Hydrogen Atom
The hydrogen atom is the simplest Coulomb 2-body system.
Definitions
This model is described with the time-independent Schrödinger equation
\[ \hat{H} \psi(\pmb{r}) = E \psi(\pmb{r}),\]
and the Hamiltonian
\[ \hat{H} = - \frac{\hbar^2}{2\mu} \nabla^2 + V(r),\]
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.
Parameters
Antique.HydrogenAtom — TypeHydrogenAtom(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).
Potential
Antique.V — MethodV(model::HydrogenAtom, r)
\[\begin{aligned} V(r) &= - \frac{Ze^2}{4\pi\varepsilon_0 r} &= - \frac{e^2}{4\pi\varepsilon_0 a_0} \frac{Z}{r/a_0} &= - \frac{Z}{r/a_0} E_\mathrm{h}, \end{aligned}\]
The domain is $0\leq r \lt \infty$.
Eigen Values
Antique.E — MethodE(model::HydrogenAtom; n=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 $E_\mathrm{h}$ 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.
Eigen Functions
Antique.ψ — Methodψ(model::HydrogenAtom, r, θ, φ; n=1, l=0, m=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::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 $0\leq r \lt \infty$.
Associated Laguerre Polynomials
Antique.L — MethodL(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:
\[\begin{aligned} L_0^0(x) &= 1, \\ L_1^0(x) &= 1 - x, \\ L_1^1(x) &= 1, \\ L_2^0(x) &= 1 - 2 x + 1/2 x^2, \\ L_2^1(x) &= 2 - x, \\ L_2^2(x) &= 1, \\ L_3^0(x) &= 1 - 3 x + 3/2 x^2 - 1/6 x^3, \\ L_3^1(x) &= 3 - 3 x + 1/2 x^2, \\ L_3^2(x) &= 3 - x, \\ L_3^3(x) &= 1, \\ L_4^0(x) &= 1 - 4 x + 3 x^2 - 2/3 x^3 + 5/12 x^4, \\ L_4^1(x) &= 4 - 6 x + 2 x^2 - 1/6 x^3, \\ L_4^2(x) &= 6 - 4 x + 1/2 x^2, \\ L_4^3(x) &= 4 - x, \\ L_4^4(x) &= 1, \\ \vdots \end{aligned}\]
Spherical Harmonics
Antique.Y — MethodY(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
\[i^{|m|+m} \sqrt{\frac{(l-|m|)!}{(l+|m|)!}} P_l^{|m|} = (-1)^{\frac{|m|+m}{2}} \sqrt{\frac{(l-|m|)!}{(l+|m|)!}} P_l^{|m|} = (-1)^m \sqrt{\frac{(l-m)!}{(l+m)!}} P_l^{m}.\]
Associated Legendre Polynomials
Antique.P — MethodP(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:
\[\begin{aligned} P_{0}^{0}(x) &= 1, \\ P_{1}^{0}(x) &= x, \\ P_{1}^{1}(x) &= \left(+1\right)\sqrt{1-x^2}, \\ P_{2}^{0}(x) &= -1/2 + 3/2 x^{2}, \\ P_{2}^{1}(x) &= \left(-3 x\right)\sqrt{1-x^2}, \\ P_{2}^{2}(x) &= 3 - 6 x, \\ P_{3}^{0}(x) &= -3/2 x + 5/2 x^{3}, \\ P_{3}^{1}(x) &= \left(3/2 - 15/2 x^{2}\right)\sqrt{1-x^2}, \\ P_{3}^{2}(x) &= 15 x - 30 x^{2}, \\ P_{3}^{3}(x) &= \left(15 - 30 x\right)\sqrt{1-x^2}, \\ P_{4}^{0}(x) &= 3/8 - 15/4 x^{2} + 35/8 x^{4}, \\ P_{4}^{1}(x) &= \left(- 15/2 x + 35/2 x^{3}\right)\sqrt{1-x^2}, \\ P_{4}^{2}(x) &= -15/2 + 15 x + 105/2 x^{2} - 105 x^{3}, \\ P_{4}^{3}(x) &= \left(105 x - 210 x^{2}\right)\sqrt{1-x^2}, \\ P_{4}^{4}(x) &= 105 - 420 x + 420 x^{2}, \\ & \vdots \end{aligned}\]
References
- cpprefjp, legendre, assoc_legendre, laguerre, assoc_laguerre
- The Digital Library of Mathematical Functions (DLMF), 18.3 Table1, 18.5 Table1, 18.5.16, 18.3 Table1, 18.5 Table1, 18.5.17, 18.3 Table1, 18.5 Table1, 18.5.12
- L. D. Landau, E. M. Lifshitz, Quantum Mechanics (Pergamon Press, 1965), p.598 (c.1), p.598 (c.4), p.603 (d.13), p.603 (d.13)
- 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,
Usage & Examples
Install Antique.jl for the first use and run using Antique before each use. The energy E(), wavefunction ψ(), potential V() and some other functions are suppoted. 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.Z1julia> H.Eₕ1.0julia> H.mₑ1.0julia> H.a₀1.0julia> H.ℏ1.0
Eigen values:
julia> E(H, n=1)-0.5julia> 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 nmHyperfine 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 cmPotential 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 * 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
Wave functions (electron density in $n=5,l=2,m=1$):
using Antique
H = HydrogenAtom(Z=1, Eₕ=1.0, a₀=1.0, mₑ=1.0, ℏ=1.0)
loop(x) = x<-1 ? loop(x+2) : (1<x ? loop(x-2) : x)
myacos(x) = acos(loop(x))
r(x,y,z) = sqrt(x^2+y^2+z^2)
θ(x,y,z) = x^2+y^2<1e-9 ? 0 : myacos(z/r(x,y,z))
φ(x,y,z) = y^2<1e-9 ? 0 : sign(y)*myacos(x/sqrt(x^2+y^2))
P(x,y,z) = abs(ψ(H,r(x,y,z),θ(x,y,z),φ(x,y,z),n=5,l=2,m=1))^2
using CairoMakie
f = Figure(size=(500,500), backgroundcolor=:transparent)
a = Axis(f[1,1], aspect=1)
hidespines!(a)
hidedecorations!(a)
heatmap!(a, -40:0.1:40, -40:0.1:40, (y,z) -> P(0,y,z), colorrange=(0.0,0.00001))
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:
- 高柳和夫『朝倉物理学大系 11 原子分子物理学』(2000, 朝倉書店) pp.11-22
- Quantum Mechanics for Engineers by Leon van Dommelen
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:
- 高柳和夫『朝倉物理学大系 11 原子分子物理学』(2000, 朝倉書店) pp.11-22
- Quantum Mechanics for Engineers by Leon van Dommelen
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 $\langle H \rangle = -\langle T \rangle$.
\[\frac{1}{2} \int \psi_n^\ast(x) V(x) \psi_n(x) \mathrm{d}x = E_n\]
n | analytical | numerical
-- | ----------------- | -----------------
1 | -1.000000000000 | -1.000000000000 ✔
2 | -0.250000000000 | -0.250000000000 ✔
3 | -0.111111111111 | -0.111111111111 ✔
4 | -0.062500000000 | -0.062500000000 ✔
5 | -0.040000000000 | -0.040000000000 ✔
6 | -0.027777777778 | -0.027777777778 ✔
7 | -0.020408163265 | -0.020408163265 ✔
8 | -0.015625000000 | -0.015625000000 ✔
9 | -0.012345679012 | -0.012345679012 ✔
10 | -0.010000000000 | -0.010000000000 ✔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 ✔