Hydrogen Atom

The hydrogen atom is the simplest Coulomb 2-body system.

Definitions

Antique.HydrogenAtom — Type

Model

This model is described with the time-independent Schrödinger equation

\[ \hat{H} \psi(\pmb{r}) = E \psi(\pmb{r}),\]

and the Hamiltonian

\[ \hat{H} = - \frac{\hbar^2}{2\mu} \nabla^2 + \frac{Z}{r/a_0} E_\mathrm{h},\]

where $\mu=\left(\frac{1}{m_\mathrm{e}}+\frac{1}{m_\mathrm{p}}\right)^{-1}$ is the reduced mass of electron $\mathrm{e}$ and proton $\mathrm{p}$. $\mu = m_\mathrm{e}$ holds in the limit $m_\mathrm{p}\rightarrow\infty$. The potential includes only Coulomb interaction and it does not include fine or hyperfine interactions in this model. Parameters are specified with the following struct:

HA = HydrogenAtom(Z=1, mₑ=1.0, a₀=1.0, Eₕ=1.0, ℏ=1.0)

$Z$ is the atomic number, $m_\mathrm{e}$ is the electron mass, $a_0$ is the Bohr radius, $E_\mathrm{h}$ is the Hartree energy and $\hbar$ is the reduced Planck constant (Dirac's constant).

References

Main:

The Digital Library of Mathematical Functions (DLMF), 18.3 Table1, 18.5 Table1, 18.5.16, 18.5.17
cpprefjp, assoc_legendre, assoc_laguerre
A. Messiah, Quanfum Mechanics VOLUME Ⅰ (North-Holland Publishing Company, 1961), p.412 (XI.3), p.419 (XI.18) (XI.18a) (XI.18b), p.483 (B.12), p.493 (B.71) (B.72), p.494 (B.81), p495 (B.93)

Supplemental:

cpprefjp, legendre, assoc_legendre, laguerre, assoc_laguerre
The Digital Library of Mathematical Functions (DLMF), 18.3 Table1, 18.5 Table1, 18.5.16, 18.5.17, 18.5.12
L. D. Landau, E. M. Lifshitz, Quantum Mechanics (Pergamon Press, 1965), p.598 (c.1), p.598 (c.4), p.603 (d.13)
L. I. Schiff, Quantum Mechanics (McGraw-Hill Book Company, 1968), p.79 (14.12), p.93 (16.19)
A. Messiah, Quanfum Mechanics (Dover Publications, 1999), p.493 (B.72), p.494 Table, p.493 (B.72), p.483 (B.12), p.483 (B.12)
W. Greiner, Quantum Mechanics: An Introduction Third Edition (Springer, 1994), p.83 (4), p.83 (5), p.149 (21)
D. J. Griffiths, Introduction to Quantum Mechanics (Prentice Hall, 1995), p.126 (4.28), p.96 Table3.1, p.126 (4.27), p.139 (4.88), p.140 Table4.4, p.139 (4.87), p.140 Table4.5
D. A. McQuarrie, J. D. Simon, Physical Chemistry: A Molecular Approach (University Science Books, 1997), p.195 Table6.1, p.196 (6.26), p.196 Table6.2, p.207 Table6.4
P. W. Atkins, J. De Paula, Atkins' Physical Chemistry, 8th edition (W. H. Freeman, 2008), p.234
J. J. Sakurai, J. Napolitano, Modern Quantum Mechanics Third Edition (Cambridge University Press, 2021), p.245 Problem 3.30.b,

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Potential

Antique.V — Method

V(model::HydrogenAtom, r)

\[\begin{aligned} V(r) &= - \frac{Ze^2}{4\pi\varepsilon_0 r} &= - \frac{e^2}{4\pi\varepsilon_0 a_0} \frac{Z}{r/a_0} &= - \frac{Z}{r/a_0} E_\mathrm{h}, \end{aligned}\]

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Eigenvalues

Antique.E — Method

E(model::HydrogenAtom; n::Int=1)

\[E_n = -\frac{m_\mathrm{e} e^4 Z^2}{2n^2(4\pi\varepsilon_0)^2\hbar^2} = -\frac{Z^2}{2n^2} E_\mathrm{h},\]

where $E_\mathrm{h} = \frac{\hbar^2}{m_\mathrm{e}{a_0}^2} = \frac{e^2}{4\pi\varepsilon_0a_0} = \frac{m_\mathrm{e}e^4}{\left(4\pi\varepsilon_0\right)^2\hbar^2}$ is the Hartree energy, one of atomic unit. About atomic units, see section 3.9.2 of the IUPAC GreenBook. In other units, $E_\mathrm{h} = 27.211~386~245~988(53)~\mathrm{eV}$ from here.

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Eigenfunctions

Antique.ψ — Method

ψ(model::HydrogenAtom, r, θ, φ; n::Int=1, l::Int=0, m::Int=0)

\[\psi_{nlm}(\pmb{r}) = R_{nl}(r) Y_{lm}(\theta,\varphi)\]

The domain is $0\leq r \lt \infty, 0\leq \theta \lt \pi, 0\leq \varphi \lt 2\pi$.

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Radial Functions

Antique.R — Method

R(model::HydrogenAtom, r; n=1, l=0)

\[R_{nl}(r) = -\sqrt{\frac{(n-l-1)!}{2n(n+l)!} \left(\frac{2Z}{n a_0}\right)^3} \left(\frac{2Zr}{n a_0}\right)^l \exp \left(-\frac{Zr}{n a_0}\right) L_{n+l}^{2l+1} \left(\frac{2Zr}{n a_0}\right),\]

where Laguerre polynomials are defined as $L_n(x) = \frac{1}{n!} \mathrm{e}^x \frac{\mathrm{d}^n}{\mathrm{d}x ^n} \left( \mathrm{e}^{-x} x^n \right)$, and associated Laguerre polynomials are defined as $L_n^{k}(x) = \frac{\mathrm{d}^k}{\mathrm{d}x^k} L_n(x)$. Note that replace $2n(n+l)!$ with $2n[(n+l)!]^3$ if Laguerre polynomials are defined as $L_n(x) = \mathrm{e}^x \frac{\mathrm{d}^n}{\mathrm{d}x ^n} \left( \mathrm{e}^{-x} x^n \right)$. The domain is $0\leq r \lt \infty$.

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Associated Laguerre Polynomials

Antique.L — Method

L(model::HydrogenAtom, x; n=0, k=0)

Rodrigues' formula & closed-form:

\[\begin{aligned} L_n^{k}(x) &= \frac{\mathrm{d}^k}{\mathrm{d}x^k} L_n(x) \\ &= \frac{\mathrm{d}^k}{\mathrm{d}x^k} \frac{1}{n!} \mathrm{e}^x \frac{\mathrm{d}^n}{\mathrm{d}x ^n} \left( \mathrm{e}^{-x} x^n \right) \\ &= \sum_{m=0}^{n-k} (-1)^{m+k} \frac{n!}{m!(m+k)!(n-m-k)!} x^m \\ &= (-1)^k L_{n-k}^{(k)}(x), \end{aligned}\]

where Laguerre polynomials are defined as $L_n(x)=\frac{1}{n!}\mathrm{e}^x \frac{\mathrm{d}^n}{\mathrm{d}x ^n} \left( \mathrm{e}^{-x} x^n \right)$.

Examples:

\[\begin{aligned} L_0^0(x) &= 1, \\ L_1^0(x) &= 1 - x, \\ L_1^1(x) &= 1, \\ L_2^0(x) &= 1 - 2 x + 1/2 x^2, \\ L_2^1(x) &= 2 - x, \\ L_2^2(x) &= 1, \\ L_3^0(x) &= 1 - 3 x + 3/2 x^2 - 1/6 x^3, \\ L_3^1(x) &= 3 - 3 x + 1/2 x^2, \\ L_3^2(x) &= 3 - x, \\ L_3^3(x) &= 1, \\ L_4^0(x) &= 1 - 4 x + 3 x^2 - 2/3 x^3 + 5/12 x^4, \\ L_4^1(x) &= 4 - 6 x + 2 x^2 - 1/6 x^3, \\ L_4^2(x) &= 6 - 4 x + 1/2 x^2, \\ L_4^3(x) &= 4 - x, \\ L_4^4(x) &= 1, \\ \vdots \end{aligned}\]

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Spherical Harmonics

Antique.Y — Method

Y(model::HydrogenAtom, 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

\[i^{|m|+m} \sqrt{\frac{(l-|m|)!}{(l+|m|)!}} P_l^{|m|} = (-1)^{\frac{|m|+m}{2}} \sqrt{\frac{(l-|m|)!}{(l+|m|)!}} P_l^{|m|} = (-1)^m \sqrt{\frac{(l-m)!}{(l+m)!}} P_l^{m}.\]

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Associated Legendre Polynomials

Antique.P — Method

P(model::HydrogenAtom, 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:

\[\begin{aligned} P_{0}^{0}(x) &= 1, \\ P_{1}^{0}(x) &= x, \\ P_{1}^{1}(x) &= \left(+1\right)\sqrt{1-x^2}, \\ P_{2}^{0}(x) &= -1/2 + 3/2 x^{2}, \\ P_{2}^{1}(x) &= \left(-3 x\right)\sqrt{1-x^2}, \\ P_{2}^{2}(x) &= 3 - 6 x, \\ P_{3}^{0}(x) &= -3/2 x + 5/2 x^{3}, \\ P_{3}^{1}(x) &= \left(3/2 - 15/2 x^{2}\right)\sqrt{1-x^2}, \\ P_{3}^{2}(x) &= 15 x - 30 x^{2}, \\ P_{3}^{3}(x) &= \left(15 - 30 x\right)\sqrt{1-x^2}, \\ P_{4}^{0}(x) &= 3/8 - 15/4 x^{2} + 35/8 x^{4}, \\ P_{4}^{1}(x) &= \left(- 15/2 x + 35/2 x^{3}\right)\sqrt{1-x^2}, \\ P_{4}^{2}(x) &= -15/2 + 15 x + 105/2 x^{2} - 105 x^{3}, \\ P_{4}^{3}(x) &= \left(105 x - 210 x^{2}\right)\sqrt{1-x^2}, \\ P_{4}^{4}(x) &= 105 - 420 x + 420 x^{2}, \\ & \vdots \end{aligned}\]

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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 HydrogenAtom and several parameters Z, Eₕ, mₑ, a₀ and ℏ are set as optional arguments.

using Antique
H = HydrogenAtom(Z=1, Eₕ=1.0, a₀=1.0, mₑ=1.0, ℏ=1.0)

Parameters:

julia> H.Z1
julia> H.Eₕ1.0
julia> H.mₑ1.0
julia> H.a₀1.0
julia> H.ℏ1.0

Eigenvalues:

julia> E(H, n=1)-0.5
julia> E(H, n=2)-0.125

Wave length ($n=2\rightarrow1$, the first line of the Lyman series):

Eₕ2nm⁻¹ = 2.1947463136320e-2 # https://physics.nist.gov/cgi-bin/cuu/CCValue?hrminv
println("ΔE = ", E(H,n=2) - E(H,n=1), " Eₕ")
println("λ  = ", ((E(H,n=2)-E(H,n=1))*Eₕ2nm⁻¹)^-1, " nm")

ΔE = 0.375 Eₕ
λ  = 121.50227341098497 nm

Hyperfine Splitting:

# E. Tiesinga, et al., Rev. Mod. Phys. 93, 025010 (2021) https://doi.org/10.1103/RevModPhys.93.025010
e  = 1.602176634e-19    # C      https://physics.nist.gov/cgi-bin/cuu/Value?e
h  = 6.62607015e-34     # J Hz-1 https://physics.nist.gov/cgi-bin/cuu/Value?h
c  = 299792458          # m s-1  https://physics.nist.gov/cgi-bin/cuu/Value?c
a0 = 5.29177210903e-11  # m      https://physics.nist.gov/cgi-bin/cuu/Value?bohrrada0
μ0 = 1.25663706212e-6   # N A-2  https://physics.nist.gov/cgi-bin/cuu/Value?mu0
μB = 9.2740100783e-24   # J T-1  https://physics.nist.gov/cgi-bin/cuu/Value?mub
μN = 5.0507837461e-27   # J T-1  https://physics.nist.gov/cgi-bin/cuu/Value?mun
ge = 2.00231930436256   #        https://physics.nist.gov/cgi-bin/cuu/Value?gem
gp = 5.5856946893       #        https://physics.nist.gov/cgi-bin/cuu/Value?gp

# D. J. Griffiths, Am. J. Phys. 50, 698 (1982) https://doi.org/10.1119/1.12733
δ = abs(ψ(H,0,0,0))^2
ΔE = 2 / 3 * μ0 * μN * μB * gp * ge * δ * a0^(-3)
println("1/π    = ", 1/π)
println("<δ(r)> = ", δ, " a₀⁻³")
println("<δ(r)> = ", δ * a0^(-3), " m⁻³")
println("ΔE = ", ΔE, " J")
println("ν = ΔE/h = ", ΔE / h * 1e-6, " MHz")
println("λ = hc/ΔE = ", h*c/ΔE*100, " cm")

1/π    = 0.3183098861837907
<δ(r)> = 0.3183098861837908 a₀⁻³
<δ(r)> = 2.1480615849063944e30 m⁻³
ΔE = 9.427622831641132e-25 J
ν = ΔE/h = 1422.8075794882932 MHz
λ = hc/ΔE = 21.070485027063118 cm

Potential energy curve:

using CairoMakie

f = Figure()
ax = Axis(f[1,1], xlabel=L"$r~/~a_0$", ylabel=L"$V(r)~/~E_\mathrm{h}$",  limits=(0.0,15.0,-2.0,0.2))
lines!(ax, 0.1:0.01:20, r -> V(H, r))
f

Radial functions:

using CairoMakie
using LaTeXStrings

# setting
f = Figure()
ax = Axis(f[1,1], xlabel=L"$r~/~a_0$", ylabel=L"$r^2|R_{nl}(r)|^2~/~a_0^{-1}$", limits=(0,20,0,0.58))

# plot
ws = []
ls = []
for n in 1:3
  for l in 0:n-1
    w = lines!(
        ax,
        0..20,
        r -> r^2 * Antique.R(H,r,n=n,l=l)^2,
        linewidth = 2,
        linestyle = [:solid,:dash,:dot,:dashdot,:dashdotdot][l+1],
        color = n,
        colormap = :tab10,
        colorrange = (1,10)
    )
    push!(ws, w)
    push!(ls, latexstring("n=$n, l=$l"))
  end
end

# legend
axislegend(ax, ws, ls, position=:rt)

f

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 |    2.000000000000 |    2.000000000000 ✔
 0 |  0 |  1 |    0.000000000000 |    0.000000000000 ✔
 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 |  7 |    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 |  6 |    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 ✔
 0 |  2 |  2 |    0.400000000000 |    0.400000000000 ✔
 0 |  2 |  3 |    0.000000000000 |    0.000000000000 ✔
 0 |  2 |  4 |    0.000000000000 |    0.000000000000 ✔
 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 |  9 |    0.000000000000 |   -0.000000000000 ✔
 0 |  3 |  0 |    0.000000000000 |   -0.000000000000 ✔
 0 |  3 |  1 |    0.000000000000 |   -0.000000000000 ✔
 0 |  3 |  2 |    0.000000000000 |    0.000000000000 ✔
 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 |  7 |    0.000000000000 |    0.000000000000 ✔
 0 |  3 |  8 |    0.000000000000 |   -0.000000000000 ✔
 0 |  3 |  9 |    0.000000000000 |   -0.000000000000 ✔
 0 |  4 |  0 |    0.000000000000 |    0.000000000000 ✔
 0 |  4 |  1 |    0.000000000000 |    0.000000000000 ✔
 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 |  6 |    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 |  2 |    0.000000000000 |    0.000000000000 ✔
 0 |  5 |  3 |    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 |  9 |    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 |  4 |    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 |  9 |    0.000000000000 |    0.000000000000 ✔
 0 |  7 |  0 |    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 |  6 |    0.000000000000 |   -0.000000000000 ✔
 0 |  7 |  7 |    0.133333333333 |    0.133333333333 ✔
 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 |  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 |  8 |    0.117647058824 |    0.117647058824 ✔
 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 ✔
 0 |  9 |  3 |    0.000000000000 |   -0.000000000000 ✔
 0 |  9 |  4 |    0.000000000000 |    0.000000000000 ✔
 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 |  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 |  7 |    0.000000000000 |    0.000000000000 ✔
 1 |  1 |  8 |    0.000000000000 |    0.000000000000 ✔
 1 |  1 |  9 |    0.000000000000 |    0.000000000000 ✔
 1 |  2 |  1 |    0.000000000000 |    0.000000000000 ✔
 1 |  2 |  2 |    2.400000000000 |    2.400000000000 ✔
 1 |  2 |  3 |    0.000000000000 |    0.000000000000 ✔
 1 |  2 |  4 |    0.000000000000 |    0.000000000000 ✔
 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 |  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 |  4 |  1 |    0.000000000000 |    0.000000000000 ✔
 1 |  4 |  2 |    0.000000000000 |    0.000000000000 ✔
 1 |  4 |  3 |    0.000000000000 |    0.000000000000 ✔
 1 |  4 |  4 |    4.444444444444 |    4.444444444444 ✔
 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 |  9 |    0.000000000000 |    0.000000000000 ✔
 1 |  5 |  1 |    0.000000000000 |    0.000000000000 ✔
 1 |  5 |  2 |    0.000000000000 |   -0.000000000000 ✔
 1 |  5 |  3 |    0.000000000000 |   -0.000000000000 ✔
 1 |  5 |  4 |    0.000000000000 |    0.000000000000 ✔
 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 |  6 |  2 |    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 |  7 |  1 |    0.000000000000 |    0.000000000000 ✔
 1 |  7 |  2 |    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 |  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 |  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 |  4 |    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 ✔
 2 |  2 |  2 |    9.600000000000 |    9.600000000000 ✔
 2 |  2 |  3 |    0.000000000000 |    0.000000000000 ✔
 2 |  2 |  4 |    0.000000000000 |    0.000000000000 ✔
 2 |  2 |  5 |    0.000000000000 |    0.000000000000 ✔
 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 |  3 |  2 |    0.000000000000 |    0.000000000000 ✔
 2 |  3 |  3 |   34.285714285714 |   34.285714285714 ✔
 2 |  3 |  4 |    0.000000000000 |    0.000000000000 ✔
 2 |  3 |  5 |    0.000000000000 |    0.000000000000 ✔
 2 |  3 |  6 |    0.000000000000 |    0.000000000000 ✔
 2 |  3 |  7 |    0.000000000000 |   -0.000000000000 ✔
 2 |  3 |  8 |    0.000000000000 |    0.000000000000 ✔
 2 |  3 |  9 |    0.000000000000 |   -0.000000000000 ✔
 2 |  4 |  2 |    0.000000000000 |    0.000000000000 ✔
 2 |  4 |  3 |    0.000000000000 |    0.000000000000 ✔
 2 |  4 |  4 |   80.000000000000 |   80.000000000000 ✔
 2 |  4 |  5 |    0.000000000000 |    0.000000000000 ✔
 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 |  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 |  8 |    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 |  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 |  6 |    0.000000000000 |    0.000000000000 ✔
 2 |  7 |  7 |  403.200000000000 |  403.200000000000 ✔
 2 |  7 |  8 |    0.000000000000 |   -0.000000000000 ✔
 2 |  7 |  9 |    0.000000000000 |   -0.000000000000 ✔
 2 |  8 |  2 |    0.000000000000 |    0.000000000000 ✔
 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 |  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 |  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 |  7 |    0.000000000000 |   -0.000000000000 ✔
 2 |  9 |  8 |    0.000000000000 |    0.000000000000 ✔
 2 |  9 |  9 |  833.684210526316 |  833.684210526316 ✔
 3 |  3 |  3 |  205.714285714286 |  205.714285714286 ✔
 3 |  3 |  4 |    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 |  8 |    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 |  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 |  6 |  3 |    0.000000000000 |    0.000000000000 ✔
 3 |  6 |  4 |    0.000000000000 |    0.000000000000 ✔
 3 |  6 |  5 |    0.000000000000 |    0.000000000000 ✔
 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 |  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 |  8 |    0.000000000000 |    0.000000000000 ✔
 3 |  7 |  9 |    0.000000000000 |   -0.000000000003 ✔
 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 |  7 |    0.000000000000 |    0.000000000000 ✔
 3 |  8 |  8 | 39134.117647058825 | 39134.117647058825 ✔
 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 |  8 |    0.000000000000 |    0.000000000000 ✔
 3 |  9 |  9 | 70029.473684210534 | 70029.473684210505 ✔
 4 |  4 |  4 | 8960.000000000000 | 8960.000000000002 ✔
 4 |  4 |  5 |    0.000000000000 |   -0.000000000002 ✔
 4 |  4 |  6 |    0.000000000000 |   -0.000000000001 ✔
 4 |  4 |  7 |    0.000000000000 |   -0.000000000000 ✔
 4 |  4 |  8 |    0.000000000000 |    0.000000000007 ✔
 4 |  4 |  9 |    0.000000000000 |    0.000000000000 ✔
 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 |  6 |  5 |    0.000000000000 |   -0.000000000001 ✔
 4 |  6 |  6 | 279138.461538461561 | 279138.461538461503 ✔
 4 |  6 |  7 |    0.000000000000 |   -0.000000000018 ✔
 4 |  6 |  8 |    0.000000000000 |    0.000000000055 ✔
 4 |  6 |  9 |    0.000000000000 |    0.000000000029 ✔
 4 |  7 |  4 |    0.000000000000 |   -0.000000000000 ✔
 4 |  7 |  5 |    0.000000000000 |   -0.000000000058 ✔
 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 |  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 |  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 ✔
 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 |  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 |  7 |  5 |    0.000000000000 |    0.000000000233 ✔
 5 |  7 |  6 |    0.000000000000 |    0.000000000250 ✔
 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 |  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 |  9 |  9 | 382360926.315789461136 | 382360926.315789461136 ✔

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 |    1.000000000000 |    1.000000000000 ✔
 0 |  1 |  0 | -1 |    0.000000000000 |    0.000000000000 ✔
 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 |  0 |    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 ✔
 1 |  0 |  1 |  0 |    0.000000000000 |    0.000000000000 ✔
 1 |  1 | -1 | -1 |    1.000000000000 |    1.000000000000 ✔
 1 |  1 | -1 |  0 |    0.000000000000 |    0.000000000000 ✔
 1 |  1 | -1 |  1 |    0.000000000000 |    0.000000000000 ✔
 1 |  1 |  0 | -1 |    0.000000000000 |    0.000000000000 ✔
 1 |  1 |  0 |  0 |    1.000000000000 |    1.000000000000 ✔
 1 |  1 |  0 |  1 |    0.000000000000 |   -0.000000000000 ✔
 1 |  1 |  1 | -1 |    0.000000000000 |    0.000000000000 ✔
 1 |  1 |  1 |  0 |    0.000000000000 |   -0.000000000000 ✔
 1 |  1 |  1 |  1 |    1.000000000000 |    1.000000000000 ✔
 1 |  2 | -1 | -2 |    0.000000000000 |   -0.000000000000 ✔
 1 |  2 | -1 | -1 |    0.000000000000 |   -0.000000000000 ✔
 1 |  2 | -1 |  0 |    0.000000000000 |    0.000000000000 ✔
 1 |  2 | -1 |  1 |    0.000000000000 |   -0.000000000000 ✔
 1 |  2 | -1 |  2 |    0.000000000000 |    0.000000000000 ✔
 1 |  2 |  0 | -2 |    0.000000000000 |   -0.000000000000 ✔
 1 |  2 |  0 | -1 |    0.000000000000 |   -0.000000000000 ✔
 1 |  2 |  0 |  0 |    0.000000000000 |    0.000000000000 ✔
 1 |  2 |  0 |  1 |    0.000000000000 |    0.000000000000 ✔
 1 |  2 |  0 |  2 |    0.000000000000 |   -0.000000000000 ✔
 1 |  2 |  1 | -2 |    0.000000000000 |   -0.000000000000 ✔
 1 |  2 |  1 | -1 |    0.000000000000 |   -0.000000000000 ✔
 1 |  2 |  1 |  0 |    0.000000000000 |   -0.000000000000 ✔
 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 |  0 |  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 ✔
 2 |  1 | -2 |  1 |    0.000000000000 |   -0.000000000000 ✔
 2 |  1 | -1 | -1 |    0.000000000000 |   -0.000000000000 ✔
 2 |  1 | -1 |  0 |    0.000000000000 |   -0.000000000000 ✔
 2 |  1 | -1 |  1 |    0.000000000000 |   -0.000000000000 ✔
 2 |  1 |  0 | -1 |    0.000000000000 |    0.000000000000 ✔
 2 |  1 |  0 |  0 |    0.000000000000 |    0.000000000000 ✔
 2 |  1 |  0 |  1 |    0.000000000000 |   -0.000000000000 ✔
 2 |  1 |  1 | -1 |    0.000000000000 |   -0.000000000000 ✔
 2 |  1 |  1 |  0 |    0.000000000000 |    0.000000000000 ✔
 2 |  1 |  1 |  1 |    0.000000000000 |   -0.000000000000 ✔
 2 |  1 |  2 | -1 |    0.000000000000 |    0.000000000000 ✔
 2 |  1 |  2 |  0 |    0.000000000000 |   -0.000000000000 ✔
 2 |  1 |  2 |  1 |    0.000000000000 |    0.000000000000 ✔
 2 |  2 | -2 | -2 |    1.000000000000 |    1.000000000000 ✔
 2 |  2 | -2 | -1 |    0.000000000000 |   -0.000000000000 ✔
 2 |  2 | -2 |  0 |    0.000000000000 |    0.000000000000 ✔
 2 |  2 | -2 |  1 |    0.000000000000 |    0.000000000000 ✔
 2 |  2 | -2 |  2 |    0.000000000000 |   -0.000000000000 ✔
 2 |  2 | -1 | -2 |    0.000000000000 |   -0.000000000000 ✔
 2 |  2 | -1 | -1 |    1.000000000000 |    1.000000000000 ✔
 2 |  2 | -1 |  0 |    0.000000000000 |   -0.000000000000 ✔
 2 |  2 | -1 |  1 |    0.000000000000 |    0.000000000000 ✔
 2 |  2 | -1 |  2 |    0.000000000000 |   -0.000000000000 ✔
 2 |  2 |  0 | -2 |    0.000000000000 |    0.000000000000 ✔
 2 |  2 |  0 | -1 |    0.000000000000 |   -0.000000000000 ✔
 2 |  2 |  0 |  0 |    1.000000000000 |    1.000000000000 ✔
 2 |  2 |  0 |  1 |    0.000000000000 |    0.000000000000 ✔
 2 |  2 |  0 |  2 |    0.000000000000 |    0.000000000000 ✔
 2 |  2 |  1 | -2 |    0.000000000000 |    0.000000000000 ✔
 2 |  2 |  1 | -1 |    0.000000000000 |    0.000000000000 ✔
 2 |  2 |  1 |  0 |    0.000000000000 |    0.000000000000 ✔
 2 |  2 |  1 |  1 |    1.000000000000 |    1.000000000000 ✔
 2 |  2 |  1 |  2 |    0.000000000000 |    0.000000000000 ✔
 2 |  2 |  2 | -2 |    0.000000000000 |   -0.000000000000 ✔
 2 |  2 |  2 | -1 |    0.000000000000 |   -0.000000000000 ✔
 2 |  2 |  2 |  0 |    0.000000000000 |    0.000000000000 ✔
 2 |  2 |  2 |  1 |    0.000000000000 |    0.000000000000 ✔
 2 |  2 |  2 |  2 |    1.000000000000 |    1.000000000000 ✔

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

\[ \begin{aligned} L_n^{k}(x) &= \frac{\mathrm{d}^k}{\mathrm{d}x^k} L_n(x) \\ &= \frac{\mathrm{d}^k}{\mathrm{d}x^k} \frac{1}{n!} \mathrm{e}^x \frac{\mathrm{d}^n}{\mathrm{d}x ^n} \left( \mathrm{e}^{-x} x^n \right) \\ &= \sum_{m=0}^{n-k} (-1)^{m+k} \frac{n!}{m!(m+k)!(n-m-k)!} x^m \\ &= (-1)^k L_{n-k}^{(k)}(x) \end{aligned}\]

$n=0, k=0:$ ✔

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

$n=1, k=0:$ ✔

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

$n=1, k=1:$ ✔

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

$n=2, k=0:$ ✔

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

$n=2, k=1:$ ✔

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

$n=2, k=2:$ ✔

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

$n=3, k=0:$ ✔

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

$n=3, k=1:$ ✔

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

$n=3, k=2:$ ✔

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

$n=3, k=3:$ ✔

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

$n=4, k=0:$ ✔

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

$n=4, k=1:$ ✔

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

$n=4, k=2:$ ✔

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

$n=4, k=3:$ ✔

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

$n=4, k=4:$ ✔

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

Normalization & Orthogonality of $L_n^{k}(x)$

\[\int_{0}^{\infty} \mathrm{e}^{-x} x^k L_i^k(x) L_j^k(x) \mathrm{d}x = \frac{i!}{(i-k)!} \delta_{ij}\]

Replace $n+k$ with $n$ for the definition of Wolfram MathWorld.

 i |  j |  k |        analytical |         numerical 
-- | -- | -- | ----------------- | ----------------- 
 0 |  0 |  0 |    1.000000000000 |    1.000000000000 ✔
 0 |  1 |  0 |    0.000000000000 |    0.000000000000 ✔
 0 |  2 |  0 |    0.000000000000 |    0.000000000000 ✔
 0 |  3 |  0 |    0.000000000000 |    0.000000000000 ✔
 0 |  4 |  0 |    0.000000000000 |    0.000000000000 ✔
 0 |  5 |  0 |    0.000000000000 |   -0.000000000000 ✔
 0 |  6 |  0 |    0.000000000000 |   -0.000000000000 ✔
 0 |  7 |  0 |    0.000000000000 |    0.000000000000 ✔
 1 |  0 |  0 |    0.000000000000 |    0.000000000000 ✔
 1 |  1 |  0 |    1.000000000000 |    1.000000000000 ✔
 1 |  1 |  1 |    1.000000000000 |    1.000000000000 ✔
 1 |  2 |  0 |    0.000000000000 |   -0.000000000000 ✔
 1 |  2 |  1 |    0.000000000000 |    0.000000000000 ✔
 1 |  3 |  0 |    0.000000000000 |   -0.000000000000 ✔
 1 |  3 |  1 |    0.000000000000 |   -0.000000000000 ✔
 1 |  4 |  0 |    0.000000000000 |   -0.000000000000 ✔
 1 |  4 |  1 |    0.000000000000 |    0.000000000000 ✔
 1 |  5 |  0 |    0.000000000000 |    0.000000000000 ✔
 1 |  5 |  1 |    0.000000000000 |   -0.000000000000 ✔
 1 |  6 |  0 |    0.000000000000 |    0.000000000000 ✔
 1 |  6 |  1 |    0.000000000000 |   -0.000000000000 ✔
 1 |  7 |  0 |    0.000000000000 |   -0.000000000000 ✔
 1 |  7 |  1 |    0.000000000000 |    0.000000000000 ✔
 2 |  0 |  0 |    0.000000000000 |    0.000000000000 ✔
 2 |  1 |  0 |    0.000000000000 |    0.000000000000 ✔
 2 |  1 |  1 |    0.000000000000 |    0.000000000000 ✔
 2 |  2 |  0 |    1.000000000000 |    1.000000000000 ✔
 2 |  2 |  1 |    2.000000000000 |    2.000000000000 ✔
 2 |  2 |  2 |    2.000000000000 |    2.000000000000 ✔
 2 |  3 |  0 |    0.000000000000 |    0.000000000000 ✔
 2 |  3 |  1 |    0.000000000000 |   -0.000000000000 ✔
 2 |  3 |  2 |    0.000000000000 |   -0.000000000000 ✔
 2 |  4 |  0 |    0.000000000000 |    0.000000000000 ✔
 2 |  4 |  1 |    0.000000000000 |   -0.000000000000 ✔
 2 |  4 |  2 |    0.000000000000 |   -0.000000000000 ✔
 2 |  5 |  0 |    0.000000000000 |    0.000000000000 ✔
 2 |  5 |  1 |    0.000000000000 |    0.000000000000 ✔
 2 |  5 |  2 |    0.000000000000 |    0.000000000000 ✔
 2 |  6 |  0 |    0.000000000000 |   -0.000000000000 ✔
 2 |  6 |  1 |    0.000000000000 |    0.000000000000 ✔
 2 |  6 |  2 |    0.000000000000 |   -0.000000000000 ✔
 2 |  7 |  0 |    0.000000000000 |    0.000000000000 ✔
 2 |  7 |  1 |    0.000000000000 |   -0.000000000000 ✔
 2 |  7 |  2 |    0.000000000000 |    0.000000000000 ✔
 3 |  0 |  0 |    0.000000000000 |    0.000000000000 ✔
 3 |  1 |  0 |    0.000000000000 |   -0.000000000000 ✔
 3 |  1 |  1 |    0.000000000000 |   -0.000000000000 ✔
 3 |  2 |  0 |    0.000000000000 |    0.000000000000 ✔
 3 |  2 |  1 |    0.000000000000 |   -0.000000000000 ✔
 3 |  2 |  2 |    0.000000000000 |   -0.000000000000 ✔
 3 |  3 |  0 |    1.000000000000 |    1.000000000000 ✔
 3 |  3 |  1 |    3.000000000000 |    3.000000000000 ✔
 3 |  3 |  2 |    6.000000000000 |    6.000000000000 ✔
 3 |  3 |  3 |    6.000000000000 |    6.000000000000 ✔
 3 |  4 |  0 |    0.000000000000 |    0.000000000000 ✔
 3 |  4 |  1 |    0.000000000000 |    0.000000000000 ✔
 3 |  4 |  2 |    0.000000000000 |   -0.000000000000 ✔
 3 |  4 |  3 |    0.000000000000 |   -0.000000000000 ✔
 3 |  5 |  0 |    0.000000000000 |   -0.000000000000 ✔
 3 |  5 |  1 |    0.000000000000 |   -0.000000000000 ✔
 3 |  5 |  2 |    0.000000000000 |   -0.000000000000 ✔
 3 |  5 |  3 |    0.000000000000 |    0.000000000000 ✔
 3 |  6 |  0 |    0.000000000000 |    0.000000000000 ✔
 3 |  6 |  1 |    0.000000000000 |   -0.000000000000 ✔
 3 |  6 |  2 |    0.000000000000 |    0.000000000000 ✔
 3 |  6 |  3 |    0.000000000000 |    0.000000000000 ✔
 3 |  7 |  0 |    0.000000000000 |   -0.000000000000 ✔
 3 |  7 |  1 |    0.000000000000 |    0.000000000000 ✔
 3 |  7 |  2 |    0.000000000000 |   -0.000000000000 ✔
 3 |  7 |  3 |    0.000000000000 |   -0.000000000000 ✔
 4 |  0 |  0 |    0.000000000000 |    0.000000000000 ✔
 4 |  1 |  0 |    0.000000000000 |   -0.000000000000 ✔
 4 |  1 |  1 |    0.000000000000 |    0.000000000000 ✔
 4 |  2 |  0 |    0.000000000000 |    0.000000000000 ✔
 4 |  2 |  1 |    0.000000000000 |   -0.000000000000 ✔
 4 |  2 |  2 |    0.000000000000 |   -0.000000000000 ✔
 4 |  3 |  0 |    0.000000000000 |    0.000000000000 ✔
 4 |  3 |  1 |    0.000000000000 |    0.000000000000 ✔
 4 |  3 |  2 |    0.000000000000 |    0.000000000000 ✔
 4 |  3 |  3 |    0.000000000000 |   -0.000000000000 ✔
 4 |  4 |  0 |    1.000000000000 |    1.000000000000 ✔
 4 |  4 |  1 |    4.000000000000 |    4.000000000000 ✔
 4 |  4 |  2 |   12.000000000000 |   12.000000000000 ✔
 4 |  4 |  3 |   24.000000000000 |   24.000000000000 ✔
 4 |  4 |  4 |   24.000000000000 |   24.000000000000 ✔
 4 |  5 |  0 |    0.000000000000 |    0.000000000000 ✔
 4 |  5 |  1 |    0.000000000000 |    0.000000000000 ✔
 4 |  5 |  2 |    0.000000000000 |    0.000000000000 ✔
 4 |  5 |  3 |    0.000000000000 |    0.000000000000 ✔
 4 |  5 |  4 |    0.000000000000 |   -0.000000000000 ✔
 4 |  6 |  0 |    0.000000000000 |   -0.000000000000 ✔
 4 |  6 |  1 |    0.000000000000 |    0.000000000000 ✔
 4 |  6 |  2 |    0.000000000000 |   -0.000000000000 ✔
 4 |  6 |  3 |    0.000000000000 |   -0.000000000000 ✔
 4 |  6 |  4 |    0.000000000000 |    0.000000000000 ✔
 4 |  7 |  0 |    0.000000000000 |    0.000000000000 ✔
 4 |  7 |  1 |    0.000000000000 |   -0.000000000000 ✔
 4 |  7 |  2 |    0.000000000000 |    0.000000000000 ✔
 4 |  7 |  3 |    0.000000000000 |    0.000000000000 ✔
 4 |  7 |  4 |    0.000000000000 |    0.000000000000 ✔
 5 |  0 |  0 |    0.000000000000 |   -0.000000000000 ✔
 5 |  1 |  0 |    0.000000000000 |    0.000000000000 ✔
 5 |  1 |  1 |    0.000000000000 |   -0.000000000000 ✔
 5 |  2 |  0 |    0.000000000000 |    0.000000000000 ✔
 5 |  2 |  1 |    0.000000000000 |    0.000000000000 ✔
 5 |  2 |  2 |    0.000000000000 |    0.000000000000 ✔
 5 |  3 |  0 |    0.000000000000 |   -0.000000000000 ✔
 5 |  3 |  1 |    0.000000000000 |   -0.000000000000 ✔
 5 |  3 |  2 |    0.000000000000 |   -0.000000000000 ✔
 5 |  3 |  3 |    0.000000000000 |    0.000000000000 ✔
 5 |  4 |  0 |    0.000000000000 |    0.000000000000 ✔
 5 |  4 |  1 |    0.000000000000 |    0.000000000000 ✔
 5 |  4 |  2 |    0.000000000000 |    0.000000000000 ✔
 5 |  4 |  3 |    0.000000000000 |    0.000000000000 ✔
 5 |  4 |  4 |    0.000000000000 |   -0.000000000000 ✔
 5 |  5 |  0 |    1.000000000000 |    1.000000000000 ✔
 5 |  5 |  1 |    5.000000000000 |    4.999999999999 ✔
 5 |  5 |  2 |   20.000000000000 |   20.000000000000 ✔
 5 |  5 |  3 |   60.000000000000 |   60.000000000000 ✔
 5 |  5 |  4 |  120.000000000000 |  120.000000000000 ✔
 5 |  5 |  5 |  120.000000000000 |  120.000000000000 ✔
 5 |  6 |  0 |    0.000000000000 |    0.000000000000 ✔
 5 |  6 |  1 |    0.000000000000 |   -0.000000000000 ✔
 5 |  6 |  2 |    0.000000000000 |    0.000000000000 ✔
 5 |  6 |  3 |    0.000000000000 |    0.000000000000 ✔
 5 |  6 |  4 |    0.000000000000 |    0.000000000000 ✔
 5 |  6 |  5 |    0.000000000000 |    0.000000000000 ✔
 5 |  7 |  0 |    0.000000000000 |   -0.000000000000 ✔
 5 |  7 |  1 |    0.000000000000 |   -0.000000000000 ✔
 5 |  7 |  2 |    0.000000000000 |   -0.000000000000 ✔
 5 |  7 |  3 |    0.000000000000 |   -0.000000000000 ✔
 5 |  7 |  4 |    0.000000000000 |   -0.000000000000 ✔
 5 |  7 |  5 |    0.000000000000 |   -0.000000000000 ✔
 6 |  0 |  0 |    0.000000000000 |   -0.000000000000 ✔
 6 |  1 |  0 |    0.000000000000 |    0.000000000000 ✔
 6 |  1 |  1 |    0.000000000000 |   -0.000000000000 ✔
 6 |  2 |  0 |    0.000000000000 |   -0.000000000000 ✔
 6 |  2 |  1 |    0.000000000000 |    0.000000000000 ✔
 6 |  2 |  2 |    0.000000000000 |   -0.000000000000 ✔
 6 |  3 |  0 |    0.000000000000 |    0.000000000000 ✔
 6 |  3 |  1 |    0.000000000000 |   -0.000000000000 ✔
 6 |  3 |  2 |    0.000000000000 |    0.000000000000 ✔
 6 |  3 |  3 |    0.000000000000 |    0.000000000000 ✔
 6 |  4 |  0 |    0.000000000000 |   -0.000000000000 ✔
 6 |  4 |  1 |    0.000000000000 |    0.000000000000 ✔
 6 |  4 |  2 |    0.000000000000 |   -0.000000000000 ✔
 6 |  4 |  3 |    0.000000000000 |   -0.000000000000 ✔
 6 |  4 |  4 |    0.000000000000 |    0.000000000000 ✔
 6 |  5 |  0 |    0.000000000000 |    0.000000000000 ✔
 6 |  5 |  1 |    0.000000000000 |   -0.000000000000 ✔
 6 |  5 |  2 |    0.000000000000 |    0.000000000000 ✔
 6 |  5 |  3 |    0.000000000000 |    0.000000000000 ✔
 6 |  5 |  4 |    0.000000000000 |    0.000000000000 ✔
 6 |  5 |  5 |    0.000000000000 |    0.000000000000 ✔
 6 |  6 |  0 |    1.000000000000 |    1.000000000000 ✔
 6 |  6 |  1 |    6.000000000000 |    6.000000000000 ✔
 6 |  6 |  2 |   30.000000000000 |   30.000000000000 ✔
 6 |  6 |  3 |  120.000000000000 |  119.999999999978 ✔
 6 |  6 |  4 |  360.000000000000 |  359.999999999996 ✔
 6 |  6 |  5 |  720.000000000000 |  720.000000000000 ✔
 6 |  6 |  6 |  720.000000000000 |  720.000000000000 ✔
 6 |  7 |  0 |    0.000000000000 |    0.000000000000 ✔
 6 |  7 |  1 |    0.000000000000 |    0.000000000000 ✔
 6 |  7 |  2 |    0.000000000000 |   -0.000000000000 ✔
 6 |  7 |  3 |    0.000000000000 |    0.000000000000 ✔
 6 |  7 |  4 |    0.000000000000 |    0.000000000000 ✔
 6 |  7 |  5 |    0.000000000000 |    0.000000000000 ✔
 6 |  7 |  6 |    0.000000000000 |    0.000000000000 ✔
 7 |  0 |  0 |    0.000000000000 |    0.000000000000 ✔
 7 |  1 |  0 |    0.000000000000 |   -0.000000000000 ✔
 7 |  1 |  1 |    0.000000000000 |    0.000000000000 ✔
 7 |  2 |  0 |    0.000000000000 |    0.000000000000 ✔
 7 |  2 |  1 |    0.000000000000 |   -0.000000000000 ✔
 7 |  2 |  2 |    0.000000000000 |    0.000000000000 ✔
 7 |  3 |  0 |    0.000000000000 |   -0.000000000000 ✔
 7 |  3 |  1 |    0.000000000000 |    0.000000000000 ✔
 7 |  3 |  2 |    0.000000000000 |   -0.000000000000 ✔
 7 |  3 |  3 |    0.000000000000 |   -0.000000000000 ✔
 7 |  4 |  0 |    0.000000000000 |    0.000000000000 ✔
 7 |  4 |  1 |    0.000000000000 |   -0.000000000000 ✔
 7 |  4 |  2 |    0.000000000000 |    0.000000000000 ✔
 7 |  4 |  3 |    0.000000000000 |    0.000000000000 ✔
 7 |  4 |  4 |    0.000000000000 |    0.000000000000 ✔
 7 |  5 |  0 |    0.000000000000 |   -0.000000000000 ✔
 7 |  5 |  1 |    0.000000000000 |   -0.000000000000 ✔
 7 |  5 |  2 |    0.000000000000 |   -0.000000000000 ✔
 7 |  5 |  3 |    0.000000000000 |   -0.000000000000 ✔
 7 |  5 |  4 |    0.000000000000 |   -0.000000000000 ✔
 7 |  5 |  5 |    0.000000000000 |   -0.000000000000 ✔
 7 |  6 |  0 |    0.000000000000 |    0.000000000000 ✔
 7 |  6 |  1 |    0.000000000000 |   -0.000000000000 ✔
 7 |  6 |  2 |    0.000000000000 |   -0.000000000000 ✔
 7 |  6 |  3 |    0.000000000000 |    0.000000000000 ✔
 7 |  6 |  4 |    0.000000000000 |    0.000000000000 ✔
 7 |  6 |  5 |    0.000000000000 |   -0.000000000000 ✔
 7 |  6 |  6 |    0.000000000000 |    0.000000000000 ✔
 7 |  7 |  0 |    1.000000000000 |    1.000000000000 ✔
 7 |  7 |  1 |    7.000000000000 |    7.000000000000 ✔
 7 |  7 |  2 |   42.000000000000 |   42.000000000000 ✔
 7 |  7 |  3 |  210.000000000000 |  210.000000000000 ✔
 7 |  7 |  4 |  840.000000000000 |  840.000000000000 ✔
 7 |  7 |  5 | 2520.000000000000 | 2519.999999999775 ✔
 7 |  7 |  6 | 5040.000000000000 | 5039.999999999985 ✔
 7 |  7 |  7 | 5040.000000000000 | 5040.000000000000 ✔

Normalization of $R_{nl}(r)$

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

 n |  l |        analytical |         numerical 
-- | -- | ----------------- | ----------------- 
 1 |  0 |    1.000000000000 |    1.000000000000 ✔
 2 |  0 |    1.000000000000 |    1.000000000000 ✔
 2 |  1 |    1.000000000000 |    1.000000000000 ✔
 3 |  0 |    1.000000000000 |    1.000000000000 ✔
 3 |  1 |    1.000000000000 |    0.999999999999 ✔
 3 |  2 |    1.000000000000 |    1.000000000000 ✔
 4 |  0 |    1.000000000000 |    1.000000000000 ✔
 4 |  1 |    1.000000000000 |    1.000000000000 ✔
 4 |  2 |    1.000000000000 |    1.000000000000 ✔
 4 |  3 |    1.000000000000 |    1.000000000000 ✔
 5 |  0 |    1.000000000000 |    1.000000000000 ✔
 5 |  1 |    1.000000000000 |    1.000000000000 ✔
 5 |  2 |    1.000000000000 |    1.000000000000 ✔
 5 |  3 |    1.000000000000 |    1.000000000000 ✔
 5 |  4 |    1.000000000000 |    1.000000000000 ✔
 6 |  0 |    1.000000000000 |    1.000000000000 ✔
 6 |  1 |    1.000000000000 |    1.000000000000 ✔
 6 |  2 |    1.000000000000 |    1.000000000000 ✔
 6 |  3 |    1.000000000000 |    1.000000000000 ✔
 6 |  4 |    1.000000000000 |    1.000000000000 ✔
 6 |  5 |    1.000000000000 |    1.000000000000 ✔
 7 |  0 |    1.000000000000 |    1.000000000000 ✔
 7 |  1 |    1.000000000000 |    1.000000000000 ✔
 7 |  2 |    1.000000000000 |    1.000000000000 ✔
 7 |  3 |    1.000000000000 |    1.000000000000 ✔
 7 |  4 |    1.000000000000 |    1.000000000000 ✔
 7 |  5 |    1.000000000000 |    1.000000000000 ✔
 7 |  6 |    1.000000000000 |    1.000000000000 ✔
 8 |  0 |    1.000000000000 |    1.000000000000 ✔
 8 |  1 |    1.000000000000 |    1.000000000000 ✔
 8 |  2 |    1.000000000000 |    1.000000000000 ✔
 8 |  3 |    1.000000000000 |    1.000000000000 ✔
 8 |  4 |    1.000000000000 |    1.000000000000 ✔
 8 |  5 |    1.000000000000 |    1.000000000000 ✔
 8 |  6 |    1.000000000000 |    1.000000000000 ✔
 8 |  7 |    1.000000000000 |    1.000000000000 ✔
 9 |  0 |    1.000000000000 |    1.000000000000 ✔
 9 |  1 |    1.000000000000 |    1.000000000000 ✔
 9 |  2 |    1.000000000000 |    1.000000000000 ✔
 9 |  3 |    1.000000000000 |    1.000000000000 ✔
 9 |  4 |    1.000000000000 |    1.000000000000 ✔
 9 |  5 |    1.000000000000 |    1.000000000000 ✔
 9 |  6 |    1.000000000000 |    1.000000000000 ✔
 9 |  7 |    1.000000000000 |    1.000000000000 ✔
 9 |  8 |    1.000000000000 |    1.000000000000 ✔

Expected Value of $r$

\[\langle r \rangle = \int r |R_{n_1 l_1}(r)|^2 r^2 \mathrm{d}r = \frac{a_\mu}{2Z} \left[ 3n^2 - l(l+1) \right] \\ a_\mu = a_0 \frac{m_\mathrm{e}}{\mu} \\ \frac{1}{\mu} = \frac{1}{m_\mathrm{e}} + \frac{1}{m_\mathrm{p}}\]

Reference:

 n |  l |        analytical |         numerical 
-- | -- | ----------------- | ----------------- 
 1 |  0 |    1.500000000000 |    1.500000000000 ✔
 2 |  0 |    6.000000000000 |    6.000000000000 ✔
 2 |  1 |    5.000000000000 |    5.000000000000 ✔
 3 |  0 |   13.500000000000 |   13.500000000000 ✔
 3 |  1 |   12.500000000000 |   12.500000000000 ✔
 3 |  2 |   10.500000000000 |   10.500000000000 ✔
 4 |  0 |   24.000000000000 |   23.999999999999 ✔
 4 |  1 |   23.000000000000 |   22.999999999999 ✔
 4 |  2 |   21.000000000000 |   21.000000000000 ✔
 4 |  3 |   18.000000000000 |   18.000000000000 ✔
 5 |  0 |   37.500000000000 |   37.500000000000 ✔
 5 |  1 |   36.500000000000 |   36.500000000000 ✔
 5 |  2 |   34.500000000000 |   34.500000000000 ✔
 5 |  3 |   31.500000000000 |   31.500000000000 ✔
 5 |  4 |   27.500000000000 |   27.499999999943 ✔
 6 |  0 |   54.000000000000 |   54.000000000001 ✔
 6 |  1 |   53.000000000000 |   53.000000000001 ✔
 6 |  2 |   51.000000000000 |   51.000000000000 ✔
 6 |  3 |   48.000000000000 |   48.000000000000 ✔
 6 |  4 |   44.000000000000 |   44.000000000000 ✔
 6 |  5 |   39.000000000000 |   39.000000000000 ✔
 7 |  0 |   73.500000000000 |   73.500000000000 ✔
 7 |  1 |   72.500000000000 |   72.500000000000 ✔
 7 |  2 |   70.500000000000 |   70.500000000000 ✔
 7 |  3 |   67.500000000000 |   67.500000000000 ✔
 7 |  4 |   63.500000000000 |   63.500000000000 ✔
 7 |  5 |   58.500000000000 |   58.500000000000 ✔
 7 |  6 |   52.500000000000 |   52.499999999992 ✔
 8 |  0 |   96.000000000000 |   96.000000000001 ✔
 8 |  1 |   95.000000000000 |   94.999999999999 ✔
 8 |  2 |   93.000000000000 |   93.000000000000 ✔
 8 |  3 |   90.000000000000 |   90.000000000000 ✔
 8 |  4 |   86.000000000000 |   86.000000000000 ✔
 8 |  5 |   81.000000000000 |   81.000000000000 ✔
 8 |  6 |   75.000000000000 |   75.000000000000 ✔
 8 |  7 |   68.000000000000 |   68.000000000000 ✔
 9 |  0 |  121.500000000000 |  121.500000000001 ✔
 9 |  1 |  120.500000000000 |  120.500000000000 ✔
 9 |  2 |  118.500000000000 |  118.500000000001 ✔
 9 |  3 |  115.500000000000 |  115.500000000000 ✔
 9 |  4 |  111.500000000000 |  111.499999999998 ✔
 9 |  5 |  106.500000000000 |  106.499999999999 ✔
 9 |  6 |  100.500000000000 |  100.500000000000 ✔
 9 |  7 |   93.500000000000 |   93.500000000000 ✔
 9 |  8 |   85.500000000000 |   85.500000000000 ✔

Expected Value of $r^2$

\[\langle r^2 \rangle = \int r^2 |R_{n_1 l_1}(r)|^2 r^2 \mathrm{d}r = \frac{a_\mu^2}{2Z^2} n^2 \left[ 5n^2 + 1 - 3l(l+1) \right] \\ a_\mu = a_0 \frac{m_\mathrm{e}}{\mu} \\ \frac{1}{\mu} = \frac{1}{m_\mathrm{e}} + \frac{1}{m_\mathrm{p}}\]

Reference:

 n |  l |        analytical |         numerical 
-- | -- | ----------------- | ----------------- 
 1 |  0 |    3.000000000000 |    3.000000000000 ✔
 2 |  0 |   42.000000000000 |   42.000000000000 ✔
 2 |  1 |   30.000000000000 |   30.000000000000 ✔
 3 |  0 |  207.000000000000 |  207.000000000000 ✔
 3 |  1 |  180.000000000000 |  180.000000000000 ✔
 3 |  2 |  126.000000000000 |  126.000000000000 ✔
 4 |  0 |  648.000000000000 |  647.999999999903 ✔
 4 |  1 |  600.000000000000 |  599.999999999936 ✔
 4 |  2 |  504.000000000000 |  503.999999999975 ✔
 4 |  3 |  360.000000000000 |  359.999999999996 ✔
 5 |  0 | 1575.000000000000 | 1574.999999999999 ✔
 5 |  1 | 1500.000000000000 | 1499.999999999998 ✔
 5 |  2 | 1350.000000000000 | 1350.000000000000 ✔
 5 |  3 | 1125.000000000000 | 1125.000000000003 ✔
 5 |  4 |  825.000000000000 |  825.000000000000 ✔
 6 |  0 | 3258.000000000000 | 3257.999999999997 ✔
 6 |  1 | 3150.000000000000 | 3149.999999999992 ✔
 6 |  2 | 2934.000000000000 | 2933.999999999998 ✔
 6 |  3 | 2610.000000000000 | 2610.000000000033 ✔
 6 |  4 | 2178.000000000000 | 2178.000000000008 ✔
 6 |  5 | 1638.000000000000 | 1638.000000000000 ✔
 7 |  0 | 6027.000000000000 | 6026.999999999992 ✔
 7 |  1 | 5880.000000000000 | 5880.000000000003 ✔
 7 |  2 | 5586.000000000000 | 5585.999999999990 ✔
 7 |  3 | 5145.000000000000 | 5144.999999999992 ✔
 7 |  4 | 4557.000000000000 | 4556.999999999997 ✔
 7 |  5 | 3822.000000000000 | 3821.999999999999 ✔
 7 |  6 | 2940.000000000000 | 2940.000000000001 ✔
 8 |  0 | 10272.000000000000 | 10272.000000000029 ✔
 8 |  1 | 10080.000000000000 | 10079.999999999995 ✔
 8 |  2 | 9696.000000000000 | 9695.999999999993 ✔
 8 |  3 | 9120.000000000000 | 9120.000000000011 ✔
 8 |  4 | 8352.000000000000 | 8352.000000000002 ✔
 8 |  5 | 7392.000000000000 | 7392.000000000010 ✔
 8 |  6 | 6240.000000000000 | 6240.000000000000 ✔
 8 |  7 | 4896.000000000000 | 4896.000000000008 ✔
 9 |  0 | 16443.000000000000 | 16443.000000000102 ✔
 9 |  1 | 16200.000000000000 | 16200.000000000040 ✔
 9 |  2 | 15714.000000000000 | 15714.000000000149 ✔
 9 |  3 | 14985.000000000000 | 14984.999999999918 ✔
 9 |  4 | 14013.000000000000 | 14012.999999999545 ✔
 9 |  5 | 12798.000000000000 | 12797.999999999807 ✔
 9 |  6 | 11340.000000000000 | 11339.999999999945 ✔
 9 |  7 | 9639.000000000000 | 9638.999999999991 ✔
 9 |  8 | 7695.000000000000 | 7694.999999999998 ✔

Virial Theorem

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

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

HydrogenAtom(1, 1.0, 1.0, 1.0, 1.0)
 n |        analytical |         numerical 
-- | ----------------- | ----------------- 
 1 | -5.000000000000e-01 | -5.000000000000e-01 ✔
 2 | -1.250000000000e-01 | -1.250000000000e-01 ✔
 3 | -5.555555555556e-02 | -5.555555555556e-02 ✔
 4 | -3.125000000000e-02 | -3.125000000000e-02 ✔
 5 | -2.000000000000e-02 | -2.000000000000e-02 ✔
 6 | -1.388888888889e-02 | -1.388888888889e-02 ✔
 7 | -1.020408163265e-02 | -1.020408163265e-02 ✔
 8 | -7.812500000000e-03 | -7.812500000000e-03 ✔
 9 | -6.172839506173e-03 | -6.172839506173e-03 ✔
10 | -5.000000000000e-03 | -5.000000000000e-03 ✔

HydrogenAtom(1, 1.0, 2.0, 1.0, 1.0)
 n |        analytical |         numerical 
-- | ----------------- | ----------------- 
 1 | -5.000000000000e-01 | -5.000000000000e-01 ✔
 2 | -1.250000000000e-01 | -1.250000000000e-01 ✔
 3 | -5.555555555556e-02 | -5.555555555556e-02 ✔
 4 | -3.125000000000e-02 | -3.125000000000e-02 ✔
 5 | -2.000000000000e-02 | -2.000000000000e-02 ✔
 6 | -1.388888888889e-02 | -1.388888888889e-02 ✔
 7 | -1.020408163265e-02 | -1.020408163265e-02 ✔
 8 | -7.812500000000e-03 | -7.812500000000e-03 ✔
 9 | -6.172839506173e-03 | -6.172839506173e-03 ✔
10 | -5.000000000000e-03 | -5.000000000000e-03 ✔

HydrogenAtom(2, 1.0, 1.0, 1.0, 1.0)
 n |        analytical |         numerical 
-- | ----------------- | ----------------- 
 1 | -2.000000000000e+00 | -2.000000000000e+00 ✔
 2 | -5.000000000000e-01 | -5.000000000000e-01 ✔
 3 | -2.222222222222e-01 | -2.222222222222e-01 ✔
 4 | -1.250000000000e-01 | -1.250000000000e-01 ✔
 5 | -8.000000000000e-02 | -8.000000000000e-02 ✔
 6 | -5.555555555556e-02 | -5.555555555556e-02 ✔
 7 | -4.081632653061e-02 | -4.081632653061e-02 ✔
 8 | -3.125000000000e-02 | -3.125000000000e-02 ✔
 9 | -2.469135802469e-02 | -2.469135802469e-02 ✔
10 | -2.000000000000e-02 | -2.000000000000e-02 ✔

HydrogenAtom(2, 1.0, 1.0, 27.211386245988, 1.0)
 n |        analytical |         numerical 
-- | ----------------- | ----------------- 
 1 | -5.442277249198e+01 | -5.442277249198e+01 ✔
 2 | -1.360569312299e+01 | -1.360569312299e+01 ✔
 3 | -6.046974721331e+00 | -6.046974721331e+00 ✔
 4 | -3.401423280748e+00 | -3.401423280749e+00 ✔
 5 | -2.176910899679e+00 | -2.176910899679e+00 ✔
 6 | -1.511743680333e+00 | -1.511743680333e+00 ✔
 7 | -1.110668826367e+00 | -1.110668826367e+00 ✔
 8 | -8.503558201871e-01 | -8.503558201871e-01 ✔
 9 | -6.718860801479e-01 | -6.718860801478e-01 ✔
10 | -5.442277249198e-01 | -5.442277249197e-01 ✔

HydrogenAtom(2, 9.1093837015e-31, 5.29177210903e-11, 4.3597447222071e-18, 1.054571817e-34)
 n |        analytical |         numerical 
-- | ----------------- | ----------------- 
 1 | -8.719489444414e-18 | -8.719489444414e-18 ✔
 2 | -2.179872361104e-18 | -2.179872361104e-18 ✔
 3 | -9.688321604905e-19 | -9.688321604905e-19 ✔
 4 | -5.449680902759e-19 | -5.449680902759e-19 ✔
 5 | -3.487795777766e-19 | -3.487795777766e-19 ✔
 6 | -2.422080401226e-19 | -2.422080401226e-19 ✔
 7 | -1.779487641717e-19 | -1.779487641717e-19 ✔
 8 | -1.362420225690e-19 | -1.362420225690e-19 ✔
 9 | -1.076480178323e-19 | -1.076480178323e-19 ✔
10 | -8.719489444414e-20 | -8.719489444414e-20 ✔

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 
-- | -- | -- | -- | -- | -- | ----------------- | ----------------- 
 1 |  1 |  0 |  0 |  0 |  0 |    1.000000000000 |    1.000000000252 ✔
 1 |  2 |  0 |  0 |  0 |  0 |    0.000000000000 |   -0.000000011223 ✔
 1 |  2 |  0 |  1 |  0 | -1 |    0.000000000000 |   -0.000000000000 ✔
 1 |  2 |  0 |  1 |  0 |  0 |    0.000000000000 |   -0.000000000000 ✔
 1 |  2 |  0 |  1 |  0 |  1 |    0.000000000000 |    0.000000000000 ✔
 1 |  3 |  0 |  0 |  0 |  0 |    0.000000000000 |   -0.000000045661 ✔
 1 |  3 |  0 |  1 |  0 | -1 |    0.000000000000 |    0.000000000000 ✔
 1 |  3 |  0 |  1 |  0 |  0 |    0.000000000000 |   -0.000000000000 ✔
 1 |  3 |  0 |  1 |  0 |  1 |    0.000000000000 |   -0.000000000000 ✔
 1 |  3 |  0 |  2 |  0 | -2 |    0.000000000000 |    0.000000000000 ✔
 1 |  3 |  0 |  2 |  0 | -1 |    0.000000000000 |   -0.000000000000 ✔
 1 |  3 |  0 |  2 |  0 |  0 |    0.000000000000 |   -0.000000000000 ✔
 1 |  3 |  0 |  2 |  0 |  1 |    0.000000000000 |    0.000000000000 ✔
 1 |  3 |  0 |  2 |  0 |  2 |    0.000000000000 |    0.000000000000 ✔
 2 |  1 |  0 |  0 |  0 |  0 |    0.000000000000 |   -0.000000011223 ✔
 2 |  1 |  1 |  0 | -1 |  0 |    0.000000000000 |   -0.000000000000 ✔
 2 |  1 |  1 |  0 |  0 |  0 |    0.000000000000 |   -0.000000000000 ✔
 2 |  1 |  1 |  0 |  1 |  0 |    0.000000000000 |    0.000000000000 ✔
 2 |  2 |  0 |  0 |  0 |  0 |    1.000000000000 |    1.000006970517 ✔
 2 |  2 |  0 |  1 |  0 | -1 |    0.000000000000 |    0.000000000000 ✔
 2 |  2 |  0 |  1 |  0 |  0 |    0.000000000000 |    0.000000000000 ✔
 2 |  2 |  0 |  1 |  0 |  1 |    0.000000000000 |   -0.000000000000 ✔
 2 |  2 |  1 |  0 | -1 |  0 |    0.000000000000 |    0.000000000000 ✔
 2 |  2 |  1 |  0 |  0 |  0 |    0.000000000000 |    0.000000000000 ✔
 2 |  2 |  1 |  0 |  1 |  0 |    0.000000000000 |   -0.000000000000 ✔
 2 |  2 |  1 |  1 | -1 | -1 |    1.000000000000 |    1.000002301351 ✔
 2 |  2 |  1 |  1 | -1 |  0 |    0.000000000000 |   -0.000000000000 ✔
 2 |  2 |  1 |  1 | -1 |  1 |    0.000000000000 |   -0.000000000000 ✔
 2 |  2 |  1 |  1 |  0 | -1 |    0.000000000000 |   -0.000000000000 ✔
 2 |  2 |  1 |  1 |  0 |  0 |    1.000000000000 |    1.000002301351 ✔
 2 |  2 |  1 |  1 |  0 |  1 |    0.000000000000 |    0.000000000000 ✔
 2 |  2 |  1 |  1 |  1 | -1 |    0.000000000000 |   -0.000000000000 ✔
 2 |  2 |  1 |  1 |  1 |  0 |    0.000000000000 |    0.000000000000 ✔
 2 |  2 |  1 |  1 |  1 |  1 |    1.000000000000 |    1.000002301351 ✔
 2 |  3 |  0 |  0 |  0 |  0 |    0.000000000000 |    0.000088519421 ✔
 2 |  3 |  0 |  1 |  0 | -1 |    0.000000000000 |   -0.000000000000 ✔
 2 |  3 |  0 |  1 |  0 |  0 |    0.000000000000 |   -0.000000000000 ✔
 2 |  3 |  0 |  1 |  0 |  1 |    0.000000000000 |    0.000000000000 ✔
 2 |  3 |  0 |  2 |  0 | -2 |    0.000000000000 |   -0.000000000000 ✔
 2 |  3 |  0 |  2 |  0 | -1 |    0.000000000000 |    0.000000000000 ✔
 2 |  3 |  0 |  2 |  0 |  0 |    0.000000000000 |   -0.000000000000 ✔
 2 |  3 |  0 |  2 |  0 |  1 |    0.000000000000 |   -0.000000000000 ✔
 2 |  3 |  0 |  2 |  0 |  2 |    0.000000000000 |   -0.000000000000 ✔
 2 |  3 |  1 |  0 | -1 |  0 |    0.000000000000 |    0.000000000000 ✔
 2 |  3 |  1 |  0 |  0 |  0 |    0.000000000000 |   -0.000000000000 ✔
 2 |  3 |  1 |  0 |  1 |  0 |    0.000000000000 |    0.000000000000 ✔
 2 |  3 |  1 |  1 | -1 | -1 |    0.000000000000 |    0.000038730338 ✔
 2 |  3 |  1 |  1 | -1 |  0 |    0.000000000000 |    0.000000000000 ✔
 2 |  3 |  1 |  1 | -1 |  1 |    0.000000000000 |   -0.000000000000 ✔
 2 |  3 |  1 |  1 |  0 | -1 |    0.000000000000 |   -0.000000000000 ✔
 2 |  3 |  1 |  1 |  0 |  0 |    0.000000000000 |    0.000038730338 ✔
 2 |  3 |  1 |  1 |  0 |  1 |    0.000000000000 |    0.000000000000 ✔
 2 |  3 |  1 |  1 |  1 | -1 |    0.000000000000 |   -0.000000000000 ✔
 2 |  3 |  1 |  1 |  1 |  0 |    0.000000000000 |   -0.000000000000 ✔
 2 |  3 |  1 |  1 |  1 |  1 |    0.000000000000 |    0.000038730338 ✔
 2 |  3 |  1 |  2 | -1 | -2 |    0.000000000000 |   -0.000000000000 ✔
 2 |  3 |  1 |  2 | -1 | -1 |    0.000000000000 |   -0.000000000000 ✔
 2 |  3 |  1 |  2 | -1 |  0 |    0.000000000000 |    0.000000000000 ✔
 2 |  3 |  1 |  2 | -1 |  1 |    0.000000000000 |    0.000000000000 ✔
 2 |  3 |  1 |  2 | -1 |  2 |    0.000000000000 |   -0.000000000272 ✔
 2 |  3 |  1 |  2 |  0 | -2 |    0.000000000000 |    0.000000000000 ✔
 2 |  3 |  1 |  2 |  0 | -1 |    0.000000000000 |    0.000000000000 ✔
 2 |  3 |  1 |  2 |  0 |  0 |    0.000000000000 |   -0.000000000000 ✔
 2 |  3 |  1 |  2 |  0 |  1 |    0.000000000000 |   -0.000000000000 ✔
 2 |  3 |  1 |  2 |  0 |  2 |    0.000000000000 |    0.000000000000 ✔
 2 |  3 |  1 |  2 |  1 | -2 |    0.000000000000 |    0.000000000272 ✔
 2 |  3 |  1 |  2 |  1 | -1 |    0.000000000000 |    0.000000000000 ✔
 2 |  3 |  1 |  2 |  1 |  0 |    0.000000000000 |   -0.000000000000 ✔
 2 |  3 |  1 |  2 |  1 |  1 |    0.000000000000 |   -0.000000000000 ✔
 2 |  3 |  1 |  2 |  1 |  2 |    0.000000000000 |    0.000000000000 ✔
 3 |  1 |  0 |  0 |  0 |  0 |    0.000000000000 |   -0.000000045661 ✔
 3 |  1 |  1 |  0 | -1 |  0 |    0.000000000000 |    0.000000000000 ✔
 3 |  1 |  1 |  0 |  0 |  0 |    0.000000000000 |   -0.000000000000 ✔
 3 |  1 |  1 |  0 |  1 |  0 |    0.000000000000 |    0.000000000000 ✔
 3 |  1 |  2 |  0 | -2 |  0 |    0.000000000000 |    0.000000000000 ✔
 3 |  1 |  2 |  0 | -1 |  0 |    0.000000000000 |    0.000000000000 ✔
 3 |  1 |  2 |  0 |  0 |  0 |    0.000000000000 |   -0.000000000000 ✔
 3 |  1 |  2 |  0 |  1 |  0 |    0.000000000000 |   -0.000000000000 ✔
 3 |  1 |  2 |  0 |  2 |  0 |    0.000000000000 |    0.000000000000 ✔
 3 |  2 |  0 |  0 |  0 |  0 |    0.000000000000 |    0.000088519421 ✔
 3 |  2 |  0 |  1 |  0 | -1 |    0.000000000000 |    0.000000000000 ✔
 3 |  2 |  0 |  1 |  0 |  0 |    0.000000000000 |    0.000000000000 ✔
 3 |  2 |  0 |  1 |  0 |  1 |    0.000000000000 |   -0.000000000000 ✔
 3 |  2 |  1 |  0 | -1 |  0 |    0.000000000000 |   -0.000000000000 ✔
 3 |  2 |  1 |  0 |  0 |  0 |    0.000000000000 |   -0.000000000000 ✔
 3 |  2 |  1 |  0 |  1 |  0 |    0.000000000000 |    0.000000000000 ✔
 3 |  2 |  1 |  1 | -1 | -1 |    0.000000000000 |    0.000038730338 ✔
 3 |  2 |  1 |  1 | -1 |  0 |    0.000000000000 |   -0.000000000000 ✔
 3 |  2 |  1 |  1 | -1 |  1 |    0.000000000000 |   -0.000000000000 ✔
 3 |  2 |  1 |  1 |  0 | -1 |    0.000000000000 |    0.000000000000 ✔
 3 |  2 |  1 |  1 |  0 |  0 |    0.000000000000 |    0.000038730338 ✔
 3 |  2 |  1 |  1 |  0 |  1 |    0.000000000000 |   -0.000000000000 ✔
 3 |  2 |  1 |  1 |  1 | -1 |    0.000000000000 |   -0.000000000000 ✔
 3 |  2 |  1 |  1 |  1 |  0 |    0.000000000000 |    0.000000000000 ✔
 3 |  2 |  1 |  1 |  1 |  1 |    0.000000000000 |    0.000038730338 ✔
 3 |  2 |  2 |  0 | -2 |  0 |    0.000000000000 |   -0.000000000000 ✔
 3 |  2 |  2 |  0 | -1 |  0 |    0.000000000000 |    0.000000000000 ✔
 3 |  2 |  2 |  0 |  0 |  0 |    0.000000000000 |   -0.000000000000 ✔
 3 |  2 |  2 |  0 |  1 |  0 |    0.000000000000 |   -0.000000000000 ✔
 3 |  2 |  2 |  0 |  2 |  0 |    0.000000000000 |   -0.000000000000 ✔
 3 |  2 |  2 |  1 | -2 | -1 |    0.000000000000 |   -0.000000000000 ✔
 3 |  2 |  2 |  1 | -2 |  0 |    0.000000000000 |    0.000000000000 ✔
 3 |  2 |  2 |  1 | -2 |  1 |    0.000000000000 |    0.000000000272 ✔
 3 |  2 |  2 |  1 | -1 | -1 |    0.000000000000 |   -0.000000000000 ✔
 3 |  2 |  2 |  1 | -1 |  0 |    0.000000000000 |    0.000000000000 ✔
 3 |  2 |  2 |  1 | -1 |  1 |    0.000000000000 |    0.000000000000 ✔
 3 |  2 |  2 |  1 |  0 | -1 |    0.000000000000 |    0.000000000000 ✔
 3 |  2 |  2 |  1 |  0 |  0 |    0.000000000000 |    0.000000000000 ✔
 3 |  2 |  2 |  1 |  0 |  1 |    0.000000000000 |   -0.000000000000 ✔
 3 |  2 |  2 |  1 |  1 | -1 |    0.000000000000 |    0.000000000000 ✔
 3 |  2 |  2 |  1 |  1 |  0 |    0.000000000000 |   -0.000000000000 ✔
 3 |  2 |  2 |  1 |  1 |  1 |    0.000000000000 |   -0.000000000000 ✔
 3 |  2 |  2 |  1 |  2 | -1 |    0.000000000000 |   -0.000000000272 ✔
 3 |  2 |  2 |  1 |  2 |  0 |    0.000000000000 |    0.000000000000 ✔
 3 |  2 |  2 |  1 |  2 |  1 |    0.000000000000 |    0.000000000000 ✔
 3 |  3 |  0 |  0 |  0 |  0 |    1.000000000000 |    1.002052594504 ✔
 3 |  3 |  0 |  1 |  0 | -1 |    0.000000000000 |    0.000000000000 ✔
 3 |  3 |  0 |  1 |  0 |  0 |    0.000000000000 |    0.000000000000 ✔
 3 |  3 |  0 |  1 |  0 |  1 |    0.000000000000 |   -0.000000000000 ✔
 3 |  3 |  0 |  2 |  0 | -2 |    0.000000000000 |    0.000000000000 ✔
 3 |  3 |  0 |  2 |  0 | -1 |    0.000000000000 |   -0.000000000000 ✔
 3 |  3 |  0 |  2 |  0 |  0 |    0.000000000000 |    0.000000000000 ✔
 3 |  3 |  0 |  2 |  0 |  1 |    0.000000000000 |    0.000000000000 ✔
 3 |  3 |  0 |  2 |  0 |  2 |    0.000000000000 |    0.000000000000 ✔
 3 |  3 |  1 |  0 | -1 |  0 |    0.000000000000 |    0.000000000000 ✔
 3 |  3 |  1 |  0 |  0 |  0 |    0.000000000000 |    0.000000000000 ✔
 3 |  3 |  1 |  0 |  1 |  0 |    0.000000000000 |   -0.000000000000 ✔
 3 |  3 |  1 |  1 | -1 | -1 |    1.000000000000 |    1.001223346388 ✔
 3 |  3 |  1 |  1 | -1 |  0 |    0.000000000000 |   -0.000000000000 ✔
 3 |  3 |  1 |  1 | -1 |  1 |    0.000000000000 |   -0.000000000000 ✔
 3 |  3 |  1 |  1 |  0 | -1 |    0.000000000000 |   -0.000000000000 ✔
 3 |  3 |  1 |  1 |  0 |  0 |    1.000000000000 |    1.001223346388 ✔
 3 |  3 |  1 |  1 |  0 |  1 |    0.000000000000 |    0.000000000000 ✔
 3 |  3 |  1 |  1 |  1 | -1 |    0.000000000000 |   -0.000000000000 ✔
 3 |  3 |  1 |  1 |  1 |  0 |    0.000000000000 |    0.000000000000 ✔
 3 |  3 |  1 |  1 |  1 |  1 |    1.000000000000 |    1.001223346388 ✔
 3 |  3 |  1 |  2 | -1 | -2 |    0.000000000000 |   -0.000000000000 ✔
 3 |  3 |  1 |  2 | -1 | -1 |    0.000000000000 |    0.000000000000 ✔
 3 |  3 |  1 |  2 | -1 |  0 |    0.000000000000 |    0.000000000000 ✔
 3 |  3 |  1 |  2 | -1 |  1 |    0.000000000000 |    0.000000000000 ✔
 3 |  3 |  1 |  2 | -1 |  2 |    0.000000000000 |    0.000000000308 ✔
 3 |  3 |  1 |  2 |  0 | -2 |    0.000000000000 |   -0.000000000000 ✔
 3 |  3 |  1 |  2 |  0 | -1 |    0.000000000000 |    0.000000000000 ✔
 3 |  3 |  1 |  2 |  0 |  0 |    0.000000000000 |   -0.000000000000 ✔
 3 |  3 |  1 |  2 |  0 |  1 |    0.000000000000 |   -0.000000000000 ✔
 3 |  3 |  1 |  2 |  0 |  2 |    0.000000000000 |   -0.000000000000 ✔
 3 |  3 |  1 |  2 |  1 | -2 |    0.000000000000 |   -0.000000000308 ✔
 3 |  3 |  1 |  2 |  1 | -1 |    0.000000000000 |    0.000000000000 ✔
 3 |  3 |  1 |  2 |  1 |  0 |    0.000000000000 |    0.000000000000 ✔
 3 |  3 |  1 |  2 |  1 |  1 |    0.000000000000 |    0.000000000000 ✔
 3 |  3 |  1 |  2 |  1 |  2 |    0.000000000000 |    0.000000000000 ✔
 3 |  3 |  2 |  0 | -2 |  0 |    0.000000000000 |    0.000000000000 ✔
 3 |  3 |  2 |  0 | -1 |  0 |    0.000000000000 |   -0.000000000000 ✔
 3 |  3 |  2 |  0 |  0 |  0 |    0.000000000000 |   -0.000000000000 ✔
 3 |  3 |  2 |  0 |  1 |  0 |    0.000000000000 |    0.000000000000 ✔
 3 |  3 |  2 |  0 |  2 |  0 |    0.000000000000 |    0.000000000000 ✔
 3 |  3 |  2 |  1 | -2 | -1 |    0.000000000000 |   -0.000000000000 ✔
 3 |  3 |  2 |  1 | -2 |  0 |    0.000000000000 |   -0.000000000000 ✔
 3 |  3 |  2 |  1 | -2 |  1 |    0.000000000000 |   -0.000000000308 ✔
 3 |  3 |  2 |  1 | -1 | -1 |    0.000000000000 |    0.000000000000 ✔
 3 |  3 |  2 |  1 | -1 |  0 |    0.000000000000 |    0.000000000000 ✔
 3 |  3 |  2 |  1 | -1 |  1 |    0.000000000000 |    0.000000000000 ✔
 3 |  3 |  2 |  1 |  0 | -1 |    0.000000000000 |   -0.000000000000 ✔
 3 |  3 |  2 |  1 |  0 |  0 |    0.000000000000 |   -0.000000000000 ✔
 3 |  3 |  2 |  1 |  0 |  1 |    0.000000000000 |    0.000000000000 ✔
 3 |  3 |  2 |  1 |  1 | -1 |    0.000000000000 |    0.000000000000 ✔
 3 |  3 |  2 |  1 |  1 |  0 |    0.000000000000 |   -0.000000000000 ✔
 3 |  3 |  2 |  1 |  1 |  1 |    0.000000000000 |    0.000000000000 ✔
 3 |  3 |  2 |  1 |  2 | -1 |    0.000000000000 |    0.000000000308 ✔
 3 |  3 |  2 |  1 |  2 |  0 |    0.000000000000 |   -0.000000000000 ✔
 3 |  3 |  2 |  1 |  2 |  1 |    0.000000000000 |    0.000000000000 ✔
 3 |  3 |  2 |  2 | -2 | -2 |    1.000000000000 |    1.000300628566 ✔
 3 |  3 |  2 |  2 | -2 | -1 |    0.000000000000 |   -0.000000000000 ✔
 3 |  3 |  2 |  2 | -2 |  0 |    0.000000000000 |   -0.000000000000 ✔
 3 |  3 |  2 |  2 | -2 |  1 |    0.000000000000 |    0.000000000000 ✔
 3 |  3 |  2 |  2 | -2 |  2 |    0.000000000000 |    0.000000193779 ✔
 3 |  3 |  2 |  2 | -1 | -2 |    0.000000000000 |   -0.000000000000 ✔
 3 |  3 |  2 |  2 | -1 | -1 |    1.000000000000 |    1.000300628559 ✔
 3 |  3 |  2 |  2 | -1 |  0 |    0.000000000000 |   -0.000000000000 ✔
 3 |  3 |  2 |  2 | -1 |  1 |    0.000000000000 |   -0.000000000000 ✔
 3 |  3 |  2 |  2 | -1 |  2 |    0.000000000000 |    0.000000000000 ✔
 3 |  3 |  2 |  2 |  0 | -2 |    0.000000000000 |   -0.000000000000 ✔
 3 |  3 |  2 |  2 |  0 | -1 |    0.000000000000 |   -0.000000000000 ✔
 3 |  3 |  2 |  2 |  0 |  0 |    1.000000000000 |    1.000300628572 ✔
 3 |  3 |  2 |  2 |  0 |  1 |    0.000000000000 |    0.000000000000 ✔
 3 |  3 |  2 |  2 |  0 |  2 |    0.000000000000 |   -0.000000000000 ✔
 3 |  3 |  2 |  2 |  1 | -2 |    0.000000000000 |   -0.000000000000 ✔
 3 |  3 |  2 |  2 |  1 | -1 |    0.000000000000 |   -0.000000000000 ✔
 3 |  3 |  2 |  2 |  1 |  0 |    0.000000000000 |    0.000000000000 ✔
 3 |  3 |  2 |  2 |  1 |  1 |    1.000000000000 |    1.000300628559 ✔
 3 |  3 |  2 |  2 |  1 |  2 |    0.000000000000 |    0.000000000000 ✔
 3 |  3 |  2 |  2 |  2 | -2 |    0.000000000000 |    0.000000193779 ✔
 3 |  3 |  2 |  2 |  2 | -1 |    0.000000000000 |   -0.000000000000 ✔
 3 |  3 |  2 |  2 |  2 |  0 |    0.000000000000 |   -0.000000000000 ✔
 3 |  3 |  2 |  2 |  2 |  1 |    0.000000000000 |    0.000000000000 ✔
 3 |  3 |  2 |  2 |  2 |  2 |    1.000000000000 |    1.000300628566 ✔