Morse Potential
The Morse potential is a model for inter-nuclear anharmonic vibration in a diatomic molecule.
Definitions
This model is described with the time-independent Schrödinger equation
\[ \hat{H} \psi(r) = E \psi(r),\]
and the Hamiltonian
\[ \hat{H} = - \frac{\hbar^2}{2\mu} \frac{\mathrm{d}^2}{\mathrm{d}r ^2} + V(r)\]
Parameters are specified with the following struct.
Parameters
Antique.MorsePotential — TypeMP = MorsePotential(rₑ=2.0, Dₑ=0.1, k=0.1, µ=918.1, ℏ=1.0)
$r_\mathrm{e}$ is the equilibrium bond distance, $D_\mathrm{e}$ is the the well depth , $k$ is the force constant, $\mu$ is the reduced mass and $\hbar$ is the reduced Planck constant (Dirac's constant).
Potential
Antique.V — MethodV(model::MorsePotential, r)
\[V(r) = D_\mathrm{e} \left( \mathrm{e}^{-2a(r-r_e)} - 2\mathrm{e}^{-a(r-r_e)} \right),\]
where $a = \sqrt{\frac{k}{2Dₑ}}$ is defined. The domain is $0\leq r \lt \infty$.
Eigen Values
Antique.E — MethodE(model::MorsePotential; n=0)
\[E_n = - D_\mathrm{e} + \hbar \omega \left( n + \frac{1}{2} \right) - \chi \hbar \omega \left( n + \frac{1}{2} \right)^2,\]
where $\omega = \sqrt{k/µ}$ and $\chi = \frac{\hbar\omega}{4D_\mathrm{e}}$ are defined.
Number of Bound States
Antique.nₘₐₓ — Methodnₘₐₓ(model::MorsePotential)
\[n_\mathrm{max} = \left\lfloor \frac{2 D_e - \omega}{\omega} \right\rfloor,\]
where $\omega = \sqrt{k/µ}$ is defined.
Eigen Functions
Antique.ψ — Methodψ(model::MorsePotential, r; n=0)
\[\psi_n(r) = N_n z^{\lambda-n-1/2} \mathrm{e}^{-z/2} L_n^{(2\lambda-2n-1)}(\xi),\]
$N_n = \sqrt{\frac{n!(2\lambda-2n-1)a}{\Gamma(2\lambda-n)}}$, $\lambda = \frac{\sqrt{2\mu D_\mathrm{e}}}{a\hbar}$, $a = \sqrt{\frac{k}{2Dₑ}}$, $L_n^{(\alpha)}(x) = \frac{x^{-\alpha} \mathrm{e}^x}{n !} \frac{\mathrm{d}^n}{\mathrm{d} x^n}\left(\mathrm{e}^{-x} x^{n+\alpha}\right)$, $\xi := 2\lambda\mathrm{e}^{-a(r-r_e)}$ are defined. The domain is $0\leq r \lt \infty$.
Generalized Laguerre Polynomials
The generalized Laguerre polynomials $L_n^{(\alpha)}(x)$, not the associated Laguerre polynomials $L_n^{k}(x)$, are used in this model.
Antique.L — MethodL(model::MorsePotential, x; n=0, α=0)
Rodrigues' formula & closed-form:
\[\begin{aligned} L_n^{(\alpha)}(x) &= \frac{x^{-\alpha}e^x}{n!} \frac{d^n}{dx^n}\left(x^{n+\alpha}e^{-x}\right) \\ &= \sum_{k=0}^n(-1)^k \left(\begin{array}{l} n+\alpha \\ n-k \end{array}\right) \frac{x^k}{k !} \\ &= \sum_{k=0}^n(-1)^k \frac{\Gamma(\alpha+n+1)}{\Gamma(\alpha+k+1)\Gamma(n-k+1)} \frac{x^k}{k !}. \end{aligned}\]
Examples:
\[\begin{aligned} L_0^{(0)}(x) &= 1, \\ L_1^{(0)}(x) &= 1 - x, \\ L_1^{(1)}(x) &= 2 - x, \\ L_2^{(0)}(x) &= 1 - 2 x + 1/2 x^{2}, \\ L_2^{(1)}(x) &= 3 - 3 x + 1/2 x^{2}, \\ L_2^{(2)}(x) &= 6 - 4 x + 1/2 x^{2}, \\ L_3^{(0)}(x) &= 1 - 3 x + 3/2 x^{2} - 1/6 x^{3}, \\ L_3^{(1)}(x) &= 4 - 6 x + 2 x^{2} - 1/6 x^{3}, \\ L_3^{(2)}(x) &= 10 - 10 x + 5/2 x^{2} - 1/6 x^{3}, \\ L_3^{(3)}(x) &= 20 - 15 x + 3 x^{2} - 1/6 x^{3}, \\ L_4^{(0)}(x) &= 1 - 4 x + 3 x^{2} - 2/3 x^{3} + 1/24 x^{4}, \\ L_4^{(1)}(x) &= 5 - 10 x + 5 x^{2} - 5/6 x^{3} + 1/24 x^{4}, \\ L_4^{(2)}(x) &= 15 - 20 x + 15/2 x^{2} - 1 x^{3} + 1/24 x^{4}, \\ L_4^{(3)}(x) &= 35 - 35 x + 21/2 x^{2} - 7/6 x^{3} + 1/24 x^{4}, \\ L_4^{(4)}(x) &= 70 - 56 x + 14 x^{2} - 4/3 x^{3} + 1/24 x^{4}, \\ \vdots \end{aligned}\]
References
- P. M. Morse, Phys. Rev. 34, 57 (1929)
- J. P. Dahl, M. Springborg, J. Chem. Phys. 88, 4535 (1988). (62), (63)
- W. K. Shao, Y. He, J. Pan, J. Nonlinear Sci. Appl., 9, 5, 3388 (2016). (1.6)
- The Digital Library of Mathematical Functions (DLMF) 18.3 Table1, 18.5 Table1, 18.5.12
Usage & Examples
Install Antique.jl for the first use and run using Antique before each use. The energy E(), wavefunction ψ(), potential V() and some other functions are suppoted. In this system, the model is generated by MorsePotential and several parameters rₑ, Dₑ, k, µ and ℏ are set as optional arguments.
# Parameters for H₂⁺
# https://doi.org/10.1002/slct.202102509
# https://doi.org/10.5281/zenodo.5047817
# https://physics.nist.gov/cgi-bin/cuu/Value?mpsme
rₑ = 1.997193319969992120068298141276
Dₑ = - 0.5 - (-0.602634619106539878727562156289)
k = 2*((-1.1026342144949464615+1/2.00) - (-0.602634619106539878727562156289)) / (2.00 - rₑ)^2
µ = 1/(1/1836.15267343 + 1/1836.15267343)
ℏ = 1.0
using Antique
MP = MorsePotential(rₑ=rₑ, Dₑ=Dₑ, k=k, µ=µ, ℏ=ℏ)Parameters:
julia> MP.rₑ1.997193319969992julia> MP.Dₑ0.10263461910653993julia> MP.k0.1027265041900817julia> MP.µ918.076336715julia> MP.ℏ1.0
Number of bound states:
julia> nₘₐₓ(MP)18
Eigen values:
julia> E(MP, n=0)-0.09741377794418261julia> E(MP, n=1)-0.08738092406760907
Potential energy curve:
using CairoMakie
f = Figure()
ax = Axis(f[1,1], xlabel=L"$r$", ylabel=L"$V(r)$", limits=(0.0,20.0,-0.11,0.1))
lines!(ax, 0.1:0.01:20, r -> V(MP, r))
f
Wave functions:
using CairoMakie
# setting
f = Figure()
ax = Axis(f[1,1], xlabel=L"$r$", ylabel=L"$\psi(r)$")
# plot
w0 = lines!(ax, 0..5, x -> ψ(MP, x, n=0))
w1 = lines!(ax, 0..5, x -> ψ(MP, x, n=1))
w2 = lines!(ax, 0..5, x -> ψ(MP, x, n=2))
w3 = lines!(ax, 0..5, x -> ψ(MP, x, n=3))
w4 = lines!(ax, 0..5, x -> ψ(MP, x, n=4))
# legend
axislegend(ax, [w0, w1, w2, w3, w4], [L"n=0", L"n=1", L"n=2", L"n=3", L"n=4"], position=:lb)
f
Potential energy curve, Energy levels, Comparison with harmonic oscillator:
using Antique
# https://physics.nist.gov/cgi-bin/cuu/Value?mmusme
m = 206.7682830
μ = 1 / (1/m + 1/m)
MP = MorsePotential(μ=μ)
# @show nₘₐₓ(MP)
using CairoMakie
# settings
f = Figure()
ax = Axis(f[1,1], xlabel=L"$r$", ylabel=L"$V(r),~E_n,~\psi_n(r) \times 5 + E_n$", aspect=1, limits=(0.5,10,-0.105,0.007))
# hidespines!(ax)
# hidedecorations!(ax)
println(nₘₐₓ(MP))
for n in 0:nₘₐₓ(MP)
# energy
EMP = E(MP, n=n)
lines!(ax, 0.1:0.01:15, r -> EMP > V(MP, r) ? EMP : NaN, color=:black, linewidth=2)
hlines!(ax, E(MP, n=n), color=:black, linewidth=1, linestyle=:dash)
# wave function
lines!(ax, 0..10, x -> E(MP,n=n) + 0.0065*ψ(MP,x,n=n), linewidth=2)
end
#potential
lines!(ax, 0..10, x -> V(MP, x), color=:black, linewidth=2)
f5
where, the potential of harmonic oscillator is defined as $V(r) \simeq \frac{1}{2} k (r - r_\mathrm{e})^2 + V_0$.
Testing
Unit testing and Integration testing were done using computer algebra system (Symbolics.jl) and numerical integration (QuadGK.jl). The test script is here.
Generalized Laguerre Polynomials $L_n^{(\alpha)}(x)$
\[ \begin{aligned} L_n^{(\alpha)}(x) &= \frac{x^{-\alpha}e^x}{n!} \frac{d^n}{dx^n}\left(x^{n+\alpha}e^{-x}\right) \\ &= \sum_{k=0}^n(-1)^k \frac{\Gamma(\alpha+n+1)}{\Gamma(\alpha+k+1)\Gamma(n-k+1)} \frac{x^k}{k !}. \end{aligned}\]
$n=0, α=0:$ ✔
\[\begin{aligned} L_{0}^{(0)}(x) = e^{ - x} e^{x} &= 1 \\ &= 1 \end{aligned}\]
$n=1, α=0:$ ✔
\[\begin{aligned} L_{1}^{(0)}(x) = \frac{\mathrm{d}}{\mathrm{d}x} x e^{ - x} e^{x} &= 1 - x \\ &= 1 - x \end{aligned}\]
$n=1, α=1:$ ✔
\[\begin{aligned} L_{1}^{(1)}(x) = \frac{e^{x} \frac{\mathrm{d}}{\mathrm{d}x} x^{2} e^{ - x}}{x} &= 2 - x \\ &= 2 - x \end{aligned}\]
$n=2, α=0:$ ✔
\[\begin{aligned} L_{2}^{(0)}(x) = \frac{1}{2} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} x^{2} e^{ - x} e^{x} &= 1 - 2 x + \frac{1}{2} x^{2} \\ &= 1 - 2 x + \frac{1}{2} x^{2} \end{aligned}\]
$n=2, α=1:$ ✔
\[\begin{aligned} L_{2}^{(1)}(x) = \frac{\frac{1}{2} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} x^{3} e^{ - x} e^{x}}{x} &= 3 - 3 x + \frac{1}{2} x^{2} \\ &= 3 - 3 x + \frac{1}{2} x^{2} \end{aligned}\]
$n=2, α=2:$ ✔
\[\begin{aligned} L_{2}^{(2)}(x) = \frac{\frac{1}{2} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} x^{4} e^{ - x} e^{x}}{x^{2}} &= 6 - 4 x + \frac{1}{2} x^{2} \\ &= 6 - 4 x + \frac{1}{2} x^{2} \end{aligned}\]
$n=3, α=0:$ ✔
\[\begin{aligned} L_{3}^{(0)}(x) = \frac{1}{6} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} x^{3} e^{ - x} e^{x} &= 1 - 3 x + \frac{3}{2} x^{2} - \frac{1}{6} x^{3} \\ &= 1 - 3 x + \frac{3}{2} x^{2} - \frac{1}{6} x^{3} \end{aligned}\]
$n=3, α=1:$ ✔
\[\begin{aligned} L_{3}^{(1)}(x) = \frac{\frac{1}{6} e^{x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} x^{4} e^{ - x}}{x} &= 4 - 6 x + 2 x^{2} - \frac{1}{6} x^{3} \\ &= 4 - 6 x + 2 x^{2} - \frac{1}{6} x^{3} \end{aligned}\]
$n=3, α=2:$ ✔
\[\begin{aligned} L_{3}^{(2)}(x) = \frac{\frac{1}{6} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} x^{5} e^{ - x} e^{x}}{x^{2}} &= 10 - 10 x + \frac{5}{2} x^{2} - \frac{1}{6} x^{3} \\ &= 10 - 10 x + \frac{5}{2} x^{2} - \frac{1}{6} x^{3} \end{aligned}\]
$n=3, α=3:$ ✔
\[\begin{aligned} L_{3}^{(3)}(x) = \frac{\frac{1}{6} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} x^{6} e^{ - x} e^{x}}{x^{3}} &= 20 - 15 x + 3 x^{2} - \frac{1}{6} x^{3} \\ &= 20 - 15 x + 3 x^{2} - \frac{1}{6} x^{3} \end{aligned}\]
$n=4, α=0:$ ✔
\[\begin{aligned} L_{4}^{(0)}(x) = \frac{1}{24} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} x^{4} e^{ - x} e^{x} &= 1 - 4 x + 3 x^{2} - \frac{2}{3} x^{3} + \frac{1}{24} x^{4} \\ &= 1 - 4 x + 3 x^{2} - \frac{2}{3} x^{3} + \frac{1}{24} x^{4} \end{aligned}\]
$n=4, α=1:$ ✔
\[\begin{aligned} L_{4}^{(1)}(x) = \frac{\frac{1}{24} e^{x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} x^{5} e^{ - x}}{x} &= 5 - 10 x + 5 x^{2} - \frac{5}{6} x^{3} + \frac{1}{24} x^{4} \\ &= 5 - 10 x + 5 x^{2} - \frac{5}{6} x^{3} + \frac{1}{24} x^{4} \end{aligned}\]
$n=4, α=2:$ ✔
\[\begin{aligned} L_{4}^{(2)}(x) = \frac{\frac{1}{24} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} x^{6} e^{ - x} e^{x}}{x^{2}} &= 15 - 20 x + \frac{15}{2} x^{2} - x^{3} + \frac{1}{24} x^{4} \\ &= 15 - 20 x + \frac{15}{2} x^{2} - x^{3} + \frac{1}{24} x^{4} \end{aligned}\]
$n=4, α=3:$ ✔
\[\begin{aligned} L_{4}^{(3)}(x) = \frac{\frac{1}{24} e^{x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} x^{7} e^{ - x}}{x^{3}} &= 35 - 35 x + \frac{21}{2} x^{2} - \frac{7}{6} x^{3} + \frac{1}{24} x^{4} \\ &= 35 - 35 x + \frac{21}{2} x^{2} - \frac{7}{6} x^{3} + \frac{1}{24} x^{4} \end{aligned}\]
$n=4, α=4:$ ✔
\[\begin{aligned} L_{4}^{(4)}(x) = \frac{\frac{1}{24} e^{x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} x^{8} e^{ - x}}{x^{4}} &= 70 - 56 x + 14 x^{2} - \frac{4}{3} x^{3} + \frac{1}{24} x^{4} \\ &= 70 - 56 x + 14 x^{2} - \frac{4}{3} x^{3} + \frac{1}{24} x^{4} \end{aligned}\]
Normalization & Orthogonality of $L_n^{(\alpha)}(x)$
\[\int_0^\infty L_i^{(\alpha)}(x) L_j^{(\alpha)}(x) x^\alpha \mathrm{e}^{-x} \mathrm{d}x = \frac{\Gamma(n+\alpha+1)}{n!} \delta_{ij}\]
α | i | j | analytical | numerical
---- | -- | -- | ----------------- | -----------------
0.01 | 0 | 0 | 0.994325851192 | 0.994325852936 ✔
0.01 | 0 | 1 | 0.000000000000 | 0.000000000000 ✔
0.01 | 0 | 2 | 0.000000000000 | 0.000000000000 ✔
0.01 | 0 | 3 | 0.000000000000 | 0.000000000000 ✔
0.01 | 0 | 4 | 0.000000000000 | 0.000000000000 ✔
0.01 | 0 | 5 | 0.000000000000 | 0.000000000000 ✔
0.01 | 0 | 6 | 0.000000000000 | 0.000000000001 ✔
0.01 | 0 | 7 | 0.000000000000 | 0.000000000001 ✔
0.01 | 0 | 8 | 0.000000000000 | 0.000000000001 ✔
0.01 | 0 | 9 | 0.000000000000 | 0.000000000001 ✔
0.01 | 1 | 0 | 0.000000000000 | 0.000000000000 ✔
0.01 | 1 | 1 | 1.004269109703 | 1.004269111483 ✔
0.01 | 1 | 2 | 0.000000000000 | 0.000000000000 ✔
0.01 | 1 | 3 | 0.000000000000 | 0.000000000000 ✔
0.01 | 1 | 4 | 0.000000000000 | 0.000000000001 ✔
0.01 | 1 | 5 | 0.000000000000 | 0.000000000001 ✔
0.01 | 1 | 6 | 0.000000000000 | 0.000000000002 ✔
0.01 | 1 | 7 | 0.000000000000 | 0.000000000002 ✔
0.01 | 1 | 8 | 0.000000000000 | 0.000000000002 ✔
0.01 | 1 | 9 | 0.000000000000 | 0.000000000003 ✔
0.01 | 2 | 0 | 0.000000000000 | 0.000000000000 ✔
0.01 | 2 | 1 | 0.000000000000 | 0.000000000000 ✔
0.01 | 2 | 2 | 1.009290455252 | 1.009290456144 ✔
0.01 | 2 | 3 | 0.000000000000 | 0.000000000001 ✔
0.01 | 2 | 4 | 0.000000000000 | 0.000000000001 ✔
0.01 | 2 | 5 | 0.000000000000 | 0.000000000003 ✔
0.01 | 2 | 6 | 0.000000000000 | 0.000000000002 ✔
0.01 | 2 | 7 | 0.000000000000 | 0.000000000003 ✔
0.01 | 2 | 8 | 0.000000000000 | 0.000000000003 ✔
0.01 | 2 | 9 | 0.000000000000 | 0.000000000007 ✔
0.01 | 3 | 0 | 0.000000000000 | 0.000000000000 ✔
0.01 | 3 | 1 | 0.000000000000 | 0.000000000000 ✔
0.01 | 3 | 2 | 0.000000000000 | 0.000000000001 ✔
0.01 | 3 | 3 | 1.012654756769 | 1.012654758579 ✔
0.01 | 3 | 4 | 0.000000000000 | 0.000000000003 ✔
0.01 | 3 | 5 | 0.000000000000 | 0.000000000003 ✔
0.01 | 3 | 6 | 0.000000000000 | 0.000000000003 ✔
0.01 | 3 | 7 | 0.000000000000 | 0.000000000007 ✔
0.01 | 3 | 8 | 0.000000000000 | 0.000000000014 ✔
0.01 | 3 | 9 | 0.000000000000 | 0.000000000014 ✔
0.01 | 4 | 0 | 0.000000000000 | 0.000000000000 ✔
0.01 | 4 | 1 | 0.000000000000 | 0.000000000001 ✔
0.01 | 4 | 2 | 0.000000000000 | 0.000000000001 ✔
0.01 | 4 | 3 | 0.000000000000 | 0.000000000003 ✔
0.01 | 4 | 4 | 1.015186393661 | 1.015186394564 ✔
0.01 | 4 | 5 | 0.000000000000 | 0.000000000002 ✔
0.01 | 4 | 6 | 0.000000000000 | 0.000000000007 ✔
0.01 | 4 | 7 | 0.000000000000 | 0.000000000014 ✔
0.01 | 4 | 8 | 0.000000000000 | 0.000000000014 ✔
0.01 | 4 | 9 | 0.000000000000 | 0.000000000014 ✔
0.01 | 5 | 0 | 0.000000000000 | 0.000000000000 ✔
0.01 | 5 | 1 | 0.000000000000 | 0.000000000001 ✔
0.01 | 5 | 2 | 0.000000000000 | 0.000000000003 ✔
0.01 | 5 | 3 | 0.000000000000 | 0.000000000003 ✔
0.01 | 5 | 4 | 0.000000000000 | 0.000000000002 ✔
0.01 | 5 | 5 | 1.017216766449 | 1.017216768275 ✔
0.01 | 5 | 6 | 0.000000000000 | 0.000000000014 ✔
0.01 | 5 | 7 | 0.000000000000 | 0.000000000014 ✔
0.01 | 5 | 8 | 0.000000000000 | 0.000000000014 ✔
0.01 | 5 | 9 | 0.000000000000 | 0.000000000028 ✔
0.01 | 6 | 0 | 0.000000000000 | 0.000000000001 ✔
0.01 | 6 | 1 | 0.000000000000 | 0.000000000002 ✔
0.01 | 6 | 2 | 0.000000000000 | 0.000000000002 ✔
0.01 | 6 | 3 | 0.000000000000 | 0.000000000003 ✔
0.01 | 6 | 4 | 0.000000000000 | 0.000000000007 ✔
0.01 | 6 | 5 | 0.000000000000 | 0.000000000014 ✔
0.01 | 6 | 6 | 1.018912127726 | 1.018912128636 ✔
0.01 | 6 | 7 | 0.000000000000 | 0.000000000014 ✔
0.01 | 6 | 8 | 0.000000000000 | 0.000000000028 ✔
0.01 | 6 | 9 | 0.000000000000 | 0.000000000028 ✔
0.01 | 7 | 0 | 0.000000000000 | 0.000000000001 ✔
0.01 | 7 | 1 | 0.000000000000 | 0.000000000002 ✔
0.01 | 7 | 2 | 0.000000000000 | 0.000000000003 ✔
0.01 | 7 | 3 | 0.000000000000 | 0.000000000007 ✔
0.01 | 7 | 4 | 0.000000000000 | 0.000000000014 ✔
0.01 | 7 | 5 | 0.000000000000 | 0.000000000014 ✔
0.01 | 7 | 6 | 0.000000000000 | 0.000000000014 ✔
0.01 | 7 | 7 | 1.020367716480 | 1.020367717392 ✔
0.01 | 7 | 8 | 0.000000000000 | 0.000000000028 ✔
0.01 | 7 | 9 | 0.000000000000 | 0.000000000028 ✔
0.01 | 8 | 0 | 0.000000000000 | 0.000000000001 ✔
0.01 | 8 | 1 | 0.000000000000 | 0.000000000002 ✔
0.01 | 8 | 2 | 0.000000000000 | 0.000000000003 ✔
0.01 | 8 | 3 | 0.000000000000 | 0.000000000014 ✔
0.01 | 8 | 4 | 0.000000000000 | 0.000000000014 ✔
0.01 | 8 | 5 | 0.000000000000 | 0.000000000014 ✔
0.01 | 8 | 6 | 0.000000000000 | 0.000000000028 ✔
0.01 | 8 | 7 | 0.000000000000 | 0.000000000028 ✔
0.01 | 8 | 8 | 1.021643176126 | 1.021643177967 ✔
0.01 | 8 | 9 | 0.000000000000 | 0.000000000028 ✔
0.01 | 9 | 0 | 0.000000000000 | 0.000000000001 ✔
0.01 | 9 | 1 | 0.000000000000 | 0.000000000003 ✔
0.01 | 9 | 2 | 0.000000000000 | 0.000000000007 ✔
0.01 | 9 | 3 | 0.000000000000 | 0.000000000014 ✔
0.01 | 9 | 4 | 0.000000000000 | 0.000000000014 ✔
0.01 | 9 | 5 | 0.000000000000 | 0.000000000028 ✔
0.01 | 9 | 6 | 0.000000000000 | 0.000000000028 ✔
0.01 | 9 | 7 | 0.000000000000 | 0.000000000028 ✔
0.01 | 9 | 8 | 0.000000000000 | 0.000000000028 ✔
0.01 | 9 | 9 | 1.022778335210 | 1.022778336127 ✔
0.05 | 0 | 0 | 0.973504265563 | 0.973504267703 ✔
0.05 | 0 | 1 | 0.000000000000 | 0.000000000000 ✔
0.05 | 0 | 2 | 0.000000000000 | 0.000000000000 ✔
0.05 | 0 | 3 | 0.000000000000 | 0.000000000000 ✔
0.05 | 0 | 4 | 0.000000000000 | 0.000000000000 ✔
0.05 | 0 | 5 | 0.000000000000 | 0.000000000001 ✔
0.05 | 0 | 6 | 0.000000000000 | 0.000000000001 ✔
0.05 | 0 | 7 | 0.000000000000 | 0.000000000002 ✔
0.05 | 0 | 8 | 0.000000000000 | 0.000000000002 ✔
0.05 | 0 | 9 | 0.000000000000 | 0.000000000002 ✔
0.05 | 1 | 0 | 0.000000000000 | 0.000000000000 ✔
0.05 | 1 | 1 | 1.022179478841 | 1.022179479980 ✔
0.05 | 1 | 2 | 0.000000000000 | 0.000000000000 ✔
0.05 | 1 | 3 | 0.000000000000 | 0.000000000001 ✔
0.05 | 1 | 4 | 0.000000000000 | 0.000000000002 ✔
0.05 | 1 | 5 | 0.000000000000 | 0.000000000002 ✔
0.05 | 1 | 6 | 0.000000000000 | 0.000000000002 ✔
0.05 | 1 | 7 | 0.000000000000 | 0.000000000002 ✔
0.05 | 1 | 8 | 0.000000000000 | 0.000000000004 ✔
0.05 | 1 | 9 | 0.000000000000 | 0.000000000007 ✔
0.05 | 2 | 0 | 0.000000000000 | 0.000000000000 ✔
0.05 | 2 | 1 | 0.000000000000 | 0.000000000000 ✔
0.05 | 2 | 2 | 1.047733965812 | 1.047733966390 ✔
0.05 | 2 | 3 | 0.000000000000 | 0.000000000002 ✔
0.05 | 2 | 4 | 0.000000000000 | 0.000000000002 ✔
0.05 | 2 | 5 | 0.000000000000 | 0.000000000004 ✔
0.05 | 2 | 6 | 0.000000000000 | 0.000000000004 ✔
0.05 | 2 | 7 | 0.000000000000 | 0.000000000008 ✔
0.05 | 2 | 8 | 0.000000000000 | 0.000000000008 ✔
0.05 | 2 | 9 | 0.000000000000 | 0.000000000008 ✔
0.05 | 3 | 0 | 0.000000000000 | 0.000000000000 ✔
0.05 | 3 | 1 | 0.000000000000 | 0.000000000001 ✔
0.05 | 3 | 2 | 0.000000000000 | 0.000000000002 ✔
0.05 | 3 | 3 | 1.065196198575 | 1.065196199813 ✔
0.05 | 3 | 4 | 0.000000000000 | 0.000000000004 ✔
0.05 | 3 | 5 | 0.000000000000 | 0.000000000004 ✔
0.05 | 3 | 6 | 0.000000000000 | 0.000000000008 ✔
0.05 | 3 | 7 | 0.000000000000 | 0.000000000008 ✔
0.05 | 3 | 8 | 0.000000000000 | 0.000000000016 ✔
0.05 | 3 | 9 | 0.000000000000 | 0.000000000015 ✔
0.05 | 4 | 0 | 0.000000000000 | 0.000000000000 ✔
0.05 | 4 | 1 | 0.000000000000 | 0.000000000002 ✔
0.05 | 4 | 2 | 0.000000000000 | 0.000000000002 ✔
0.05 | 4 | 3 | 0.000000000000 | 0.000000000004 ✔
0.05 | 4 | 4 | 1.078511151058 | 1.078511152326 ✔
0.05 | 4 | 5 | 0.000000000000 | 0.000000000008 ✔
0.05 | 4 | 6 | 0.000000000000 | 0.000000000008 ✔
0.05 | 4 | 7 | 0.000000000000 | 0.000000000017 ✔
0.05 | 4 | 8 | 0.000000000000 | 0.000000000017 ✔
0.05 | 4 | 9 | 0.000000000000 | 0.000000000036 ✔
0.05 | 5 | 0 | 0.000000000000 | 0.000000000001 ✔
0.05 | 5 | 1 | 0.000000000000 | 0.000000000002 ✔
0.05 | 5 | 2 | 0.000000000000 | 0.000000000004 ✔
0.05 | 5 | 3 | 0.000000000000 | 0.000000000004 ✔
0.05 | 5 | 4 | 0.000000000000 | 0.000000000008 ✔
0.05 | 5 | 5 | 1.089296262568 | 1.089296263862 ✔
0.05 | 5 | 6 | 0.000000000000 | 0.000000000017 ✔
0.05 | 5 | 7 | 0.000000000000 | 0.000000000034 ✔
0.05 | 5 | 8 | 0.000000000000 | 0.000000000035 ✔
0.05 | 5 | 9 | 0.000000000000 | 0.000000000034 ✔
0.05 | 6 | 0 | 0.000000000000 | 0.000000000001 ✔
0.05 | 6 | 1 | 0.000000000000 | 0.000000000002 ✔
0.05 | 6 | 2 | 0.000000000000 | 0.000000000004 ✔
0.05 | 6 | 3 | 0.000000000000 | 0.000000000008 ✔
0.05 | 6 | 4 | 0.000000000000 | 0.000000000008 ✔
0.05 | 6 | 5 | 0.000000000000 | 0.000000000017 ✔
0.05 | 6 | 6 | 1.098373731423 | 1.098373732739 ✔
0.05 | 6 | 7 | 0.000000000000 | 0.000000000035 ✔
0.05 | 6 | 8 | 0.000000000000 | 0.000000000035 ✔
0.05 | 6 | 9 | 0.000000000000 | 0.000000000035 ✔
0.05 | 7 | 0 | 0.000000000000 | 0.000000000002 ✔
0.05 | 7 | 1 | 0.000000000000 | 0.000000000002 ✔
0.05 | 7 | 2 | 0.000000000000 | 0.000000000008 ✔
0.05 | 7 | 3 | 0.000000000000 | 0.000000000008 ✔
0.05 | 7 | 4 | 0.000000000000 | 0.000000000017 ✔
0.05 | 7 | 5 | 0.000000000000 | 0.000000000034 ✔
0.05 | 7 | 6 | 0.000000000000 | 0.000000000035 ✔
0.05 | 7 | 7 | 1.106219258076 | 1.106219258720 ✔
0.05 | 7 | 8 | 0.000000000000 | 0.000000000035 ✔
0.05 | 7 | 9 | 0.000000000000 | 0.000000000036 ✔
0.05 | 8 | 0 | 0.000000000000 | 0.000000000002 ✔
0.05 | 8 | 1 | 0.000000000000 | 0.000000000004 ✔
0.05 | 8 | 2 | 0.000000000000 | 0.000000000008 ✔
0.05 | 8 | 3 | 0.000000000000 | 0.000000000016 ✔
0.05 | 8 | 4 | 0.000000000000 | 0.000000000017 ✔
0.05 | 8 | 5 | 0.000000000000 | 0.000000000035 ✔
0.05 | 8 | 6 | 0.000000000000 | 0.000000000035 ✔
0.05 | 8 | 7 | 0.000000000000 | 0.000000000035 ✔
0.05 | 8 | 8 | 1.113133128439 | 1.113133129790 ✔
0.05 | 8 | 9 | 0.000000000000 | 0.000000000074 ✔
0.05 | 9 | 0 | 0.000000000000 | 0.000000000002 ✔
0.05 | 9 | 1 | 0.000000000000 | 0.000000000007 ✔
0.05 | 9 | 2 | 0.000000000000 | 0.000000000008 ✔
0.05 | 9 | 3 | 0.000000000000 | 0.000000000015 ✔
0.05 | 9 | 4 | 0.000000000000 | 0.000000000036 ✔
0.05 | 9 | 5 | 0.000000000000 | 0.000000000034 ✔
0.05 | 9 | 6 | 0.000000000000 | 0.000000000035 ✔
0.05 | 9 | 7 | 0.000000000000 | 0.000000000036 ✔
0.05 | 9 | 8 | 0.000000000000 | 0.000000000074 ✔
0.05 | 9 | 9 | 1.119317201375 | 1.119317202034 ✔
0.10 | 0 | 0 | 0.951350769867 | 0.951350771636 ✔
0.10 | 0 | 1 | 0.000000000000 | 0.000000000000 ✔
0.10 | 0 | 2 | 0.000000000000 | 0.000000000000 ✔
0.10 | 0 | 3 | 0.000000000000 | 0.000000000000 ✔
0.10 | 0 | 4 | 0.000000000000 | 0.000000000000 ✔
0.10 | 0 | 5 | 0.000000000000 | 0.000000000001 ✔
0.10 | 0 | 6 | 0.000000000000 | 0.000000000001 ✔
0.10 | 0 | 7 | 0.000000000000 | 0.000000000001 ✔
0.10 | 0 | 8 | 0.000000000000 | 0.000000000001 ✔
0.10 | 0 | 9 | 0.000000000000 | 0.000000000001 ✔
0.10 | 1 | 0 | 0.000000000000 | 0.000000000000 ✔
0.10 | 1 | 1 | 1.046485846854 | 1.046485847852 ✔
0.10 | 1 | 2 | 0.000000000000 | 0.000000000001 ✔
0.10 | 1 | 3 | 0.000000000000 | 0.000000000001 ✔
0.10 | 1 | 4 | 0.000000000000 | 0.000000000001 ✔
0.10 | 1 | 5 | 0.000000000000 | 0.000000000001 ✔
0.10 | 1 | 6 | 0.000000000000 | 0.000000000001 ✔
0.10 | 1 | 7 | 0.000000000000 | 0.000000000003 ✔
0.10 | 1 | 8 | 0.000000000000 | 0.000000000003 ✔
0.10 | 1 | 9 | 0.000000000000 | 0.000000000006 ✔
0.10 | 2 | 0 | 0.000000000000 | 0.000000000000 ✔
0.10 | 2 | 1 | 0.000000000000 | 0.000000000001 ✔
0.10 | 2 | 2 | 1.098810139196 | 1.098810140297 ✔
0.10 | 2 | 3 | 0.000000000000 | 0.000000000001 ✔
0.10 | 2 | 4 | 0.000000000000 | 0.000000000001 ✔
0.10 | 2 | 5 | 0.000000000000 | 0.000000000003 ✔
0.10 | 2 | 6 | 0.000000000000 | 0.000000000003 ✔
0.10 | 2 | 7 | 0.000000000000 | 0.000000000006 ✔
0.10 | 2 | 8 | 0.000000000000 | 0.000000000006 ✔
0.10 | 2 | 9 | 0.000000000000 | 0.000000000006 ✔
0.10 | 3 | 0 | 0.000000000000 | 0.000000000000 ✔
0.10 | 3 | 1 | 0.000000000000 | 0.000000000001 ✔
0.10 | 3 | 2 | 0.000000000000 | 0.000000000001 ✔
0.10 | 3 | 3 | 1.135437143836 | 1.135437145012 ✔
0.10 | 3 | 4 | 0.000000000000 | 0.000000000006 ✔
0.10 | 3 | 5 | 0.000000000000 | 0.000000000003 ✔
0.10 | 3 | 6 | 0.000000000000 | 0.000000000006 ✔
0.10 | 3 | 7 | 0.000000000000 | 0.000000000006 ✔
0.10 | 3 | 8 | 0.000000000000 | 0.000000000013 ✔
0.10 | 3 | 9 | 0.000000000000 | 0.000000000013 ✔
0.10 | 4 | 0 | 0.000000000000 | 0.000000000000 ✔
0.10 | 4 | 1 | 0.000000000000 | 0.000000000001 ✔
0.10 | 4 | 2 | 0.000000000000 | 0.000000000001 ✔
0.10 | 4 | 3 | 0.000000000000 | 0.000000000006 ✔
0.10 | 4 | 4 | 1.163823072432 | 1.163823073667 ✔
0.10 | 4 | 5 | 0.000000000000 | 0.000000000006 ✔
0.10 | 4 | 6 | 0.000000000000 | 0.000000000013 ✔
0.10 | 4 | 7 | 0.000000000000 | 0.000000000013 ✔
0.10 | 4 | 8 | 0.000000000000 | 0.000000000014 ✔
0.10 | 4 | 9 | 0.000000000000 | 0.000000000029 ✔
0.10 | 5 | 0 | 0.000000000000 | 0.000000000001 ✔
0.10 | 5 | 1 | 0.000000000000 | 0.000000000001 ✔
0.10 | 5 | 2 | 0.000000000000 | 0.000000000003 ✔
0.10 | 5 | 3 | 0.000000000000 | 0.000000000003 ✔
0.10 | 5 | 4 | 0.000000000000 | 0.000000000006 ✔
0.10 | 5 | 5 | 1.187099533881 | 1.187099535166 ✔
0.10 | 5 | 6 | 0.000000000000 | 0.000000000013 ✔
0.10 | 5 | 7 | 0.000000000000 | 0.000000000029 ✔
0.10 | 5 | 8 | 0.000000000000 | 0.000000000030 ✔
0.10 | 5 | 9 | 0.000000000000 | 0.000000000030 ✔
0.10 | 6 | 0 | 0.000000000000 | 0.000000000001 ✔
0.10 | 6 | 1 | 0.000000000000 | 0.000000000001 ✔
0.10 | 6 | 2 | 0.000000000000 | 0.000000000003 ✔
0.10 | 6 | 3 | 0.000000000000 | 0.000000000006 ✔
0.10 | 6 | 4 | 0.000000000000 | 0.000000000013 ✔
0.10 | 6 | 5 | 0.000000000000 | 0.000000000013 ✔
0.10 | 6 | 6 | 1.206884526112 | 1.206884527440 ✔
0.10 | 6 | 7 | 0.000000000000 | 0.000000000030 ✔
0.10 | 6 | 8 | 0.000000000000 | 0.000000000030 ✔
0.10 | 6 | 9 | 0.000000000000 | 0.000000000065 ✔
0.10 | 7 | 0 | 0.000000000000 | 0.000000000001 ✔
0.10 | 7 | 1 | 0.000000000000 | 0.000000000003 ✔
0.10 | 7 | 2 | 0.000000000000 | 0.000000000006 ✔
0.10 | 7 | 3 | 0.000000000000 | 0.000000000006 ✔
0.10 | 7 | 4 | 0.000000000000 | 0.000000000013 ✔
0.10 | 7 | 5 | 0.000000000000 | 0.000000000029 ✔
0.10 | 7 | 6 | 0.000000000000 | 0.000000000030 ✔
0.10 | 7 | 7 | 1.224125733628 | 1.224125734265 ✔
0.10 | 7 | 8 | 0.000000000000 | 0.000000000066 ✔
0.10 | 7 | 9 | 0.000000000000 | 0.000000000031 ✔
0.10 | 8 | 0 | 0.000000000000 | 0.000000000001 ✔
0.10 | 8 | 1 | 0.000000000000 | 0.000000000003 ✔
0.10 | 8 | 2 | 0.000000000000 | 0.000000000006 ✔
0.10 | 8 | 3 | 0.000000000000 | 0.000000000013 ✔
0.10 | 8 | 4 | 0.000000000000 | 0.000000000014 ✔
0.10 | 8 | 5 | 0.000000000000 | 0.000000000030 ✔
0.10 | 8 | 6 | 0.000000000000 | 0.000000000030 ✔
0.10 | 8 | 7 | 0.000000000000 | 0.000000000066 ✔
0.10 | 8 | 8 | 1.239427305298 | 1.239427306699 ✔
0.10 | 8 | 9 | 0.000000000000 | 0.000000000067 ✔
0.10 | 9 | 0 | 0.000000000000 | 0.000000000001 ✔
0.10 | 9 | 1 | 0.000000000000 | 0.000000000006 ✔
0.10 | 9 | 2 | 0.000000000000 | 0.000000000006 ✔
0.10 | 9 | 3 | 0.000000000000 | 0.000000000013 ✔
0.10 | 9 | 4 | 0.000000000000 | 0.000000000029 ✔
0.10 | 9 | 5 | 0.000000000000 | 0.000000000030 ✔
0.10 | 9 | 6 | 0.000000000000 | 0.000000000065 ✔
0.10 | 9 | 7 | 0.000000000000 | 0.000000000031 ✔
0.10 | 9 | 8 | 0.000000000000 | 0.000000000067 ✔
0.10 | 9 | 9 | 1.253198719802 | 1.253198721234 ✔
0.50 | 0 | 0 | 0.886226925453 | 0.886226925863 ✔
0.50 | 0 | 1 | 0.000000000000 | 0.000000000000 ✔
0.50 | 0 | 2 | 0.000000000000 | 0.000000000000 ✔
0.50 | 0 | 3 | 0.000000000000 | 0.000000000000 ✔
0.50 | 0 | 4 | 0.000000000000 | 0.000000000000 ✔
0.50 | 0 | 5 | 0.000000000000 | 0.000000000000 ✔
0.50 | 0 | 6 | 0.000000000000 | 0.000000000000 ✔
0.50 | 0 | 7 | 0.000000000000 | 0.000000000000 ✔
0.50 | 0 | 8 | 0.000000000000 | 0.000000000000 ✔
0.50 | 0 | 9 | 0.000000000000 | -0.000000000000 ✔
0.50 | 1 | 0 | 0.000000000000 | 0.000000000000 ✔
0.50 | 1 | 1 | 1.329340388179 | 1.329340389103 ✔
0.50 | 1 | 2 | 0.000000000000 | 0.000000000000 ✔
0.50 | 1 | 3 | 0.000000000000 | 0.000000000000 ✔
0.50 | 1 | 4 | 0.000000000000 | 0.000000000000 ✔
0.50 | 1 | 5 | 0.000000000000 | 0.000000000000 ✔
0.50 | 1 | 6 | 0.000000000000 | 0.000000000000 ✔
0.50 | 1 | 7 | 0.000000000000 | 0.000000000000 ✔
0.50 | 1 | 8 | 0.000000000000 | 0.000000000000 ✔
0.50 | 1 | 9 | 0.000000000000 | 0.000000000000 ✔
0.50 | 2 | 0 | 0.000000000000 | 0.000000000000 ✔
0.50 | 2 | 1 | 0.000000000000 | 0.000000000000 ✔
0.50 | 2 | 2 | 1.661675485224 | 1.661675485734 ✔
0.50 | 2 | 3 | 0.000000000000 | 0.000000000000 ✔
0.50 | 2 | 4 | 0.000000000000 | 0.000000000000 ✔
0.50 | 2 | 5 | 0.000000000000 | 0.000000000000 ✔
0.50 | 2 | 6 | 0.000000000000 | 0.000000000000 ✔
0.50 | 2 | 7 | 0.000000000000 | 0.000000000000 ✔
0.50 | 2 | 8 | 0.000000000000 | 0.000000000000 ✔
0.50 | 2 | 9 | 0.000000000000 | -0.000000000000 ✔
0.50 | 3 | 0 | 0.000000000000 | 0.000000000000 ✔
0.50 | 3 | 1 | 0.000000000000 | 0.000000000000 ✔
0.50 | 3 | 2 | 0.000000000000 | 0.000000000000 ✔
0.50 | 3 | 3 | 1.938621399428 | 1.938621400123 ✔
0.50 | 3 | 4 | 0.000000000000 | 0.000000000000 ✔
0.50 | 3 | 5 | 0.000000000000 | 0.000000000001 ✔
0.50 | 3 | 6 | 0.000000000000 | 0.000000000001 ✔
0.50 | 3 | 7 | 0.000000000000 | 0.000000000001 ✔
0.50 | 3 | 8 | 0.000000000000 | 0.000000000001 ✔
0.50 | 3 | 9 | 0.000000000000 | 0.000000000001 ✔
0.50 | 4 | 0 | 0.000000000000 | 0.000000000000 ✔
0.50 | 4 | 1 | 0.000000000000 | 0.000000000000 ✔
0.50 | 4 | 2 | 0.000000000000 | 0.000000000000 ✔
0.50 | 4 | 3 | 0.000000000000 | 0.000000000000 ✔
0.50 | 4 | 4 | 2.180949074356 | 2.180949075236 ✔
0.50 | 4 | 5 | 0.000000000000 | 0.000000000001 ✔
0.50 | 4 | 6 | 0.000000000000 | 0.000000000001 ✔
0.50 | 4 | 7 | 0.000000000000 | 0.000000000001 ✔
0.50 | 4 | 8 | 0.000000000000 | 0.000000000001 ✔
0.50 | 4 | 9 | 0.000000000000 | 0.000000000003 ✔
0.50 | 5 | 0 | 0.000000000000 | 0.000000000000 ✔
0.50 | 5 | 1 | 0.000000000000 | 0.000000000000 ✔
0.50 | 5 | 2 | 0.000000000000 | 0.000000000000 ✔
0.50 | 5 | 3 | 0.000000000000 | 0.000000000001 ✔
0.50 | 5 | 4 | 0.000000000000 | 0.000000000001 ✔
0.50 | 5 | 5 | 2.399043981792 | 2.399043982856 ✔
0.50 | 5 | 6 | 0.000000000000 | 0.000000000001 ✔
0.50 | 5 | 7 | 0.000000000000 | 0.000000000001 ✔
0.50 | 5 | 8 | 0.000000000000 | 0.000000000003 ✔
0.50 | 5 | 9 | 0.000000000000 | 0.000000000001 ✔
0.50 | 6 | 0 | 0.000000000000 | 0.000000000000 ✔
0.50 | 6 | 1 | 0.000000000000 | 0.000000000000 ✔
0.50 | 6 | 2 | 0.000000000000 | 0.000000000000 ✔
0.50 | 6 | 3 | 0.000000000000 | 0.000000000001 ✔
0.50 | 6 | 4 | 0.000000000000 | 0.000000000001 ✔
0.50 | 6 | 5 | 0.000000000000 | 0.000000000001 ✔
0.50 | 6 | 6 | 2.598964313608 | 2.598964314050 ✔
0.50 | 6 | 7 | 0.000000000000 | 0.000000000003 ✔
0.50 | 6 | 8 | 0.000000000000 | 0.000000000003 ✔
0.50 | 6 | 9 | 0.000000000000 | 0.000000000008 ✔
0.50 | 7 | 0 | 0.000000000000 | 0.000000000000 ✔
0.50 | 7 | 1 | 0.000000000000 | 0.000000000000 ✔
0.50 | 7 | 2 | 0.000000000000 | 0.000000000000 ✔
0.50 | 7 | 3 | 0.000000000000 | 0.000000000001 ✔
0.50 | 7 | 4 | 0.000000000000 | 0.000000000001 ✔
0.50 | 7 | 5 | 0.000000000000 | 0.000000000001 ✔
0.50 | 7 | 6 | 0.000000000000 | 0.000000000003 ✔
0.50 | 7 | 7 | 2.784604621723 | 2.784604622230 ✔
0.50 | 7 | 8 | 0.000000000000 | 0.000000000008 ✔
0.50 | 7 | 9 | 0.000000000000 | 0.000000000009 ✔
0.50 | 8 | 0 | 0.000000000000 | 0.000000000000 ✔
0.50 | 8 | 1 | 0.000000000000 | 0.000000000000 ✔
0.50 | 8 | 2 | 0.000000000000 | 0.000000000000 ✔
0.50 | 8 | 3 | 0.000000000000 | 0.000000000001 ✔
0.50 | 8 | 4 | 0.000000000000 | 0.000000000001 ✔
0.50 | 8 | 5 | 0.000000000000 | 0.000000000003 ✔
0.50 | 8 | 6 | 0.000000000000 | 0.000000000003 ✔
0.50 | 8 | 7 | 0.000000000000 | 0.000000000008 ✔
0.50 | 8 | 8 | 2.958642410581 | 2.958642412199 ✔
0.50 | 8 | 9 | 0.000000000000 | 0.000000000009 ✔
0.50 | 9 | 0 | 0.000000000000 | -0.000000000000 ✔
0.50 | 9 | 1 | 0.000000000000 | 0.000000000000 ✔
0.50 | 9 | 2 | 0.000000000000 | -0.000000000000 ✔
0.50 | 9 | 3 | 0.000000000000 | 0.000000000001 ✔
0.50 | 9 | 4 | 0.000000000000 | 0.000000000003 ✔
0.50 | 9 | 5 | 0.000000000000 | 0.000000000001 ✔
0.50 | 9 | 6 | 0.000000000000 | 0.000000000008 ✔
0.50 | 9 | 7 | 0.000000000000 | 0.000000000009 ✔
0.50 | 9 | 8 | 0.000000000000 | 0.000000000009 ✔
0.50 | 9 | 9 | 3.123011433391 | 3.123011435194 ✔
1.00 | 0 | 0 | 1.000000000000 | 1.000000000000 ✔
1.00 | 0 | 1 | 0.000000000000 | -0.000000000000 ✔
1.00 | 0 | 2 | 0.000000000000 | -0.000000000000 ✔
1.00 | 0 | 3 | 0.000000000000 | 0.000000000000 ✔
1.00 | 0 | 4 | 0.000000000000 | 0.000000000000 ✔
1.00 | 0 | 5 | 0.000000000000 | 0.000000000000 ✔
1.00 | 0 | 6 | 0.000000000000 | 0.000000000000 ✔
1.00 | 0 | 7 | 0.000000000000 | 0.000000000000 ✔
1.00 | 0 | 8 | 0.000000000000 | 0.000000000000 ✔
1.00 | 0 | 9 | 0.000000000000 | 0.000000000000 ✔
1.00 | 1 | 0 | 0.000000000000 | -0.000000000000 ✔
1.00 | 1 | 1 | 2.000000000000 | 2.000000000000 ✔
1.00 | 1 | 2 | 0.000000000000 | 0.000000000000 ✔
1.00 | 1 | 3 | 0.000000000000 | -0.000000000000 ✔
1.00 | 1 | 4 | 0.000000000000 | -0.000000000000 ✔
1.00 | 1 | 5 | 0.000000000000 | -0.000000000000 ✔
1.00 | 1 | 6 | 0.000000000000 | -0.000000000000 ✔
1.00 | 1 | 7 | 0.000000000000 | 0.000000000000 ✔
1.00 | 1 | 8 | 0.000000000000 | -0.000000000000 ✔
1.00 | 1 | 9 | 0.000000000000 | -0.000000000000 ✔
1.00 | 2 | 0 | 0.000000000000 | -0.000000000000 ✔
1.00 | 2 | 1 | 0.000000000000 | 0.000000000000 ✔
1.00 | 2 | 2 | 3.000000000000 | 3.000000000000 ✔
1.00 | 2 | 3 | 0.000000000000 | 0.000000000000 ✔
1.00 | 2 | 4 | 0.000000000000 | 0.000000000000 ✔
1.00 | 2 | 5 | 0.000000000000 | -0.000000000000 ✔
1.00 | 2 | 6 | 0.000000000000 | 0.000000000000 ✔
1.00 | 2 | 7 | 0.000000000000 | 0.000000000000 ✔
1.00 | 2 | 8 | 0.000000000000 | 0.000000000000 ✔
1.00 | 2 | 9 | 0.000000000000 | 0.000000000000 ✔
1.00 | 3 | 0 | 0.000000000000 | 0.000000000000 ✔
1.00 | 3 | 1 | 0.000000000000 | -0.000000000000 ✔
1.00 | 3 | 2 | 0.000000000000 | 0.000000000000 ✔
1.00 | 3 | 3 | 4.000000000000 | 4.000000000000 ✔
1.00 | 3 | 4 | 0.000000000000 | 0.000000000000 ✔
1.00 | 3 | 5 | 0.000000000000 | 0.000000000000 ✔
1.00 | 3 | 6 | 0.000000000000 | 0.000000000000 ✔
1.00 | 3 | 7 | 0.000000000000 | 0.000000000000 ✔
1.00 | 3 | 8 | 0.000000000000 | -0.000000000000 ✔
1.00 | 3 | 9 | 0.000000000000 | -0.000000000001 ✔
1.00 | 4 | 0 | 0.000000000000 | 0.000000000000 ✔
1.00 | 4 | 1 | 0.000000000000 | -0.000000000000 ✔
1.00 | 4 | 2 | 0.000000000000 | 0.000000000000 ✔
1.00 | 4 | 3 | 0.000000000000 | 0.000000000000 ✔
1.00 | 4 | 4 | 5.000000000000 | 4.999999999999 ✔
1.00 | 4 | 5 | 0.000000000000 | -0.000000000000 ✔
1.00 | 4 | 6 | 0.000000000000 | -0.000000000000 ✔
1.00 | 4 | 7 | 0.000000000000 | -0.000000000000 ✔
1.00 | 4 | 8 | 0.000000000000 | 0.000000000000 ✔
1.00 | 4 | 9 | 0.000000000000 | 0.000000000000 ✔
1.00 | 5 | 0 | 0.000000000000 | 0.000000000000 ✔
1.00 | 5 | 1 | 0.000000000000 | -0.000000000000 ✔
1.00 | 5 | 2 | 0.000000000000 | -0.000000000000 ✔
1.00 | 5 | 3 | 0.000000000000 | 0.000000000000 ✔
1.00 | 5 | 4 | 0.000000000000 | -0.000000000000 ✔
1.00 | 5 | 5 | 6.000000000000 | 6.000000000000 ✔
1.00 | 5 | 6 | 0.000000000000 | -0.000000000000 ✔
1.00 | 5 | 7 | 0.000000000000 | -0.000000000000 ✔
1.00 | 5 | 8 | 0.000000000000 | -0.000000000000 ✔
1.00 | 5 | 9 | 0.000000000000 | -0.000000000000 ✔
1.00 | 6 | 0 | 0.000000000000 | 0.000000000000 ✔
1.00 | 6 | 1 | 0.000000000000 | -0.000000000000 ✔
1.00 | 6 | 2 | 0.000000000000 | 0.000000000000 ✔
1.00 | 6 | 3 | 0.000000000000 | 0.000000000000 ✔
1.00 | 6 | 4 | 0.000000000000 | -0.000000000000 ✔
1.00 | 6 | 5 | 0.000000000000 | -0.000000000000 ✔
1.00 | 6 | 6 | 7.000000000000 | 7.000000000000 ✔
1.00 | 6 | 7 | 0.000000000000 | 0.000000000000 ✔
1.00 | 6 | 8 | 0.000000000000 | 0.000000000000 ✔
1.00 | 6 | 9 | 0.000000000000 | 0.000000000000 ✔
1.00 | 7 | 0 | 0.000000000000 | 0.000000000000 ✔
1.00 | 7 | 1 | 0.000000000000 | 0.000000000000 ✔
1.00 | 7 | 2 | 0.000000000000 | 0.000000000000 ✔
1.00 | 7 | 3 | 0.000000000000 | 0.000000000000 ✔
1.00 | 7 | 4 | 0.000000000000 | -0.000000000000 ✔
1.00 | 7 | 5 | 0.000000000000 | -0.000000000000 ✔
1.00 | 7 | 6 | 0.000000000000 | 0.000000000000 ✔
1.00 | 7 | 7 | 8.000000000000 | 8.000000000000 ✔
1.00 | 7 | 8 | 0.000000000000 | 0.000000000000 ✔
1.00 | 7 | 9 | 0.000000000000 | -0.000000000000 ✔
1.00 | 8 | 0 | 0.000000000000 | 0.000000000000 ✔
1.00 | 8 | 1 | 0.000000000000 | -0.000000000000 ✔
1.00 | 8 | 2 | 0.000000000000 | 0.000000000000 ✔
1.00 | 8 | 3 | 0.000000000000 | -0.000000000000 ✔
1.00 | 8 | 4 | 0.000000000000 | 0.000000000000 ✔
1.00 | 8 | 5 | 0.000000000000 | -0.000000000000 ✔
1.00 | 8 | 6 | 0.000000000000 | 0.000000000000 ✔
1.00 | 8 | 7 | 0.000000000000 | 0.000000000000 ✔
1.00 | 8 | 8 | 9.000000000000 | 9.000000000000 ✔
1.00 | 8 | 9 | 0.000000000000 | -0.000000000000 ✔
1.00 | 9 | 0 | 0.000000000000 | 0.000000000000 ✔
1.00 | 9 | 1 | 0.000000000000 | -0.000000000000 ✔
1.00 | 9 | 2 | 0.000000000000 | 0.000000000000 ✔
1.00 | 9 | 3 | 0.000000000000 | -0.000000000001 ✔
1.00 | 9 | 4 | 0.000000000000 | 0.000000000000 ✔
1.00 | 9 | 5 | 0.000000000000 | -0.000000000000 ✔
1.00 | 9 | 6 | 0.000000000000 | 0.000000000000 ✔
1.00 | 9 | 7 | 0.000000000000 | -0.000000000000 ✔
1.00 | 9 | 8 | 0.000000000000 | -0.000000000000 ✔
1.00 | 9 | 9 | 10.000000000000 | 10.000000000002 ✔Normalization & Orthogonality of $\psi_n(r)$
\[\int_0^\infty \psi_i^\ast(r) \psi_j(r) \mathrm{d}r = \delta_{ij}\]
i | j | analytical | numerical
-- | -- | ----------------- | -----------------
0 | 0 | 1.000000000000 | 1.000000000000 ✔
0 | 1 | 0.000000000000 | 0.000000000000 ✔
0 | 2 | 0.000000000000 | -0.000000000000 ✔
0 | 3 | 0.000000000000 | 0.000000000000 ✔
0 | 4 | 0.000000000000 | 0.000000000000 ✔
0 | 5 | 0.000000000000 | -0.000000000000 ✔
0 | 6 | 0.000000000000 | -0.000000000000 ✔
0 | 7 | 0.000000000000 | 0.000000000002 ✔
0 | 8 | 0.000000000000 | -0.000000000026 ✔
0 | 9 | 0.000000000000 | -0.000000000104 ✔
1 | 0 | 0.000000000000 | 0.000000000000 ✔
1 | 1 | 1.000000000000 | 1.000000000000 ✔
1 | 2 | 0.000000000000 | -0.000000000000 ✔
1 | 3 | 0.000000000000 | 0.000000000000 ✔
1 | 4 | 0.000000000000 | 0.000000000000 ✔
1 | 5 | 0.000000000000 | -0.000000000000 ✔
1 | 6 | 0.000000000000 | 0.000000000000 ✔
1 | 7 | 0.000000000000 | 0.000000000001 ✔
1 | 8 | 0.000000000000 | -0.000000000022 ✔
1 | 9 | 0.000000000000 | -0.000000000067 ✔
2 | 0 | 0.000000000000 | -0.000000000000 ✔
2 | 1 | 0.000000000000 | -0.000000000000 ✔
2 | 2 | 1.000000000000 | 1.000000000000 ✔
2 | 3 | 0.000000000000 | -0.000000000000 ✔
2 | 4 | 0.000000000000 | 0.000000000000 ✔
2 | 5 | 0.000000000000 | -0.000000000000 ✔
2 | 6 | 0.000000000000 | 0.000000000000 ✔
2 | 7 | 0.000000000000 | 0.000000000000 ✔
2 | 8 | 0.000000000000 | -0.000000000009 ✔
2 | 9 | 0.000000000000 | -0.000000000030 ✔
3 | 0 | 0.000000000000 | 0.000000000000 ✔
3 | 1 | 0.000000000000 | 0.000000000000 ✔
3 | 2 | 0.000000000000 | -0.000000000000 ✔
3 | 3 | 1.000000000000 | 1.000000000000 ✔
3 | 4 | 0.000000000000 | -0.000000000000 ✔
3 | 5 | 0.000000000000 | -0.000000000000 ✔
3 | 6 | 0.000000000000 | 0.000000000000 ✔
3 | 7 | 0.000000000000 | -0.000000000001 ✔
3 | 8 | 0.000000000000 | -0.000000000002 ✔
3 | 9 | 0.000000000000 | -0.000000000006 ✔
4 | 0 | 0.000000000000 | 0.000000000000 ✔
4 | 1 | 0.000000000000 | 0.000000000000 ✔
4 | 2 | 0.000000000000 | 0.000000000000 ✔
4 | 3 | 0.000000000000 | -0.000000000000 ✔
4 | 4 | 1.000000000000 | 1.000000000000 ✔
4 | 5 | 0.000000000000 | 0.000000000000 ✔
4 | 6 | 0.000000000000 | 0.000000000000 ✔
4 | 7 | 0.000000000000 | 0.000000000000 ✔
4 | 8 | 0.000000000000 | -0.000000000001 ✔
4 | 9 | 0.000000000000 | 0.000000000001 ✔
5 | 0 | 0.000000000000 | -0.000000000000 ✔
5 | 1 | 0.000000000000 | -0.000000000000 ✔
5 | 2 | 0.000000000000 | -0.000000000000 ✔
5 | 3 | 0.000000000000 | -0.000000000000 ✔
5 | 4 | 0.000000000000 | 0.000000000000 ✔
5 | 5 | 1.000000000000 | 1.000000000000 ✔
5 | 6 | 0.000000000000 | -0.000000000000 ✔
5 | 7 | 0.000000000000 | -0.000000000001 ✔
5 | 8 | 0.000000000000 | 0.000000000000 ✔
5 | 9 | 0.000000000000 | -0.000000000001 ✔
6 | 0 | 0.000000000000 | -0.000000000000 ✔
6 | 1 | 0.000000000000 | 0.000000000000 ✔
6 | 2 | 0.000000000000 | 0.000000000000 ✔
6 | 3 | 0.000000000000 | 0.000000000000 ✔
6 | 4 | 0.000000000000 | 0.000000000000 ✔
6 | 5 | 0.000000000000 | -0.000000000000 ✔
6 | 6 | 1.000000000000 | 1.000000000000 ✔
6 | 7 | 0.000000000000 | -0.000000000000 ✔
6 | 8 | 0.000000000000 | -0.000000000002 ✔
6 | 9 | 0.000000000000 | -0.000000000003 ✔
7 | 0 | 0.000000000000 | 0.000000000002 ✔
7 | 1 | 0.000000000000 | 0.000000000001 ✔
7 | 2 | 0.000000000000 | 0.000000000000 ✔
7 | 3 | 0.000000000000 | -0.000000000001 ✔
7 | 4 | 0.000000000000 | 0.000000000000 ✔
7 | 5 | 0.000000000000 | -0.000000000001 ✔
7 | 6 | 0.000000000000 | -0.000000000000 ✔
7 | 7 | 1.000000000000 | 1.000000000000 ✔
7 | 8 | 0.000000000000 | -0.000000000000 ✔
7 | 9 | 0.000000000000 | 0.000000000004 ✔
8 | 0 | 0.000000000000 | -0.000000000026 ✔
8 | 1 | 0.000000000000 | -0.000000000022 ✔
8 | 2 | 0.000000000000 | -0.000000000009 ✔
8 | 3 | 0.000000000000 | -0.000000000002 ✔
8 | 4 | 0.000000000000 | -0.000000000001 ✔
8 | 5 | 0.000000000000 | 0.000000000000 ✔
8 | 6 | 0.000000000000 | -0.000000000002 ✔
8 | 7 | 0.000000000000 | -0.000000000000 ✔
8 | 8 | 1.000000000000 | 0.999999999995 ✔
8 | 9 | 0.000000000000 | 0.000000000000 ✔
9 | 0 | 0.000000000000 | -0.000000000104 ✔
9 | 1 | 0.000000000000 | -0.000000000067 ✔
9 | 2 | 0.000000000000 | -0.000000000030 ✔
9 | 3 | 0.000000000000 | -0.000000000006 ✔
9 | 4 | 0.000000000000 | 0.000000000001 ✔
9 | 5 | 0.000000000000 | -0.000000000001 ✔
9 | 6 | 0.000000000000 | -0.000000000003 ✔
9 | 7 | 0.000000000000 | 0.000000000004 ✔
9 | 8 | 0.000000000000 | 0.000000000000 ✔
9 | 9 | 1.000000000000 | 1.000000000015 ✔Eigen Values
\[ \begin{aligned} E_n &= \int \psi^\ast_n(r) \hat{H} \psi_n(r) \mathrm{d}x \\ &= \int \psi^\ast_n(r) \left[ \hat{V} + \hat{T} \right] \psi(r) \mathrm{d}x \\ &= \int \psi^\ast_n(r) \left[ V(r) - \frac{\hbar^2}{2m} \frac{\mathrm{d}^{2}}{\mathrm{d} r^{2}} \right] \psi(r) \mathrm{d}x \\ &\simeq \int \psi^\ast_n(r) \left[ V(r)\psi(r) -\frac{\hbar^2}{2m} \frac{\psi(r+\Delta r) - 2\psi(r) + \psi(r-\Delta r)}{\Delta r^{2}} \right] \mathrm{d}x. \end{aligned}\]
Where, the difference formula for the 2nd-order derivative:
\[\begin{aligned} % 2\psi(r) % + \frac{\mathrm{d}^{2} \psi(r)}{\mathrm{d} r^{2}} \Delta r^{2} % + O\left(\Delta r^{4}\right) % &= % \psi(r+\Delta r) % + \psi(r-\Delta r) % \\ % \frac{\mathrm{d}^{2} \psi(r)}{\mathrm{d} r^{2}} \Delta r^{2} % &= % \psi(r+\Delta r) % - 2\psi(r) % + \psi(r-\Delta r) % - O\left(\Delta r^{4}\right) % \\ % \frac{\mathrm{d}^{2} \psi(r)}{\mathrm{d} r^{2}} % &= % \frac{\psi(r+\Delta r) - 2\psi(r) + \psi(r-\Delta r)}{\Delta r^{2}} % - \frac{O\left(\Delta r^{4}\right)}{\Delta r^{2}} % \\ \frac{\mathrm{d}^{2} \psi(r)}{\mathrm{d} r^{2}} &= \frac{\psi(r+\Delta r) - 2\psi(r) + \psi(r-\Delta r)}{\Delta r^{2}} + O\left(\Delta r^{2}\right) \end{aligned}\]
are given by the sum of 2 Taylor series:
\[\begin{aligned} \psi(r+\Delta r) &= \psi(r) + \frac{\mathrm{d} \psi(r)}{\mathrm{d} r} \Delta r + \frac{1}{2!} \frac{\mathrm{d}^{2} \psi(r)}{\mathrm{d} r^{2}} \Delta r^{2} + \frac{1}{3!} \frac{\mathrm{d}^{3} \psi(r)}{\mathrm{d} r^{3}} \Delta r^{3} + O\left(\Delta r^{4}\right), \\ \psi(r-\Delta r) &= \psi(r) - \frac{\mathrm{d} \psi(r)}{\mathrm{d} r} \Delta r + \frac{1}{2!} \frac{\mathrm{d}^{2} \psi(r)}{\mathrm{d} r^{2}} \Delta r^{2} - \frac{1}{3!} \frac{\mathrm{d}^{3} \psi(r)}{\mathrm{d} r^{3}} \Delta r^{3} + O\left(\Delta r^{4}\right). \end{aligned}\]
k | n | analytical | numerical
--- | -- | ----------------- | -----------------
0.1 | 0 | -0.097482629904 | -0.097482629943 ✔
0.1 | 1 | -0.087576629073 | -0.087576629208 ✔
0.1 | 2 | -0.078201265005 | -0.078201265359 ✔
0.1 | 3 | -0.069356537702 | -0.069356538266 ✔
0.1 | 4 | -0.061042447162 | -0.061042448777 ✔
0.1 | 5 | -0.053258993386 | -0.053258996131 ✔
0.1 | 6 | -0.046006176374 | -0.046006177829 ✔
0.1 | 7 | -0.039283996126 | -0.039283997743 ✔
0.1 | 8 | -0.033092452642 | -0.033092467851 ✔
0.1 | 9 | -0.027431545922 | -0.027431467792 ✔
0.2 | 0 | -0.095387461081 | -0.095387461144 ✔
0.2 | 1 | -0.081689100176 | -0.081689100427 ✔
0.2 | 2 | -0.069052012799 | -0.069052013380 ✔
0.2 | 3 | -0.057476198949 | -0.057476199867 ✔
0.2 | 4 | -0.046961658628 | -0.046961660317 ✔
0.2 | 5 | -0.037508391834 | -0.037508393202 ✔
0.2 | 6 | -0.029116398568 | -0.029116400340 ✔
0.2 | 7 | -0.021785678830 | -0.021785684062 ✔
0.2 | 8 | -0.015516232619 | -0.015516237539 ✔
0.2 | 9 | -0.010308059937 | -0.010308062755 ✔
0.3 | 0 | -0.093795214605 | -0.093795214695 ✔
0.3 | 1 | -0.077310338322 | -0.077310338694 ✔
0.3 | 2 | -0.062417372330 | -0.062417373167 ✔
0.3 | 3 | -0.049116316630 | -0.049116318029 ✔
0.3 | 4 | -0.037407171221 | -0.037407173073 ✔
0.3 | 5 | -0.027289936105 | -0.027289938027 ✔
0.3 | 6 | -0.018764611280 | -0.018764613693 ✔
0.3 | 7 | -0.011831196747 | -0.011831198102 ✔
0.3 | 8 | -0.006489692505 | -0.006489694275 ✔
0.3 | 9 | -0.002740098556 | -0.002740100893 ✔
0.1 | 0 | -0.097413777944 | -0.097413777967 ✔
0.1 | 1 | -0.087380924068 | -0.087380924205 ✔
0.1 | 2 | -0.077893174789 | -0.077893175145 ✔
0.1 | 3 | -0.068950530107 | -0.068950530660 ✔
0.1 | 4 | -0.060552990023 | -0.060552989095 ✔
0.1 | 5 | -0.052700554537 | -0.052700557255 ✔
0.1 | 6 | -0.045393223648 | -0.045393222818 ✔
0.1 | 7 | -0.038630997356 | -0.038631017157 ✔
0.1 | 8 | -0.032413875662 | -0.032413886246 ✔
0.1 | 9 | -0.026741858566 | -0.026742018376 ✔