1 | 1 | 1 | 1 | 1 | 0.000000000 | -0.000030502 ✔
2 | 1 | 2 | 0 | -2 | 0 | 0.000000000 | -0.000019162 ✔
2 | 1 | 2 | 0 | -1 | 0 | 0.000000000 | 0.000004609 ✔
2 | 1 | 2 | 0 | 0 | 0 | 0.000000000 | 0.000027271 ✔
2 | 1 | 2 | 0 | 1 | 0 | 0.000000000 | -0.000004609 ✔
2 | 1 | 2 | 0 | 2 | 0 | 0.000000000 | -0.000019162 ✔
2 | 1 | 2 | 1 | -2 | -1 | 0.000000000 | 0.000018054 ✔
2 | 1 | 2 | 1 | -2 | 0 | 0.000000000 | 0.000000000 ✔
2 | 1 | 2 | 1 | -2 | 1 | 0.000000000 | -0.002185307 ✔
2 | 1 | 2 | 1 | -1 | -1 | 0.000000000 | 0.000000121 ✔
2 | 1 | 2 | 1 | -1 | 0 | 0.000000000 | 0.000009501 ✔
2 | 1 | 2 | 1 | -1 | 1 | 0.000000000 | -0.000000000 ✔
2 | 1 | 2 | 1 | 0 | -1 | 0.000000000 | 0.000258496 ✔
2 | 1 | 2 | 1 | 0 | 0 | 0.000000000 | -0.000000213 ✔
2 | 1 | 2 | 1 | 0 | 1 | 0.000000000 | -0.000258496 ✔
2 | 1 | 2 | 1 | 1 | -1 | 0.000000000 | -0.000000000 ✔
2 | 1 | 2 | 1 | 1 | 0 | 0.000000000 | -0.000009501 ✔
2 | 1 | 2 | 1 | 1 | 1 | 0.000000000 | 0.000000121 ✔
2 | 1 | 2 | 1 | 2 | -1 | 0.000000000 | 0.002185307 ✔
2 | 1 | 2 | 1 | 2 | 0 | 0.000000000 | 0.000000000 ✔
2 | 1 | 2 | 1 | 2 | 1 | 0.000000000 | -0.000018054 ✔
2 | 2 | 0 | 0 | 0 | 0 | 1.000000000 | 1.000004762 ✔
2 | 2 | 0 | 1 | 0 | -1 | 0.000000000 | -0.000014548 ✔
2 | 2 | 0 | 1 | 0 | 0 | 0.000000000 | -0.000000000 ✔
2 | 2 | 0 | 1 | 0 | 1 | 0.000000000 | 0.000014548 ✔
2 | 2 | 0 | 2 | 0 | -2 | 0.000000000 | -0.000040789 ✔
2 | 2 | 0 | 2 | 0 | -1 | 0.000000000 | -0.000000000 ✔
2 | 2 | 0 | 2 | 0 | 0 | 0.000000000 | -0.000100816 ✔
2 | 2 | 0 | 2 | 0 | 1 | 0.000000000 | 0.000000000 ✔
2 | 2 | 0 | 2 | 0 | 2 | 0.000000000 | -0.000040789 ✔
2 | 2 | 1 | 0 | -1 | 0 | 0.000000000 | -0.000014548 ✔
2 | 2 | 1 | 0 | 0 | 0 | 0.000000000 | -0.000000000 ✔
2 | 2 | 1 | 0 | 1 | 0 | 0.000000000 | 0.000014548 ✔
2 | 2 | 1 | 1 | -1 | -1 | 1.000000000 | 1.000050475 ✔
2 | 2 | 1 | 1 | -1 | 0 | 0.000000000 | 0.000243319 ✔
2 | 2 | 1 | 1 | -1 | 1 | 0.000000000 | 0.003526936 ✔
2 | 2 | 1 | 1 | 0 | -1 | 0.000000000 | 0.000243319 ✔
2 | 2 | 1 | 1 | 0 | 0 | 1.000000000 | 1.000005999 ✔
2 | 2 | 1 | 1 | 0 | 1 | 0.000000000 | -0.000243319 ✔
2 | 2 | 1 | 1 | 1 | -1 | 0.000000000 | 0.003526936 ✔
2 | 2 | 1 | 1 | 1 | 0 | 0.000000000 | -0.000243319 ✔
2 | 2 | 1 | 1 | 1 | 1 | 1.000000000 | 1.000050475 ✔
2 | 2 | 1 | 2 | -1 | -2 | 0.000000000 | -0.000126948 ✔
2 | 2 | 1 | 2 | -1 | -1 | 0.000000000 | -0.000000523 ✔
2 | 2 | 1 | 2 | -1 | 0 | 0.000000000 | -0.000094855 ✔
2 | 2 | 1 | 2 | -1 | 1 | 0.000000000 | -0.000000000 ✔
2 | 2 | 1 | 2 | -1 | 2 | 0.000000000 | 0.009644367 ✔
2 | 2 | 1 | 2 | 0 | -2 | 0.000000000 | 0.000000000 ✔
2 | 2 | 1 | 2 | 0 | -1 | 0.000000000 | -0.000065719 ✔
2 | 2 | 1 | 2 | 0 | 0 | 0.000000000 | -0.000001438 ✔
2 | 2 | 1 | 2 | 0 | 1 | 0.000000000 | 0.000065719 ✔
2 | 2 | 1 | 2 | 0 | 2 | 0.000000000 | 0.000000000 ✔
2 | 2 | 1 | 2 | 1 | -2 | 0.000000000 | -0.009644367 ✔
2 | 2 | 1 | 2 | 1 | -1 | 0.000000000 | -0.000000000 ✔
2 | 2 | 1 | 2 | 1 | 0 | 0.000000000 | 0.000094855 ✔
2 | 2 | 1 | 2 | 1 | 1 | 0.000000000 | -0.000000523 ✔
2 | 2 | 1 | 2 | 1 | 2 | 0.000000000 | 0.000126948 ✔
2 | 2 | 2 | 0 | -2 | 0 | 0.000000000 | -0.000040789 ✔
2 | 2 | 2 | 0 | -1 | 0 | 0.000000000 | -0.000000000 ✔
2 | 2 | 2 | 0 | 0 | 0 | 0.000000000 | -0.000100816 ✔
2 | 2 | 2 | 0 | 1 | 0 | 0.000000000 | 0.000000000 ✔
2 | 2 | 2 | 0 | 2 | 0 | 0.000000000 | -0.000040789 ✔
2 | 2 | 2 | 1 | -2 | -1 | 0.000000000 | -0.000126948 ✔
2 | 2 | 2 | 1 | -2 | 0 | 0.000000000 | 0.000000000 ✔
2 | 2 | 2 | 1 | -2 | 1 | 0.000000000 | 0.009644368 ✔
2 | 2 | 2 | 1 | -1 | -1 | 0.000000000 | -0.000000523 ✔
2 | 2 | 2 | 1 | -1 | 0 | 0.000000000 | -0.000065719 ✔
2 | 2 | 2 | 1 | -1 | 1 | 0.000000000 | -0.000000000 ✔
2 | 2 | 2 | 1 | 0 | -1 | 0.000000000 | -0.000094855 ✔
2 | 2 | 2 | 1 | 0 | 0 | 0.000000000 | -0.000001438 ✔
2 | 2 | 2 | 1 | 0 | 1 | 0.000000000 | 0.000094855 ✔
2 | 2 | 2 | 1 | 1 | -1 | 0.000000000 | -0.000000000 ✔
2 | 2 | 2 | 1 | 1 | 0 | 0.000000000 | 0.000065719 ✔
2 | 2 | 2 | 1 | 1 | 1 | 0.000000000 | -0.000000523 ✔
2 | 2 | 2 | 1 | 2 | -1 | 0.000000000 | -0.009644368 ✔
2 | 2 | 2 | 1 | 2 | 0 | 0.000000000 | 0.000000000 ✔
2 | 2 | 2 | 1 | 2 | 1 | 0.000000000 | 0.000126948 ✔
2 | 2 | 2 | 2 | -2 | -2 | 1.000000000 | 0.999687053 ✔
2 | 2 | 2 | 2 | -2 | -1 | 0.000000000 | -0.000000000 ✔
2 | 2 | 2 | 2 | -2 | 0 | 0.000000000 | 0.008326706 ✔
2 | 2 | 2 | 2 | -2 | 1 | 0.000000000 | 0.000000000 ✔
2 | 2 | 2 | 2 | -2 | 2 | 0.000000000 | -0.013843039 ✔
2 | 2 | 2 | 2 | -1 | -2 | 0.000000000 | -0.000000000 ✔
2 | 2 | 2 | 2 | -1 | -1 | 1.000000000 | 0.999593326 ✔
2 | 2 | 2 | 2 | -1 | 0 | 0.000000000 | -0.000000000 ✔
2 | 2 | 2 | 2 | -1 | 1 | 0.000000000 | 0.011709066 ✔
2 | 2 | 2 | 2 | -1 | 2 | 0.000000000 | -0.000000000 ✔
2 | 2 | 2 | 2 | 0 | -2 | 0.000000000 | 0.008326706 ✔
2 | 2 | 2 | 2 | 0 | -1 | 0.000000000 | -0.000000000 ✔
2 | 2 | 2 | 2 | 0 | 0 | 1.000000000 | 0.999699379 ✔
2 | 2 | 2 | 2 | 0 | 1 | 0.000000000 | 0.000000000 ✔
2 | 2 | 2 | 2 | 0 | 2 | 0.000000000 | 0.008326706 ✔
2 | 2 | 2 | 2 | 1 | -2 | 0.000000000 | 0.000000000 ✔
2 | 2 | 2 | 2 | 1 | -1 | 0.000000000 | 0.011709066 ✔
2 | 2 | 2 | 2 | 1 | 0 | 0.000000000 | 0.000000000 ✔
2 | 2 | 2 | 2 | 1 | 1 | 1.000000000 | 0.999593326 ✔
2 | 2 | 2 | 2 | 1 | 2 | 0.000000000 | 0.000000000 ✔
2 | 2 | 2 | 2 | 2 | -2 | 0.000000000 | -0.013843039 ✔
2 | 2 | 2 | 2 | 2 | -1 | 0.000000000 | -0.000000000 ✔
2 | 2 | 2 | 2 | 2 | 0 | 0.000000000 | 0.008326706 ✔
2 | 2 | 2 | 2 | 2 | 1 | 0.000000000 | 0.000000000 ✔
2 | 2 | 2 | 2 | 2 | 2 | 1.000000000 | 0.999687053 ✔