Coulomb 2-Body System
This is the model of two particles interacting through Coulomb forces such as positronium, muonium, hydrogen atoms, deuterium atoms, etc.
Definitions
Antique.CoulombTwoBody — TypeModel
This model is described with the time-independent Schrödinger equation
\[ \hat{H} \psi(\pmb{r}) = E \psi(\pmb{r}),\]
and the Hamiltonian
\[ \hat{H} = - \frac{\hbar^2}{2\mu} \nabla^2 + \frac{z_1 z_2}{r/a_0} E_\mathrm{h},\]
where $\mu=\left(\frac{1}{m_1}+\frac{1}{m_2}\right)^{-1}$ is the reduced mass of particle 1 and particle 2. 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:
CTB = CoulombTwoBody(z₁=-1, z₂=1, m₁=1.0, m₂=1.0, mₑ=1.0, a₀=1.0, Eₕ=1.0, ℏ=1.0)$z₁$ is the charge number of particle 1, $z₂$ is the charge number of particle 2, $m₁$ is the mass of particle 1, $m₂$ is the mass of particle 2, $m_\mathrm{e}$ is the electron mass (use the same unit as $m₁$ and $m₂$. For example of hydrogen atom, use $m_\mathrm{e}=9.1093837139\times10^{-31}\mathrm{kg}$, $m_1=9.1093837139\times10^{-31}\mathrm{kg}$ and $m_2=1.67262192595\times10^{-27}\mathrm{kg}$ in the IS unit system, use $~m_\mathrm{e}=1.0~m_\mathrm{e}$, $m_1=1.0~m_\mathrm{e}$ and $m_2=1836.152673426~m_\mathrm{e}$ in the atomic unit.), $a_0$ is the Bohr radius, $E_\mathrm{h}$ is the Hartree energy and $\hbar$ is the reduced Planck constant (Dirac's constant).
References
- 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 I. THE HYDROGEN ATOM
- D. J. Griffiths, D. F. Schroeter, Introduction to Quantum Mechanics Third Edition (Cambridge University Press, 2018) p.143 4.2 THE HYDROGEN ATOM, p.200 Problem 5.1, p.200 Problem 5.2
- W. Greiner, Quantum Mechanics: An Introduction Forth Edition (Springer, 2001) p.217 The Hydrogen Atom, p.236 9.5 Spectrum of a Diatomic Molecule
Potential
Antique.V — MethodV(model::CoulombTwoBody, r)
\[\begin{aligned} V(r) &= - \frac{z_1 z_2 e^2}{4\pi\varepsilon_0 r} &= - \frac{e^2}{4\pi\varepsilon_0 a_0} \frac{z_1 z_2}{r/a_0} &= - \frac{z_1 z_2}{r/a_0} E_\mathrm{h}, \end{aligned}\]
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. The domain is $0\leq r \lt \infty$.
Eigenvalues
Antique.E — MethodE(model::CoulombTwoBody; n::Int=1)
\[E_n = -\frac{(z_1 z_2)^2}{2n^2} \frac{\mu}{m_\mathrm{e}} E_\mathrm{h},\]
where $\mu=\left(\frac{1}{m_1}+\frac{1}{m_2}\right)^{-1}$ is the reduced mass of particle 1 and particle 2, $E_\mathrm{h} = \frac{\hbar^2}{m_\mathrm{e}{a_0}^2} = \frac{e^2}{4\pi\varepsilon_0a_0} = \frac{m_\mathrm{e}e^4}{\left(4\pi\varepsilon_0\right)^2\hbar^2}$ 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.
Eigenfunctions
Antique.ψ — Methodψ(model::CoulombTwoBody, 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 $0\leq r \lt \infty, 0\leq \theta \lt \pi, 0\leq \varphi \lt 2\pi$.
Radial Functions
Antique.R — MethodR(model::CoulombTwoBody, r; n=1, l=0)
\[R_{nl}(r) = -\sqrt{\frac{(n-l-1)!}{2n(n+l)!} \left(\frac{2Z}{n a_\mu}\right)^3} \left(\frac{2Zr}{n a_\mu}\right)^l \exp \left(-\frac{Zr}{n a_\mu}\right) L_{n+l}^{2l+1} \left(\frac{2Zr}{n a_\mu}\right),\]
where $\frac{1}{\mu} = \frac{1}{m_1}+\frac{1}{m_2}$, $a_\mu = a_0 \frac{m_\mathrm{e}}{\mu}$, $Z = - z_1 z_2$, 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::CoulombTwoBody, 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::CoulombTwoBody, 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::CoulombTwoBody, x; n=0, m=0)
Rodrigues' formula & closed-form:
\[\begin{aligned} P_n^m(x) &= \left( 1-x^2 \right)^{m/2} \frac{\mathrm{d}^m}{\mathrm{d}x^m} P_n(x) \\ &= \left( 1-x^2 \right)^{m/2} \frac{\mathrm{d}^m}{\mathrm{d}x^m} \frac{1}{2^n n!} \frac{\mathrm{d}^n}{\mathrm{d}x ^n} \left[ \left( x^2-1 \right)^n \right] \\ &= \frac{1}{2^n} (1-x^2)^{m/2} \sum_{j=0}^{\left\lfloor\frac{n-m}{2}\right\rfloor} (-1)^j \frac{(2n-2j)!}{j! (n-j)! (n-2j-m)!} x^{(n-2j-m)}. \end{aligned},\]
where Legendre polynomials are defined as $P_n(x) = \frac{1}{2^n n!} \frac{\mathrm{d}^n}{\mathrm{d}x ^n} \left[ \left( x^2-1 \right)^n \right]$. Note that $P_l^{-m} = (-1)^m \frac{(l-m)!}{(l+m)!} P_l^m$ for $m<0$. (It is not compatible with $P_k^m(t) = (-1)^m\left( 1-t^2 \right)^{m/2} \frac{\mathrm{d}^m P_k(t)}{\mathrm{d}t^m}$ caused by $(-1)^m$.) The specific formulae are given below.
Examples:
\[\begin{aligned} P_{0}^{0}(x) &= 1, \\ P_{1}^{0}(x) &= x, \\ P_{1}^{1}(x) &= \left(+1\right)\sqrt{1-x^2}, \\ P_{2}^{0}(x) &= -1/2 + 3/2 x^{2}, \\ P_{2}^{1}(x) &= \left(-3 x\right)\sqrt{1-x^2}, \\ P_{2}^{2}(x) &= 3 - 6 x, \\ P_{3}^{0}(x) &= -3/2 x + 5/2 x^{3}, \\ P_{3}^{1}(x) &= \left(3/2 - 15/2 x^{2}\right)\sqrt{1-x^2}, \\ P_{3}^{2}(x) &= 15 x - 30 x^{2}, \\ P_{3}^{3}(x) &= \left(15 - 30 x\right)\sqrt{1-x^2}, \\ P_{4}^{0}(x) &= 3/8 - 15/4 x^{2} + 35/8 x^{4}, \\ P_{4}^{1}(x) &= \left(- 15/2 x + 35/2 x^{3}\right)\sqrt{1-x^2}, \\ P_{4}^{2}(x) &= -15/2 + 15 x + 105/2 x^{2} - 105 x^{3}, \\ P_{4}^{3}(x) &= \left(105 x - 210 x^{2}\right)\sqrt{1-x^2}, \\ P_{4}^{4}(x) &= 105 - 420 x + 420 x^{2}, \\ & \vdots \end{aligned}\]
Usage & Examples
Install Antique.jl for the first use and run using Antique before each use. The energy E(), wavefunction ψ() and potential V() will be exported. In this system, the model is generated by CoulombTwoBody and several parameters z₁, z₂, m₁, m₂, mₑ, a₀, Eₕ and ℏ are set as optional arguments.
using Antique
Ps = CoulombTwoBody(z₁=-1, z₂=1, m₁=1.0, m₂=1.0, mₑ=1.0, a₀=1.0, Eₕ=1.0, ℏ=1.0)Parameters
julia> Ps.z₁-1julia> Ps.z₂1julia> Ps.m₁1.0julia> Ps.m₂1.0julia> Ps.mₑ1.0julia> Ps.a₀1.0julia> Ps.Eₕ1.0julia> Ps.ℏ1.0
Eigenvalues
Examples of positronium:
julia> E(Ps, n=1)-0.25julia> E(Ps, n=2)-0.0625
Mass and Charge Dependence
The values of masses are cited from the 2018 CODATA recommended values, E. Tiesinga, et al., Rev. Mod. Phys. 93, 025010 (2021).
me = 1.0 # me #
mµ = 206.7682830 # me # https://physics.nist.gov/cgi%2Dbin/cuu/Value?mmusme
mp = 1836.15267343 # me # https://physics.nist.gov/cgi%2Dbin/cuu/Value?mpsme
md = 3670.48296788 # me # https://physics.nist.gov/cgi%2Dbin/cuu/Value?mdsme
mt = 5496.92153573 # me # https://physics.nist.gov/cgi%2Dbin/cuu/Value?mtsme
mh = 5495.88528007 # me # https://physics.nist.gov/cgi-bin/cuu/Value?mhsme
ma = 7294.29954142 # me # https://physics.nist.gov/cgi-bin/cuu/Value?malsme
Ps = CoulombTwoBody(m₁=me, m₂=me)
Mu = CoulombTwoBody(m₁=me, m₂=mµ)
H = CoulombTwoBody(m₁=me, m₂=mp)
D = CoulombTwoBody(m₁=me, m₂=md)
T = CoulombTwoBody(m₁=me, m₂=mt)
BO = CoulombTwoBody(m₁=me, m₂=Inf)
He3⁺ = CoulombTwoBody(z₁=-1, z₂=+2, m₁=me, m₂=mh)
He4⁺ = CoulombTwoBody(z₁=-1, z₂=+2, m₁=me, m₂=ma)
He∞⁺ = CoulombTwoBody(z₁=-1, z₂=+2, m₁=me, m₂=Inf)
pμ = CoulombTwoBody(z₁=-1, z₂=+1, m₁=mµ, m₂=mp)
dμ = CoulombTwoBody(z₁=-1, z₂=+1, m₁=mµ, m₂=md)
tμ = CoulombTwoBody(z₁=-1, z₂=+1, m₁=mµ, m₂=mt)
bμ = CoulombTwoBody(z₁=-1, z₂=+1, m₁=mµ, m₂=Inf)
hμ = CoulombTwoBody(z₁=-1, z₂=+2, m₁=mµ, m₂=mh)
αμ = CoulombTwoBody(z₁=-1, z₂=+2, m₁=mµ, m₂=ma)
println(" \tE / Eₕ")
println("Ps \t", E(Ps))
println("Mu \t", E(Mu))
println("H \t", E(H))
println("D \t", E(D))
println("T \t", E(T))
println("∞H \t", E(BO))
println("³He⁺\t", E(He3⁺))
println("⁴He⁺\t", E(He4⁺))
println("∞He⁺\t", E(He∞⁺))
println("pμ \t", E(pμ))
println("dμ \t", E(dμ))
println("tμ \t", E(tμ))
println("bμ \t", E(bμ))
println("hμ \t", E(hμ))
println("αμ \t", E(αμ)) E / Eₕ
Ps -0.25
Mu -0.49759347291713435
H -0.49972783971238144
D -0.4998638152473063
T -0.49990905654132184
∞H -0.5
³He⁺ -1.9996361575877797
⁴He⁺ -1.9997258508730662
∞He⁺ -2.0
pμ -92.92041731130719
dμ -97.87081258624124
tμ -99.63629368450574
bμ -103.38414150000001
hμ -398.5424505827022
αμ -402.13735621933824println(" \t<δ³(r)> / a₀⁻³")
println("1/8π =\t", 1/8/π)
println("Ps \t", abs(ψ(Ps,0,0,0))^2)
println("Mu \t", abs(ψ(Mu,0,0,0))^2)
println("H \t", abs(ψ(H ,0,0,0))^2)
println("D \t", abs(ψ(D ,0,0,0))^2)
println("T \t", abs(ψ(T ,0,0,0))^2)
println("∞H \t", abs(ψ(BO,0,0,0))^2)
println("1/π = \t", 1/π) <δ³(r)> / a₀⁻³
1/8π = 0.039788735772973836
Ps 0.03978873577297385
Mu 0.3137358439360387
H 0.3177903812026296
D 0.3180498633001772
T 0.31813622856178475
∞H 0.3183098861837908
1/π = 0.3183098861837907Lifetime of Positronium
The lifetime $\tau$ of positronium (Ps, $\mathrm{e}^+\mathrm{e}^-$) is written as
\[\tau = \frac{1}{\Gamma},\]
\[\Gamma = 4 \pi \alpha^4 c {a_0}^2 \langle\delta^3(\pmb{r})\rangle.\]
where $\langle\delta^3(\pmb{r})\rangle = \langle\psi|\delta^3(\pmb{r})|\psi\rangle = |\psi(\pmb{0})|^2 = \frac{1}{8\pi} a_0^{-3} \simeq 2.685\times10^{29}~\mathrm{m}^{-3}$ is the value of probability density at the origin ($r=0$).
# (7.169) in D. J. Griffiths, Introduction to Elementary Particles (John Wiley & Sons, Inc. 1987) ISBN 0-471-60386-4
# S. Berko, H. N. Pendleton, Annual Review of Nuclear and Particle Science, 30, 543 (1980) https://doi.org/10.1146/annurev.ns.30.120180.002551)
# A. M. Frolov, S. I. Kryuchkov, and V. H. Smith, Jr., Phys. Rev. A, 51, 4514 (1995) https://doi.org/10.1103/PhysRevA.51.4514
α = 7.2973525693e-3 # # https://physics.nist.gov/cgi-bin/cuu/Value?alph
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
Ps = CoulombTwoBody(z₁=1, z₂=-1, m₁=1.0, m₂=1.0, mₑ=1.0, a₀=1.0, Eₕ=1.0, ℏ=1.0)
δ = abs(ψ(Ps,0,0,0))^2 * a₀^(-3)
Γ = 4 * π * α^4 * c * a₀^2 * δ
τ = 1/Γ
println("<δ> = ", abs(ψ(Ps,0,0,0))^2, " a₀⁻³")
println(" = ", δ, " m⁻³")
println("Γ = ", Γ / 1e9, " GHz")
println("τ = ", τ / 1e-12, " ps")<δ> = 0.03978873577297385 a₀⁻³
= 2.685076981132993e29 m⁻³
Γ = 8.0325029283017 GHz
τ = 124.49419675610734 psHyperfine Splitting
The hyperfine splitting of hydrogen atoms is given as
\[\Delta E (\mathrm{H}) = -\frac{2}{3} \mu_0 \gamma_\mathrm{p} \gamma_\mathrm{e} \hbar^2 \langle\delta^3(\pmb{r})\rangle\]
in Griffiths(1982). Note that this fomula is not available for deuterium (D) and positronium (Ps) because of the different spin between the proton and the deuteron for D, and the contribution of positron-electron pair annihilation for Ps. And also note that the mass of proton is used for the definitions of gyromagnetic ratio in all nucleons and nuclei:
\[\begin{aligned} &\gamma_\mathrm{e} = \frac{-e}{2 m_\mathrm{e}} g_\mathrm{e}, & &\gamma_\mathrm{e^+} = \frac{+e}{2 m_\mathrm{e}} g_\mathrm{e}, & &\gamma_\mathrm{\mu} = \frac{-e}{2 m_\mathrm{\mu}} g_\mathrm{\mu}, \\ &\gamma_\mathrm{p} = \frac{+e}{2 m_\mathrm{p}} g_\mathrm{p}, & &\gamma_\mathrm{d} = \frac{+e}{2 m_\mathrm{p}} g_\mathrm{d}, & &\gamma_\mathrm{t} = \frac{+e}{2 m_\mathrm{p}} g_\mathrm{t}, & &\gamma_\mathrm{h} = \frac{+2e}{2 m_\mathrm{p}} g_\mathrm{h}. & \end{aligned}\]
The value of probability density at the origin is $\langle\delta^3(\pmb{r})\rangle = \langle\psi|\delta^3(\pmb{r})|\psi\rangle = |\psi(\pmb{0})|^2 \simeq \frac{1}{\pi} a_0^{-3} \simeq 2.148\times10^{30}~\mathrm{m}^{-3}$ in Mu, H, D and T. This values are very different in Ps, $^3\mathrm{He}^+$ and muonic hydrogen ($\mathrm{p\mu}$) due to the difference of reduced masses and charges. The energy can be converted to frequency (Hz) by $v = \Delta E / h$.
a₀ = 5.29177210903e-11 # m # https://physics.nist.gov/cgi-bin/cuu/Value?bohrrada0
Eₕ = 4.3597447222071e-18 # J # https://physics.nist.gov/cgi-bin/cuu/Value?hr
ℏ = 1.054571817e-34 # J s # https://physics.nist.gov/cgi-bin/cuu/Value?hbar
me = 9.1093837015e-31 # kg # https://physics.nist.gov/cgi-bin/cuu/Value?me
mµ = 1.883531627e-28 # kg # https://physics.nist.gov/cgi-bin/cuu/Value?mmu
mp = 1.67262192369e-27 # kg # https://physics.nist.gov/cgi-bin/cuu/Value?mp
md = 3.3435837724e-27 # kg # https://physics.nist.gov/cgi-bin/cuu/Value?md
mt = 5.0073567446e-27 # kg # https://physics.nist.gov/cgi-bin/cuu/Value?mt
mh = 5.0064127796e-27 # kg # https://physics.nist.gov/cgi-bin/cuu/Value?mh
e = 1.602176634e-19 # C # https://physics.nist.gov/cgi-bin/cuu/Value?e
µ₀ = 1.25663706212e-6 # N A-2 # https://physics.nist.gov/cgi-bin/cuu/Value?mu0
h = 6.62607015e-34 # J Hz-1 # https://physics.nist.gov/cgi-bin/cuu/Value?h
eV = 1.602176634e-19 # J # https://physics.nist.gov/cgi-bin/cuu/Value?evj
ge = 2.00231930436256 # https://physics.nist.gov/cgi-bin/cuu/Value?gem
gµ = 2.0023318418 # https://physics.nist.gov/cgi-bin/cuu/Value?gmum
gp = 5.5856946893 # https://physics.nist.gov/cgi-bin/cuu/Value?gp
gd = 0.8574382338 # https://physics.nist.gov/cgi-bin/cuu/Value?gdn
gt = 5.957924931 # https://physics.nist.gov/cgi-bin/cuu/Value?gtn
gh = -4.255250615 # https://physics.nist.gov/cgi-bin/cuu/Value?ghn
Ps = CoulombTwoBody(z₁=-1, z₂=+1, m₁=me, m₂=me, mₑ=me, a₀=a₀, Eₕ=Eₕ, ℏ=ℏ)
Mu = CoulombTwoBody(z₁=-1, z₂=+1, m₁=me, m₂=mµ, mₑ=me, a₀=a₀, Eₕ=Eₕ, ℏ=ℏ)
H = CoulombTwoBody(z₁=-1, z₂=+1, m₁=me, m₂=mp, mₑ=me, a₀=a₀, Eₕ=Eₕ, ℏ=ℏ)
D = CoulombTwoBody(z₁=-1, z₂=+1, m₁=me, m₂=md, mₑ=me, a₀=a₀, Eₕ=Eₕ, ℏ=ℏ)
T = CoulombTwoBody(z₁=-1, z₂=+1, m₁=me, m₂=mt, mₑ=me, a₀=a₀, Eₕ=Eₕ, ℏ=ℏ)
he = CoulombTwoBody(z₁=-1, z₂=+2, m₁=me, m₂=mh, mₑ=me, a₀=a₀, Eₕ=Eₕ, ℏ=ℏ)
pμ = CoulombTwoBody(z₁=-1, z₂=+1, m₁=mµ, m₂=mp, mₑ=me, a₀=a₀, Eₕ=Eₕ, ℏ=ℏ)
ΔE_H = 2/3 / 4 * µ₀ * ℏ^2 * e^2 * ge / me * gp / mp * abs(ψ(H,0,0,0))^2
ΔE_D = 1 / 4 * µ₀ * ℏ^2 * e^2 * ge / me * gd / mp * abs(ψ(D,0,0,0))^2
ΔE_T = 2/3 / 4 * µ₀ * ℏ^2 * e^2 * ge / me * gt / mp * abs(ψ(H,0,0,0))^2
ΔE_Ps = 7/6 / 4 * µ₀ * ℏ^2 * e^2 * ge / me * ge / me * abs(ψ(Ps,0,0,0))^2
ΔE_Mu = 2/3 / 4 * µ₀ * ℏ^2 * e^2 * ge / me * gµ / mµ * abs(ψ(Mu,0,0,0))^2
ΔE_he = 2/3 / 4 * µ₀ * ℏ^2 * e^2 * ge / me * gh / mp * abs(ψ(he,0,0,0))^2
ΔE_pµ = 2/3 / 4 * µ₀ * ℏ^2 * e^2 * gµ / mµ * gp / mp * abs(ψ(pµ,0,0,0))^2
# Karshenboim(2001) https://doi.org/10.48550/arXiv.hep-ph/0109128
# Karshenboim(2003) https://doi.org/10.48550/arXiv.hep-ph/0310099
# Griffiths(1982) https://doi.org/10.1119/1.12733
# Adamczak(2012) https://doi.org/10.1016/j.nimb.2012.04.001
println("H \t", ΔE_H / h / 1e6, " MHz\t Antique.jl + CODATA2018")
println(" \t", "1420.405751768(1) MHz\t Karshenboim(2001)")
println("D \t", ΔE_D / h / 1e6, " MHz\t Antique.jl + CODATA2018")
println(" \t", "327.384352522(2) MHz\t Karshenboim(2001)")
println("T \t", ΔE_T / h / 1e6, " MHz\t Antique.jl + CODATA2018")
println(" \t", "1516.701470773(8) MHz\t Karshenboim(2001)")
println("Ps\t", ΔE_Ps / h / 1e6, " MHz\t Antique.jl + CODATA2018")
println(" \t", "203391.7(6) MHz\t Karshenboim(2003)")
println("Mu\t", ΔE_Mu / h / 1e6, " MHz\t Antique.jl + CODATA2018")
println(" \t", "4463.30278(5) MHz\t Karshenboim(2001)")
println("³He⁺\t", ΔE_he / h / 1e6, " MHz\t Antique.jl + CODATA2018")
println(" \t", "-8665.649867(10) MHz\t Karshenboim(2001)")
println("µp\t", ΔE_pµ / h / 1e12, " THz\t Antique.jl + CODATA2018")
println(" \t", 0.182725*eV / h / 1e12 , " THz\t Griffiths(1982), Adamczak(2012)")H 1420.4854518754269 MHz Antique.jl + CODATA2018
1420.405751768(1) MHz Karshenboim(2001)
D 327.34684982805805 MHz Antique.jl + CODATA2018
327.384352522(2) MHz Karshenboim(2001)
T 1515.1464873408624 MHz Antique.jl + CODATA2018
1516.701470773(8) MHz Karshenboim(2001)
Ps 204860.9400441627 MHz Antique.jl + CODATA2018
203391.7(6) MHz Karshenboim(2003)
Mu 4464.202736244739 MHz Antique.jl + CODATA2018
4463.30278(5) MHz Karshenboim(2001)
³He⁺ -8666.566269930268 MHz Antique.jl + CODATA2018
-8665.649867(10) MHz Karshenboim(2001)
µp 44.16603467817586 THz Antique.jl + CODATA2018
44.18270842599667 THz Griffiths(1982), Adamczak(2012)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 $r$
\[\langle r \rangle = \int r |R_{nl}(r)|^2 r^2 \mathrm{d}r = \frac{a_\mu}{2Z} \left[ 3n^2 - l(l+1) \right] \\ a_\mu = a_0 \frac{m_\mathrm{e}}{\mu} \\ \frac{1}{\mu} = \frac{1}{m_\mathrm{e}} + \frac{1}{m_\mathrm{p}}\]
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
CoulombTwoBody(-1, 1, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0)
n | l | analytical | numerical
-- | -- | ----------------- | -----------------
1 | 0 | 3.000000000000 | 3.000000000000 ✔
2 | 0 | 12.000000000000 | 12.000000000000 ✔
2 | 1 | 10.000000000000 | 10.000000000000 ✔
3 | 0 | 27.000000000000 | 27.000000000000 ✔
3 | 1 | 25.000000000000 | 25.000000000000 ✔
3 | 2 | 21.000000000000 | 21.000000000000 ✔
4 | 0 | 48.000000000000 | 47.999999999998 ✔
4 | 1 | 46.000000000000 | 45.999999999999 ✔
4 | 2 | 42.000000000000 | 42.000000000000 ✔
4 | 3 | 36.000000000000 | 36.000000000000 ✔
5 | 0 | 75.000000000000 | 75.000000000000 ✔
5 | 1 | 73.000000000000 | 73.000000000000 ✔
5 | 2 | 69.000000000000 | 69.000000000000 ✔
5 | 3 | 63.000000000000 | 63.000000000000 ✔
5 | 4 | 55.000000000000 | 55.000000000000 ✔
6 | 0 | 108.000000000000 | 108.000000000002 ✔
6 | 1 | 106.000000000000 | 106.000000000001 ✔
6 | 2 | 102.000000000000 | 102.000000000001 ✔
6 | 3 | 96.000000000000 | 96.000000000000 ✔
6 | 4 | 88.000000000000 | 88.000000000000 ✔
6 | 5 | 78.000000000000 | 78.000000000000 ✔
7 | 0 | 147.000000000000 | 147.000000000000 ✔
7 | 1 | 145.000000000000 | 145.000000000001 ✔
7 | 2 | 141.000000000000 | 141.000000000000 ✔
7 | 3 | 135.000000000000 | 135.000000000000 ✔
7 | 4 | 127.000000000000 | 127.000000000000 ✔
7 | 5 | 117.000000000000 | 117.000000000000 ✔
7 | 6 | 105.000000000000 | 104.999999999986 ✔
8 | 0 | 192.000000000000 | 191.999999999997 ✔
8 | 1 | 190.000000000000 | 189.999999999999 ✔
8 | 2 | 186.000000000000 | 185.999999999997 ✔
8 | 3 | 180.000000000000 | 179.999999999997 ✔
8 | 4 | 172.000000000000 | 171.999999999998 ✔
8 | 5 | 162.000000000000 | 162.000000000000 ✔
8 | 6 | 150.000000000000 | 150.000000000000 ✔
8 | 7 | 136.000000000000 | 136.000000000000 ✔
9 | 0 | 243.000000000000 | 242.999999999998 ✔
9 | 1 | 241.000000000000 | 241.000000000001 ✔
9 | 2 | 237.000000000000 | 237.000000000001 ✔
9 | 3 | 231.000000000000 | 231.000000000001 ✔
9 | 4 | 223.000000000000 | 222.999999999998 ✔
9 | 5 | 213.000000000000 | 213.000000000001 ✔
9 | 6 | 201.000000000000 | 200.999999999999 ✔
9 | 7 | 187.000000000000 | 187.000000000000 ✔
9 | 8 | 171.000000000000 | 171.000000000000 ✔
CoulombTwoBody(-1, 1, 1.0, 206.768283, 1.0, 1.0, 1.0, 1.0)
n | l | analytical | numerical
-- | -- | ----------------- | -----------------
1 | 0 | 1.507254497538 | 1.507254497538 ✔
2 | 0 | 6.029017990153 | 6.029017990153 ✔
2 | 1 | 5.024181658461 | 5.024181658461 ✔
3 | 0 | 13.565290477844 | 13.565290477844 ✔
3 | 1 | 12.560454146152 | 12.560454146152 ✔
3 | 2 | 10.550781482767 | 10.550781482767 ✔
4 | 0 | 24.116071960611 | 24.116071960610 ✔
4 | 1 | 23.111235628919 | 23.111235628918 ✔
4 | 2 | 21.101562965535 | 21.101562965534 ✔
4 | 3 | 18.087053970458 | 18.087053970458 ✔
5 | 0 | 37.681362438455 | 37.681362438455 ✔
5 | 1 | 36.676526106763 | 36.676526106763 ✔
5 | 2 | 34.666853443378 | 34.666853443378 ✔
5 | 3 | 31.652344448302 | 31.652344448302 ✔
5 | 4 | 27.632999121534 | 27.632999121534 ✔
6 | 0 | 54.261161911375 | 54.261161911375 ✔
6 | 1 | 53.256325579683 | 53.256325579684 ✔
6 | 2 | 51.246652916299 | 51.246652916299 ✔
6 | 3 | 48.232143921222 | 48.232143921222 ✔
6 | 4 | 44.212798594454 | 44.212798594454 ✔
6 | 5 | 39.188616935993 | 39.188616935993 ✔
7 | 0 | 73.855470379371 | 73.855470379372 ✔
7 | 1 | 72.850634047679 | 72.850634047679 ✔
7 | 2 | 70.840961384295 | 70.840961384295 ✔
7 | 3 | 67.826452389219 | 67.826452389219 ✔
7 | 4 | 63.807107062450 | 63.807107062450 ✔
7 | 5 | 58.782925403990 | 58.782925403990 ✔
7 | 6 | 52.753907413837 | 52.753907413827 ✔
8 | 0 | 96.464287842444 | 96.464287842444 ✔
8 | 1 | 95.459451510752 | 95.459451510752 ✔
8 | 2 | 93.449778847368 | 93.449778847368 ✔
8 | 3 | 90.435269852292 | 90.435269852291 ✔
8 | 4 | 86.415924525523 | 86.415924525523 ✔
8 | 5 | 81.391742867062 | 81.391742867062 ✔
8 | 6 | 75.362724876910 | 75.362724876910 ✔
8 | 7 | 68.328870555065 | 68.328870555065 ✔
9 | 0 | 122.087614300594 | 122.087614300595 ✔
9 | 1 | 121.082777968902 | 121.082777968901 ✔
9 | 2 | 119.073105305517 | 119.073105305518 ✔
9 | 3 | 116.058596310441 | 116.058596310441 ✔
9 | 4 | 112.039250983672 | 112.039250983670 ✔
9 | 5 | 107.015069325212 | 107.015069325211 ✔
9 | 6 | 100.986051335059 | 100.986051335059 ✔
9 | 7 | 93.952197013214 | 93.952197013214 ✔
9 | 8 | 85.913506359677 | 85.913506359677 ✔
CoulombTwoBody(-1, 1, 1.0, 1836.15267343, 1.0, 1.0, 1.0, 1.0)
n | l | analytical | numerical
-- | -- | ----------------- | -----------------
1 | 0 | 1.500816925532 | 1.500816925532 ✔
2 | 0 | 6.003267702129 | 6.003267702129 ✔
2 | 1 | 5.002723085107 | 5.002723085107 ✔
3 | 0 | 13.507352329790 | 13.507352329790 ✔
3 | 1 | 12.506807712769 | 12.506807712769 ✔
3 | 2 | 10.505718478726 | 10.505718478726 ✔
4 | 0 | 24.013070808516 | 24.013070808515 ✔
4 | 1 | 23.012526191494 | 23.012526191494 ✔
4 | 2 | 21.011436957451 | 21.011436957451 ✔
4 | 3 | 18.009803106387 | 18.009803106387 ✔
5 | 0 | 37.520423138306 | 37.520423138306 ✔
5 | 1 | 36.519878521284 | 36.519878521284 ✔
5 | 2 | 34.518789287241 | 34.518789287241 ✔
5 | 3 | 31.517155436177 | 31.517155436177 ✔
5 | 4 | 27.514976968091 | 27.514976968032 ✔
6 | 0 | 54.029409319160 | 54.029409319161 ✔
6 | 1 | 53.028864702139 | 53.028864702140 ✔
6 | 2 | 51.027775468096 | 51.027775468096 ✔
6 | 3 | 48.026141617031 | 48.026141617032 ✔
6 | 4 | 44.023963148945 | 44.023963148945 ✔
6 | 5 | 39.021240063838 | 39.021240063838 ✔
7 | 0 | 73.540029351079 | 73.540029351079 ✔
7 | 1 | 72.539484734058 | 72.539484734058 ✔
7 | 2 | 70.538395500015 | 70.538395500015 ✔
7 | 3 | 67.536761648950 | 67.536761648950 ✔
7 | 4 | 63.534583180864 | 63.534583180864 ✔
7 | 5 | 58.531860095757 | 58.531860095757 ✔
7 | 6 | 52.528592393628 | 52.528592393620 ✔
8 | 0 | 96.052283234063 | 96.052283234063 ✔
8 | 1 | 95.051738617041 | 95.051738617041 ✔
8 | 2 | 93.050649382998 | 93.050649382998 ✔
8 | 3 | 90.049015531934 | 90.049015531934 ✔
8 | 4 | 86.046837063848 | 86.046837063848 ✔
8 | 5 | 81.044113978740 | 81.044113978741 ✔
8 | 6 | 75.040846276612 | 75.040846276612 ✔
8 | 7 | 68.037033957461 | 68.037033957461 ✔
9 | 0 | 121.566170968111 | 121.566170968109 ✔
9 | 1 | 120.565626351089 | 120.565626351090 ✔
9 | 2 | 118.564537117046 | 118.564537117047 ✔
9 | 3 | 115.562903265982 | 115.562903265981 ✔
9 | 4 | 111.560724797896 | 111.560724797894 ✔
9 | 5 | 106.558001712788 | 106.558001712788 ✔
9 | 6 | 100.554734010660 | 100.554734010659 ✔
9 | 7 | 93.550921691509 | 93.550921691509 ✔
9 | 8 | 85.546564755337 | 85.546564755337 ✔
CoulombTwoBody(-1, 1, 1.0, Inf, 1.0, 1.0, 1.0, 1.0)
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 ✔
CoulombTwoBody(-1, 1, 206.768283, 1836.15267343, 1.0, 1.0, 1.0, 1.0)
n | l | analytical | numerical
-- | -- | ----------------- | -----------------
1 | 0 | 0.008071423070 | 0.008071423070 ✔
2 | 0 | 0.032285692282 | 0.032285692282 ✔
2 | 1 | 0.026904743568 | 0.026904743568 ✔
3 | 0 | 0.072642807634 | 0.072642807634 ✔
3 | 1 | 0.067261858920 | 0.067261858920 ✔
3 | 2 | 0.056499961493 | 0.056499961493 ✔
4 | 0 | 0.129142769127 | 0.129142769127 ✔
4 | 1 | 0.123761820413 | 0.123761820413 ✔
4 | 2 | 0.112999922986 | 0.112999922986 ✔
4 | 3 | 0.096857076845 | 0.096857076845 ✔
5 | 0 | 0.201785576761 | 0.201785576761 ✔
5 | 1 | 0.196404628047 | 0.196404628047 ✔
5 | 2 | 0.185642730620 | 0.185642730620 ✔
5 | 3 | 0.169499884479 | 0.169499884479 ✔
5 | 4 | 0.147976089624 | 0.147976089624 ✔
6 | 0 | 0.290571230535 | 0.290571230535 ✔
6 | 1 | 0.285190281822 | 0.285190281822 ✔
6 | 2 | 0.274428384394 | 0.274428384394 ✔
6 | 3 | 0.258285538254 | 0.258285538254 ✔
6 | 4 | 0.236761743399 | 0.236761743399 ✔
6 | 5 | 0.209856999831 | 0.209856999831 ✔
7 | 0 | 0.395499730451 | 0.395499730451 ✔
7 | 1 | 0.390118781737 | 0.390118781737 ✔
7 | 2 | 0.379356884310 | 0.379356884310 ✔
7 | 3 | 0.363214038169 | 0.363214038169 ✔
7 | 4 | 0.341690243315 | 0.341690243315 ✔
7 | 5 | 0.314785499747 | 0.314785499747 ✔
7 | 6 | 0.282499807465 | 0.282499807465 ✔
8 | 0 | 0.516571076507 | 0.516571076507 ✔
8 | 1 | 0.511190127794 | 0.511190127794 ✔
8 | 2 | 0.500428230366 | 0.500428230366 ✔
8 | 3 | 0.484285384225 | 0.484285384225 ✔
8 | 4 | 0.462761589371 | 0.462761589371 ✔
8 | 5 | 0.435856845803 | 0.435856845803 ✔
8 | 6 | 0.403571153521 | 0.403571153521 ✔
8 | 7 | 0.365904512526 | 0.365904512526 ✔
9 | 0 | 0.653785268704 | 0.653785268704 ✔
9 | 1 | 0.648404319991 | 0.648404319991 ✔
9 | 2 | 0.637642422564 | 0.637642422564 ✔
9 | 3 | 0.621499576423 | 0.621499576423 ✔
9 | 4 | 0.599975781568 | 0.599975781568 ✔
9 | 5 | 0.573071038000 | 0.573071038000 ✔
9 | 6 | 0.540785345718 | 0.540785345718 ✔
9 | 7 | 0.503118704723 | 0.503118704723 ✔
9 | 8 | 0.460071115014 | 0.460071115014 ✔
CoulombTwoBody(-1, 2, 206.768283, 7294.29954142, 1.0, 1.0, 1.0, 1.0)
n | l | analytical | numerical
-- | -- | ----------------- | -----------------
1 | 0 | 0.003730068786 | 0.003730068786 ✔
2 | 0 | 0.014920275143 | 0.014920275143 ✔
2 | 1 | 0.012433562619 | 0.012433562619 ✔
3 | 0 | 0.033570619071 | 0.033570619071 ✔
3 | 1 | 0.031083906548 | 0.031083906548 ✔
3 | 2 | 0.026110481500 | 0.026110481500 ✔
4 | 0 | 0.059681100571 | 0.059681100571 ✔
4 | 1 | 0.057194388047 | 0.057194388047 ✔
4 | 2 | 0.052220963000 | 0.052220963000 ✔
4 | 3 | 0.044760825428 | 0.044760825428 ✔
5 | 0 | 0.093251719643 | 0.093251719643 ✔
5 | 1 | 0.090765007119 | 0.090765007119 ✔
5 | 2 | 0.085791582071 | 0.085791582071 ✔
5 | 3 | 0.078331444500 | 0.078331444500 ✔
5 | 4 | 0.068384594405 | 0.068384594405 ✔
6 | 0 | 0.134282476285 | 0.134282476285 ✔
6 | 1 | 0.131795763762 | 0.131795763762 ✔
6 | 2 | 0.126822338714 | 0.126822338714 ✔
6 | 3 | 0.119362201143 | 0.119362201143 ✔
6 | 4 | 0.109415351047 | 0.109415351047 ✔
6 | 5 | 0.096981788428 | 0.096981788428 ✔
7 | 0 | 0.182773370500 | 0.182773370500 ✔
7 | 1 | 0.180286657976 | 0.180286657976 ✔
7 | 2 | 0.175313232928 | 0.175313232928 ✔
7 | 3 | 0.167853095357 | 0.167853095357 ✔
7 | 4 | 0.157906245262 | 0.157906245262 ✔
7 | 5 | 0.145472682643 | 0.145472682643 ✔
7 | 6 | 0.130552407500 | 0.130552407500 ✔
8 | 0 | 0.238724402285 | 0.238724402285 ✔
8 | 1 | 0.236237689761 | 0.236237689761 ✔
8 | 2 | 0.231264264714 | 0.231264264714 ✔
8 | 3 | 0.223804127142 | 0.223804127142 ✔
8 | 4 | 0.213857277047 | 0.213857277047 ✔
8 | 5 | 0.201423714428 | 0.201423714428 ✔
8 | 6 | 0.186503439285 | 0.186503439285 ✔
8 | 7 | 0.169096451619 | 0.169096451619 ✔
9 | 0 | 0.302135571642 | 0.302135571642 ✔
9 | 1 | 0.299648859118 | 0.299648859118 ✔
9 | 2 | 0.294675434071 | 0.294675434071 ✔
9 | 3 | 0.287215296499 | 0.287215296499 ✔
9 | 4 | 0.277268446404 | 0.277268446404 ✔
9 | 5 | 0.264834883785 | 0.264834883785 ✔
9 | 6 | 0.249914608642 | 0.249914608642 ✔
9 | 7 | 0.232507620976 | 0.232507620976 ✔
9 | 8 | 0.212613920785 | 0.212613920785 ✔
Expected Value of $r^2$
\[\langle r^2 \rangle = \int r^2 |R_{nl}(r)|^2 r^2 \mathrm{d}r = \frac{a_\mu^2}{2Z^2} n^2 \left[ 5n^2 + 1 - 3l(l+1) \right] \\ a_\mu = a_0 \frac{m_\mathrm{e}}{\mu} \\ \frac{1}{\mu} = \frac{1}{m_\mathrm{e}} + \frac{1}{m_\mathrm{p}}\]
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
CoulombTwoBody(-1, 1, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0)
n | l | analytical | numerical
-- | -- | ----------------- | -----------------
1 | 0 | 12.000000000000 | 12.000000000000 ✔
2 | 0 | 168.000000000000 | 168.000000000000 ✔
2 | 1 | 120.000000000000 | 120.000000000000 ✔
3 | 0 | 828.000000000000 | 828.000000000000 ✔
3 | 1 | 720.000000000000 | 720.000000000000 ✔
3 | 2 | 504.000000000000 | 504.000000000000 ✔
4 | 0 | 2592.000000000000 | 2591.999999999680 ✔
4 | 1 | 2400.000000000000 | 2399.999999999790 ✔
4 | 2 | 2016.000000000000 | 2015.999999999917 ✔
4 | 3 | 1440.000000000000 | 1439.999999999985 ✔
5 | 0 | 6300.000000000000 | 6299.999999999992 ✔
5 | 1 | 6000.000000000000 | 5999.999999999996 ✔
5 | 2 | 5400.000000000000 | 5400.000000000001 ✔
5 | 3 | 4500.000000000000 | 4500.000000000003 ✔
5 | 4 | 3300.000000000000 | 3300.000000000000 ✔
6 | 0 | 13032.000000000000 | 13031.999999999989 ✔
6 | 1 | 12600.000000000000 | 12599.999999999935 ✔
6 | 2 | 11736.000000000000 | 11735.999999999989 ✔
6 | 3 | 10440.000000000000 | 10440.000000000115 ✔
6 | 4 | 8712.000000000000 | 8712.000000000024 ✔
6 | 5 | 6552.000000000000 | 6552.000000000002 ✔
7 | 0 | 24108.000000000000 | 24107.999999999989 ✔
7 | 1 | 23520.000000000000 | 23520.000000000102 ✔
7 | 2 | 22344.000000000000 | 22344.000000000007 ✔
7 | 3 | 20580.000000000000 | 20580.000000000000 ✔
7 | 4 | 18228.000000000000 | 18227.999999999996 ✔
7 | 5 | 15288.000000000000 | 15288.000000000000 ✔
7 | 6 | 11760.000000000000 | 11759.999999994438 ✔
8 | 0 | 41088.000000000000 | 41088.000000000058 ✔
8 | 1 | 40320.000000000000 | 40319.999999999920 ✔
8 | 2 | 38784.000000000000 | 38783.999999999418 ✔
8 | 3 | 36480.000000000000 | 36479.999999999454 ✔
8 | 4 | 33408.000000000000 | 33407.999999999607 ✔
8 | 5 | 29568.000000000000 | 29567.999999999789 ✔
8 | 6 | 24960.000000000000 | 24959.999999999960 ✔
8 | 7 | 19584.000000000000 | 19583.999999999993 ✔
9 | 0 | 65772.000000000000 | 65771.999999999854 ✔
9 | 1 | 64800.000000000000 | 64800.000000000313 ✔
9 | 2 | 62856.000000000000 | 62856.000000000386 ✔
9 | 3 | 59940.000000000000 | 59939.999999999927 ✔
9 | 4 | 56052.000000000000 | 56051.999999998654 ✔
9 | 5 | 51192.000000000000 | 51192.000000000007 ✔
9 | 6 | 45360.000000000000 | 45359.999999999891 ✔
9 | 7 | 38556.000000000000 | 38555.999999999738 ✔
9 | 8 | 30780.000000000000 | 30779.999999999967 ✔
CoulombTwoBody(-1, 1, 1.0, 206.768283, 1.0, 1.0, 1.0, 1.0)
n | l | analytical | numerical
-- | -- | ----------------- | -----------------
1 | 0 | 3.029088160465 | 3.029088160465 ✔
2 | 0 | 42.407234246517 | 42.407234246517 ✔
2 | 1 | 30.290881604655 | 30.290881604655 ✔
3 | 0 | 209.007083072118 | 209.007083072118 ✔
3 | 1 | 181.745289627929 | 181.745289627929 ✔
3 | 2 | 127.221702739550 | 127.221702739550 ✔
4 | 0 | 654.283042660544 | 654.283042660437 ✔
4 | 1 | 605.817632093097 | 605.817632093025 ✔
4 | 2 | 508.886810958201 | 508.886810958173 ✔
4 | 3 | 363.490579255858 | 363.490579255853 ✔
5 | 0 | 1590.271284244378 | 1590.271284244375 ✔
5 | 1 | 1514.544080232741 | 1514.544080232738 ✔
5 | 2 | 1363.089672209467 | 1363.089672209466 ✔
5 | 3 | 1135.908060174556 | 1135.908060174555 ✔
5 | 4 | 832.999244128008 | 832.999244128007 ✔
6 | 0 | 3289.589742265514 | 3289.589742265513 ✔
6 | 1 | 3180.542568488756 | 3180.542568488750 ✔
6 | 2 | 2962.448220935242 | 2962.448220935266 ✔
6 | 3 | 2635.306699604970 | 2635.306699605012 ✔
6 | 4 | 2199.118004497940 | 2199.118004497951 ✔
6 | 5 | 1653.882135614153 | 1653.882135614154 ✔
7 | 0 | 6085.438114375154 | 6085.438114375178 ✔
7 | 1 | 5937.012794512346 | 5937.012794512338 ✔
7 | 2 | 5640.162154786729 | 5640.162154786726 ✔
7 | 3 | 5194.886195198303 | 5194.886195198301 ✔
7 | 4 | 4601.184915747068 | 4601.184915747067 ✔
7 | 5 | 3859.058316433025 | 3859.058316433023 ✔
7 | 6 | 2968.506397256173 | 2968.506397256174 ✔
8 | 0 | 10371.597861433813 | 10371.597861433793 ✔
8 | 1 | 10177.736219164022 | 10177.736219164037 ✔
8 | 2 | 9790.012934624439 | 9790.012934624450 ✔
8 | 3 | 9208.428007815068 | 9208.428007815050 ✔
8 | 4 | 8432.981438735904 | 8432.981438735898 ✔
8 | 5 | 7463.673227386949 | 7463.673227386961 ✔
8 | 6 | 6300.503373768204 | 6300.503373768204 ✔
8 | 7 | 4943.471877879668 | 4943.471877879667 ✔
9 | 0 | 16602.432207511312 | 16602.432207511407 ✔
9 | 1 | 16357.076066513608 | 16357.076066513559 ✔
9 | 2 | 15866.363784518200 | 15866.363784518258 ✔
9 | 3 | 15130.295361525088 | 15130.295361525063 ✔
9 | 4 | 14148.870797534271 | 14148.870797533782 ✔
9 | 5 | 12922.090092545750 | 12922.090092545510 ✔
9 | 6 | 11449.953246559526 | 11449.953246559453 ✔
9 | 7 | 9732.460259575597 | 9732.460259575586 ✔
9 | 8 | 7769.611131593963 | 7769.611131593960 ✔
CoulombTwoBody(-1, 1, 1.0, 1836.15267343, 1.0, 1.0, 1.0, 1.0)
n | l | analytical | numerical
-- | -- | ----------------- | -----------------
1 | 0 | 3.003268591952 | 3.003268591952 ✔
2 | 0 | 42.045760287328 | 42.045760287328 ✔
2 | 1 | 30.032685919520 | 30.032685919520 ✔
3 | 0 | 207.225532844690 | 207.225532844690 ✔
3 | 1 | 180.196115517122 | 180.196115517121 ✔
3 | 2 | 126.137280861985 | 126.137280861985 ✔
4 | 0 | 648.706015861638 | 648.706015861539 ✔
4 | 1 | 600.653718390405 | 600.653718390341 ✔
4 | 2 | 504.549123447940 | 504.549123447915 ✔
4 | 3 | 360.392231034243 | 360.392231034239 ✔
5 | 0 | 1576.716010774813 | 1576.716010774812 ✔
5 | 1 | 1501.634295976013 | 1501.634295976010 ✔
5 | 2 | 1351.470866378412 | 1351.470866378410 ✔
5 | 3 | 1126.225721982010 | 1126.225721982012 ✔
5 | 4 | 825.898862786807 | 825.898862786807 ✔
6 | 0 | 3261.549690859900 | 3261.549690859898 ✔
6 | 1 | 3153.432021549627 | 3153.432021549621 ✔
6 | 2 | 2937.196682929081 | 2937.196682929080 ✔
6 | 3 | 2612.843674998262 | 2612.843674998296 ✔
6 | 4 | 2180.372997757171 | 2180.372997757178 ✔
6 | 5 | 1639.784651205806 | 1639.784651205807 ✔
7 | 0 | 6033.566601231620 | 6033.566601231601 ✔
7 | 1 | 5886.406440225971 | 5886.406440225976 ✔
7 | 2 | 5592.086118214672 | 5592.086118214662 ✔
7 | 3 | 5150.605635197724 | 5150.605635197723 ✔
7 | 4 | 4561.964991175128 | 4561.964991175127 ✔
7 | 5 | 3826.164186146881 | 3826.164186146881 ✔
7 | 6 | 2943.203220112985 | 2943.203220112987 ✔
8 | 0 | 10283.191658843736 | 10283.191658843743 ✔
8 | 1 | 10090.982468958808 | 10090.982468958782 ✔
8 | 2 | 9706.564089188947 | 9706.564089188942 ✔
8 | 3 | 9129.936519534158 | 9129.936519534149 ✔
8 | 4 | 8361.099759994440 | 8361.099759994429 ✔
8 | 5 | 7400.053810569791 | 7400.053810569806 ✔
8 | 6 | 6246.798671260214 | 6246.798671260213 ✔
8 | 7 | 4901.334342065707 | 4901.334342065713 ✔
9 | 0 | 16460.915152489055 | 16460.915152488851 ✔
9 | 1 | 16217.650396540941 | 16217.650396541034 ✔
9 | 2 | 15731.120884644713 | 15731.120884644752 ✔
9 | 3 | 15001.326616800370 | 15001.326616800246 ✔
9 | 4 | 14028.267593007913 | 14028.267593007509 ✔
9 | 5 | 12811.943813267344 | 12811.943813267151 ✔
9 | 6 | 11352.355277578659 | 11352.355277578601 ✔
9 | 7 | 9649.501985941861 | 9649.501985941853 ✔
9 | 8 | 7703.383938356947 | 7703.383938356944 ✔
CoulombTwoBody(-1, 1, 1.0, Inf, 1.0, 1.0, 1.0, 1.0)
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 ✔
CoulombTwoBody(-1, 1, 206.768283, 1836.15267343, 1.0, 1.0, 1.0, 1.0)
n | l | analytical | numerical
-- | -- | ----------------- | -----------------
1 | 0 | 0.000086863827 | 0.000086863827 ✔
2 | 0 | 0.001216093580 | 0.001216093580 ✔
2 | 1 | 0.000868638272 | 0.000868638272 ✔
3 | 0 | 0.005993604075 | 0.005993604075 ✔
3 | 1 | 0.005211829631 | 0.005211829631 ✔
3 | 2 | 0.003648280741 | 0.003648280741 ✔
4 | 0 | 0.018762586670 | 0.018762586670 ✔
4 | 1 | 0.017372765435 | 0.017372765435 ✔
4 | 2 | 0.014593122966 | 0.014593122966 ✔
4 | 3 | 0.010423659261 | 0.010423659261 ✔
5 | 0 | 0.045603509267 | 0.045603509267 ✔
5 | 1 | 0.043431913588 | 0.043431913588 ✔
5 | 2 | 0.039088722229 | 0.039088722229 ✔
5 | 3 | 0.032573935191 | 0.032573935191 ✔
5 | 4 | 0.023887552473 | 0.023887552473 ✔
6 | 0 | 0.094334116313 | 0.094334116313 ✔
6 | 1 | 0.091207018535 | 0.091207018535 ✔
6 | 2 | 0.084952822978 | 0.084952822978 ✔
6 | 3 | 0.075571529643 | 0.075571529643 ✔
6 | 4 | 0.063063138530 | 0.063063138530 ✔
6 | 5 | 0.047427649638 | 0.047427649638 ✔
7 | 0 | 0.174509428796 | 0.174509428796 ✔
7 | 1 | 0.170253101264 | 0.170253101264 ✔
7 | 2 | 0.161740446201 | 0.161740446201 ✔
7 | 3 | 0.148971463606 | 0.148971463606 ✔
7 | 4 | 0.131946153480 | 0.131946153480 ✔
7 | 5 | 0.110664515822 | 0.110664515822 ✔
7 | 6 | 0.085126550632 | 0.085126550632 ✔
8 | 0 | 0.297421744250 | 0.297421744250 ✔
8 | 1 | 0.291862459310 | 0.291862459310 ✔
8 | 2 | 0.280743889432 | 0.280743889432 ✔
8 | 3 | 0.264066034614 | 0.264066034614 ✔
8 | 4 | 0.241828894857 | 0.241828894857 ✔
8 | 5 | 0.214032470161 | 0.214032470161 ✔
8 | 6 | 0.180676760525 | 0.180676760525 ✔
8 | 7 | 0.141761765951 | 0.141761765951 ✔
9 | 0 | 0.476100636750 | 0.476100636750 ✔
9 | 1 | 0.469064666749 | 0.469064666749 ✔
9 | 2 | 0.454992726746 | 0.454992726746 ✔
9 | 3 | 0.433884816743 | 0.433884816743 ✔
9 | 4 | 0.405740936738 | 0.405740936738 ✔
9 | 5 | 0.370561086732 | 0.370561086732 ✔
9 | 6 | 0.328345266724 | 0.328345266724 ✔
9 | 7 | 0.279093476716 | 0.279093476716 ✔
9 | 8 | 0.222805716706 | 0.222805716706 ✔
CoulombTwoBody(-1, 2, 206.768283, 7294.29954142, 1.0, 1.0, 1.0, 1.0)
n | l | analytical | numerical
-- | -- | ----------------- | -----------------
1 | 0 | 0.000018551218 | 0.000018551218 ✔
2 | 0 | 0.000259717045 | 0.000259717045 ✔
2 | 1 | 0.000185512175 | 0.000185512175 ✔
3 | 0 | 0.001280034009 | 0.001280034009 ✔
3 | 1 | 0.001113073052 | 0.001113073052 ✔
3 | 2 | 0.000779151136 | 0.000779151136 ✔
4 | 0 | 0.004007062986 | 0.004007062986 ✔
4 | 1 | 0.003710243506 | 0.003710243506 ✔
4 | 2 | 0.003116604545 | 0.003116604545 ✔
4 | 3 | 0.002226146103 | 0.002226146103 ✔
5 | 0 | 0.009739389202 | 0.009739389202 ✔
5 | 1 | 0.009275608764 | 0.009275608764 ✔
5 | 2 | 0.008348047888 | 0.008348047888 ✔
5 | 3 | 0.006956706573 | 0.006956706573 ✔
5 | 4 | 0.005101584820 | 0.005101584820 ✔
6 | 0 | 0.020146622236 | 0.020146622236 ✔
6 | 1 | 0.019478778405 | 0.019478778405 ✔
6 | 2 | 0.018143090743 | 0.018143090743 ✔
6 | 3 | 0.016139559249 | 0.016139559249 ✔
6 | 4 | 0.013468183925 | 0.013468183925 ✔
6 | 5 | 0.010128964770 | 0.010128964770 ✔
7 | 0 | 0.037269396014 | 0.037269396014 ✔
7 | 1 | 0.036360386355 | 0.036360386355 ✔
7 | 2 | 0.034542367037 | 0.034542367037 ✔
7 | 3 | 0.031815338061 | 0.031815338061 ✔
7 | 4 | 0.028179299425 | 0.028179299425 ✔
7 | 5 | 0.023634251131 | 0.023634251131 ✔
7 | 6 | 0.018180193178 | 0.018180193178 ✔
8 | 0 | 0.063519368816 | 0.063519368816 ✔
8 | 1 | 0.062332090895 | 0.062332090895 ✔
8 | 2 | 0.059957535051 | 0.059957535051 ✔
8 | 3 | 0.056395701286 | 0.056395701286 ✔
8 | 4 | 0.051646589598 | 0.051646589598 ✔
8 | 5 | 0.045710199989 | 0.045710199989 ✔
8 | 6 | 0.038586532459 | 0.038586532459 ✔
8 | 7 | 0.030275587006 | 0.030275587006 ✔
9 | 0 | 0.101679223272 | 0.101679223272 ✔
9 | 1 | 0.100176574652 | 0.100176574652 ✔
9 | 2 | 0.097171277412 | 0.097171277412 ✔
9 | 3 | 0.092663331553 | 0.092663331553 ✔
9 | 4 | 0.086652737074 | 0.086652737074 ✔
9 | 5 | 0.079139493975 | 0.079139493975 ✔
9 | 6 | 0.070123602256 | 0.070123602256 ✔
9 | 7 | 0.059605061918 | 0.059605061918 ✔
9 | 8 | 0.047583872960 | 0.047583872960 ✔
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 $\langle H \rangle = -\langle T \rangle$.
\[\frac{1}{2} \int \psi_n^\ast(x) V(x) \psi_n(x) \mathrm{d}x = E_n\]
CoulombTwoBody(-1, 1, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0)
n | analytical | numerical
-- | ----------------- | -----------------
1 | -0.250000000000 | -0.250000000000 ✔
2 | -0.062500000000 | -0.062500000000 ✔
3 | -0.027777777778 | -0.027777777778 ✔
4 | -0.015625000000 | -0.015625000000 ✔
5 | -0.010000000000 | -0.010000000000 ✔
6 | -0.006944444444 | -0.006944444444 ✔
7 | -0.005102040816 | -0.005102040816 ✔
8 | -0.003906250000 | -0.003906250000 ✔
9 | -0.003086419753 | -0.003086419753 ✔
10 | -0.002500000000 | -0.002500000000 ✔
CoulombTwoBody(-1, 1, 1.0, 206.768283, 1.0, 1.0, 1.0, 1.0)
n | analytical | numerical
-- | ----------------- | -----------------
1 | -0.497593472917 | -0.497593472917 ✔
2 | -0.124398368229 | -0.124398368229 ✔
3 | -0.055288163657 | -0.055288163657 ✔
4 | -0.031099592057 | -0.031099592057 ✔
5 | -0.019903738917 | -0.019903738917 ✔
6 | -0.013822040914 | -0.013822040914 ✔
7 | -0.010154968835 | -0.010154968835 ✔
8 | -0.007774898014 | -0.007774898014 ✔
9 | -0.006143129295 | -0.006143129295 ✔
10 | -0.004975934729 | -0.004975934729 ✔
CoulombTwoBody(-1, 1, 1.0, 1836.15267343, 1.0, 1.0, 1.0, 1.0)
n | analytical | numerical
-- | ----------------- | -----------------
1 | -0.499727839712 | -0.499727839712 ✔
2 | -0.124931959928 | -0.124931959928 ✔
3 | -0.055525315524 | -0.055525315524 ✔
4 | -0.031232989982 | -0.031232989982 ✔
5 | -0.019989113588 | -0.019989113588 ✔
6 | -0.013881328881 | -0.013881328881 ✔
7 | -0.010198527341 | -0.010198527341 ✔
8 | -0.007808247496 | -0.007808247496 ✔
9 | -0.006169479503 | -0.006169479503 ✔
10 | -0.004997278397 | -0.004997278397 ✔
CoulombTwoBody(-1, 1, 1.0, Inf, 1.0, 1.0, 1.0, 1.0)
n | analytical | numerical
-- | ----------------- | -----------------
1 | -0.500000000000 | -0.500000000000 ✔
2 | -0.125000000000 | -0.125000000000 ✔
3 | -0.055555555556 | -0.055555555556 ✔
4 | -0.031250000000 | -0.031250000000 ✔
5 | -0.020000000000 | -0.020000000000 ✔
6 | -0.013888888889 | -0.013888888889 ✔
7 | -0.010204081633 | -0.010204081633 ✔
8 | -0.007812500000 | -0.007812500000 ✔
9 | -0.006172839506 | -0.006172839506 ✔
10 | -0.005000000000 | -0.005000000000 ✔
CoulombTwoBody(-1, 1, 206.768283, 1836.15267343, 1.0, 1.0, 1.0, 1.0)
n | analytical | numerical
-- | ----------------- | -----------------
1 | -92.920417311307 | -92.920417311307 ✔
2 | -23.230104327827 | -23.230104327827 ✔
3 | -10.324490812367 | -10.324490812367 ✔
4 | -5.807526081957 | -5.807526081957 ✔
5 | -3.716816692452 | -3.716816692452 ✔
6 | -2.581122703092 | -2.581122703092 ✔
7 | -1.896335047170 | -1.896335047170 ✔
8 | -1.451881520489 | -1.451881520489 ✔
9 | -1.147165645819 | -1.147165645819 ✔
10 | -0.929204173113 | -0.929204173113 ✔
CoulombTwoBody(-1, 2, 206.768283, 7294.29954142, 1.0, 1.0, 1.0, 1.0)
n | analytical | numerical
-- | ----------------- | -----------------
1 | -402.137356219338 | -402.137356219338 ✔
2 | -100.534339054835 | -100.534339054835 ✔
3 | -44.681928468815 | -44.681928468815 ✔
4 | -25.133584763709 | -25.133584763709 ✔
5 | -16.085494248774 | -16.085494248774 ✔
6 | -11.170482117204 | -11.170482117204 ✔
7 | -8.206884820803 | -8.206884820803 ✔
8 | -6.283396190927 | -6.283396190927 ✔
9 | -4.964658718757 | -4.964658718757 ✔
10 | -4.021373562193 | -4.021373562193 ✔
CoulombTwoBody(-1, 2, 1.0, 1.0, 1.0, 1.0, 27.211386245988, 1.0)
n | analytical | numerical
-- | ----------------- | -----------------
1 | -27.211386245988 | -27.211386245988 ✔
2 | -6.802846561497 | -6.802846561497 ✔
3 | -3.023487360665 | -3.023487360665 ✔
4 | -1.700711640374 | -1.700711640374 ✔
5 | -1.088455449840 | -1.088455449840 ✔
6 | -0.755871840166 | -0.755871840166 ✔
7 | -0.555334413183 | -0.555334413183 ✔
8 | -0.425177910094 | -0.425177910094 ✔
9 | -0.335943040074 | -0.335943040074 ✔
10 | -0.272113862460 | -0.272113862460 ✔
CoulombTwoBody(-1, 2, 9.1093837015e-31, 1.67262192595e-27, 9.1093837015e-31, 5.29177210903e-11, 4.3597447222071e-18, 1.054571817e-34)
n | analytical | numerical
-- | ----------------- | -----------------
1 | -0.000000000000 | -0.000000000000 ✔
2 | -0.000000000000 | -0.000000000000 ✔
3 | -0.000000000000 | -0.000000000000 ✔
4 | -0.000000000000 | -0.000000000000 ✔
5 | -0.000000000000 | -0.000000000000 ✔
6 | -0.000000000000 | -0.000000000000 ✔
7 | -0.000000000000 | -0.000000000000 ✔
8 | -0.000000000000 | -0.000000000000 ✔
9 | -0.000000000000 | -0.000000000000 ✔
10 | -0.000000000000 | -0.000000000000 ✔
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 | 1.000000000092 ✔
1 | 2 | 0 | 0 | 0 | 0 | 0.000000000000 | -0.000000004734 ✔
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 | -0.000000019782 ✔
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 | -0.000000004734 ✔
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 | 1.000004247415 ✔
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 | 1.000001369572 ✔
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 | 1.000001369572 ✔
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 | 1.000001369572 ✔
2 | 3 | 0 | 0 | 0 | 0 | 0.000000000000 | 0.000060895216 ✔
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 | 0.000026040451 ✔
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 | 0.000026040451 ✔
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 | 0.000026040451 ✔
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 | -0.000000019782 ✔
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 | 0.000060895216 ✔
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 | 0.000026040451 ✔
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 | 0.000026040451 ✔
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 | 0.000026040451 ✔
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 | 1.001615171730 ✔
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 | 1.000942652622 ✔
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 | 1.000942652622 ✔
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 | 1.000942652622 ✔
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 | 1.000222366721 ✔
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 | 0.000000193764 ✔
3 | 3 | 2 | 2 | -1 | -2 | 0.000000000000 | -0.000000000000 ✔
3 | 3 | 2 | 2 | -1 | -1 | 1.000000000000 | 1.000222366714 ✔
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 | 1.000222366727 ✔
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 | 1.000222366714 ✔
3 | 3 | 2 | 2 | 1 | 2 | 0.000000000000 | 0.000000000000 ✔
3 | 3 | 2 | 2 | 2 | -2 | 0.000000000000 | 0.000000193764 ✔
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 | 1.000222366721 ✔