Morse Potential

The Morse potential is a model for inter-nuclear anharmonic vibration in a diatomic molecule.

Definitions

Antique.MorsePotential — Type

Model

This model is described with the time-independent Schrödinger equation

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

and the Hamiltonian

\[ \hat{H} = - \frac{\hbar^2}{2\mu} \frac{\mathrm{d}^2}{\mathrm{d}r ^2} + D_\mathrm{e} \left( \mathrm{e}^{-2a(r-r_e)} - 2\mathrm{e}^{-a(r-r_e)} \right),\]

where $a = \sqrt{\frac{k}{2Dₑ}}$ is defined. Parameters are specified with the following struct:

MP = 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).

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, 18.5.17_5

source

Potential

Antique.V — Method

V(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 $0\leq r \lt \infty$.

source

Eigenvalues

Antique.E — Method

E(model::MorsePotential; n::Int=0, nocheck=false)

\[E_n = - D_\mathrm{e} + \hbar \omega \left( n + \frac{1}{2} \right) - \chi \hbar \omega \left( n + \frac{1}{2} \right)^2,\]

where $\omega = \sqrt{k/µ}$ and $\chi = \frac{\hbar\omega}{4D_\mathrm{e}}$ are defined.

source

Maximum Quantum Number

Antique.nₘₐₓ — Method

nₘₐₓ(model::MorsePotential)

Note

Note that the number of bound states is equal to the maximum quantum number nₘₐₓ, since we count the ground state from n=1 in this model.

\[n_\mathrm{max} = \left\lfloor \frac{2 D_e - \omega}{\omega} \right\rfloor,\]

where $\omega = \sqrt{k/µ}$ is defined.

source

Eigenfunctions

Antique.ψ — Method

ψ(model::MorsePotential, r; n::Int=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 $0\leq r \lt \infty$.

source

Generalized Laguerre Polynomials

Antique.L — Method

L(model::MorsePotential, x; n=0, α=0)

Note

The generalized Laguerre polynomials $L_n^{(\alpha)}(x)$, not the associated Laguerre polynomials $L_n^{k}(x)$, are used in this model.

Rodrigues' formula & closed-form:

\[\begin{aligned} L_n^{(\alpha)}(x) &= \frac{x^{-\alpha}e^x}{n!} \frac{d^n}{dx^n}\left(x^{n+\alpha}e^{-x}\right) \\ &= \sum_{k=0}^n(-1)^k \left(\begin{array}{l} n+\alpha \\ n-k \end{array}\right) \frac{x^k}{k !} \\ &= \sum_{k=0}^n(-1)^k \frac{\Gamma(\alpha+n+1)}{\Gamma(\alpha+k+1)\Gamma(n-k+1)} \frac{x^k}{k !}. \end{aligned}\]

Examples:

\[\begin{aligned} L_0^{(0)}(x) &= 1, \\ L_1^{(0)}(x) &= 1 - x, \\ L_1^{(1)}(x) &= 2 - x, \\ L_2^{(0)}(x) &= 1 - 2 x + 1/2 x^{2}, \\ L_2^{(1)}(x) &= 3 - 3 x + 1/2 x^{2}, \\ L_2^{(2)}(x) &= 6 - 4 x + 1/2 x^{2}, \\ L_3^{(0)}(x) &= 1 - 3 x + 3/2 x^{2} - 1/6 x^{3}, \\ L_3^{(1)}(x) &= 4 - 6 x + 2 x^{2} - 1/6 x^{3}, \\ L_3^{(2)}(x) &= 10 - 10 x + 5/2 x^{2} - 1/6 x^{3}, \\ L_3^{(3)}(x) &= 20 - 15 x + 3 x^{2} - 1/6 x^{3}, \\ L_4^{(0)}(x) &= 1 - 4 x + 3 x^{2} - 2/3 x^{3} + 1/24 x^{4}, \\ L_4^{(1)}(x) &= 5 - 10 x + 5 x^{2} - 5/6 x^{3} + 1/24 x^{4}, \\ L_4^{(2)}(x) &= 15 - 20 x + 15/2 x^{2} - 1 x^{3} + 1/24 x^{4}, \\ L_4^{(3)}(x) &= 35 - 35 x + 21/2 x^{2} - 7/6 x^{3} + 1/24 x^{4}, \\ L_4^{(4)}(x) &= 70 - 56 x + 14 x^{2} - 4/3 x^{3} + 1/24 x^{4}, \\ \vdots \end{aligned}\]

source

Usage & Examples

Install Antique.jl for the first use and run using Antique before each use. The energy E(), wave function ψ(), potential V() and nₘₐₓ() will be exported. 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.rₑ1.997193319969992
julia> MP.Dₑ0.10263461910653993
julia> MP.k0.1027265041900817
julia> MP.µ918.076336715
julia> MP.ℏ1.0

Maximum quantum number:

julia> nₘₐₓ(MP)18

Eigenvalues:

julia> E(MP, n=0)-0.09741377794418261
julia> 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
fig = Figure()
axis = Axis(fig[1,1], xlabel=L"$r$", ylabel=L"$\psi(r)$")

# plot
lines!(axis, 0..5, x -> ψ(MP, x, n=0), label=L"n=0")
lines!(axis, 0..5, x -> ψ(MP, x, n=1), label=L"n=1")
lines!(axis, 0..5, x -> ψ(MP, x, n=2), label=L"n=2")
lines!(axis, 0..5, x -> ψ(MP, x, n=3), label=L"n=3")
lines!(axis, 0..5, x -> ψ(MP, x, n=4), label=L"n=4")

# legend
axislegend(axis, position=:lb, framevisible=false)

fig

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)

@show 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)

f

nₘₐₓ(MP) = 5

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 a computer algebra system (Symbolics.jl) and numerical integration (QuadGK.jl). The test script is here.

Generalized Laguerre Polynomials $L_n^{(\alpha)}(x)$

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

$n=0, α=0:$ ✔

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

$n=1, α=0:$ ✔

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

$n=1, α=1:$ ✔

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

$n=2, α=0:$ ✔

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

$n=2, α=1:$ ✔

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

$n=2, α=2:$ ✔

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

$n=3, α=0:$ ✔

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

$n=3, α=1:$ ✔

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

$n=3, α=2:$ ✔

\[\begin{aligned} L_{3}^{(2)}(x) = \frac{\frac{1}{6} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} x^{5} e^{ - x} e^{x}}{x^{2}} &= 10 - 10 x + \frac{5}{2} x^{2} - \frac{1}{6} x^{3} \\ &= 10 - 10 x + \frac{5}{2} x^{2} - \frac{1}{6} x^{3} \end{aligned}\]

$n=3, α=3:$ ✔

\[\begin{aligned} L_{3}^{(3)}(x) = \frac{\frac{1}{6} e^{x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} x^{6} e^{ - x}}{x^{3}} &= 20 - 15 x + 3 x^{2} - \frac{1}{6} x^{3} \\ &= 20 - 15 x + 3 x^{2} - \frac{1}{6} x^{3} \end{aligned}\]

$n=4, α=0:$ ✔

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

$n=4, α=1:$ ✔

\[\begin{aligned} L_{4}^{(1)}(x) = \frac{\frac{1}{24} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} x^{5} e^{ - x} e^{x}}{x} &= 5 - 10 x + 5 x^{2} - \frac{5}{6} x^{3} + \frac{1}{24} x^{4} \\ &= 5 - 10 x + 5 x^{2} - \frac{5}{6} x^{3} + \frac{1}{24} x^{4} \end{aligned}\]

$n=4, α=2:$ ✔

\[\begin{aligned} L_{4}^{(2)}(x) = \frac{\frac{1}{24} e^{x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} x^{6} e^{ - x}}{x^{2}} &= 15 - 20 x + \frac{15}{2} x^{2} - x^{3} + \frac{1}{24} x^{4} \\ &= 15 - 20 x + \frac{15}{2} x^{2} - x^{3} + \frac{1}{24} x^{4} \end{aligned}\]

$n=4, α=3:$ ✔

\[\begin{aligned} L_{4}^{(3)}(x) = \frac{\frac{1}{24} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} x^{7} e^{ - x} e^{x}}{x^{3}} &= 35 - 35 x + \frac{21}{2} x^{2} - \frac{7}{6} x^{3} + \frac{1}{24} x^{4} \\ &= 35 - 35 x + \frac{21}{2} x^{2} - \frac{7}{6} x^{3} + \frac{1}{24} x^{4} \end{aligned}\]

$n=4, α=4:$ ✔

\[\begin{aligned} L_{4}^{(4)}(x) = \frac{\frac{1}{24} e^{x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} x^{8} e^{ - x}}{x^{4}} &= 70 - 56 x + 14 x^{2} - \frac{4}{3} x^{3} + \frac{1}{24} x^{4} \\ &= 70 - 56 x + 14 x^{2} - \frac{4}{3} x^{3} + \frac{1}{24} x^{4} \end{aligned}\]

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

\[\int_0^\infty L_i^{(\alpha)}(x) L_j^{(\alpha)}(x) x^\alpha \mathrm{e}^{-x} \mathrm{d}x = \frac{\Gamma(n+\alpha+1)}{n!} \delta_{ij}\]

   α |  i |  j |     analytical |      numerical 
---- | -- | -- | -------------- | -------------- 
0.01 |  0 |  0 |    0.994325851 |    0.994325853 ✔
0.01 |  0 |  1 |    0.000000000 |    0.000000000 ✔
0.01 |  0 |  2 |    0.000000000 |    0.000000000 ✔
0.01 |  0 |  3 |    0.000000000 |    0.000000000 ✔
0.01 |  0 |  4 |    0.000000000 |    0.000000000 ✔
0.01 |  0 |  5 |    0.000000000 |    0.000000000 ✔
0.01 |  0 |  6 |    0.000000000 |    0.000000000 ✔
0.01 |  0 |  7 |    0.000000000 |    0.000000000 ✔
0.01 |  0 |  8 |    0.000000000 |    0.000000000 ✔
0.01 |  0 |  9 |    0.000000000 |    0.000000000 ✔
0.01 |  1 |  0 |    0.000000000 |    0.000000000 ✔
0.01 |  1 |  1 |    1.004269110 |    1.004269111 ✔
0.01 |  1 |  2 |    0.000000000 |    0.000000000 ✔
0.01 |  1 |  3 |    0.000000000 |    0.000000000 ✔
0.01 |  1 |  4 |    0.000000000 |    0.000000000 ✔
0.01 |  1 |  5 |    0.000000000 |    0.000000000 ✔
0.01 |  1 |  6 |    0.000000000 |    0.000000000 ✔
0.01 |  1 |  7 |    0.000000000 |    0.000000000 ✔
0.01 |  1 |  8 |    0.000000000 |    0.000000000 ✔
0.01 |  1 |  9 |    0.000000000 |    0.000000000 ✔
0.01 |  2 |  0 |    0.000000000 |    0.000000000 ✔
0.01 |  2 |  1 |    0.000000000 |    0.000000000 ✔
0.01 |  2 |  2 |    1.009290455 |    1.009290456 ✔
0.01 |  2 |  3 |    0.000000000 |    0.000000000 ✔
0.01 |  2 |  4 |    0.000000000 |    0.000000000 ✔
0.01 |  2 |  5 |    0.000000000 |    0.000000000 ✔
0.01 |  2 |  6 |    0.000000000 |    0.000000000 ✔
0.01 |  2 |  7 |    0.000000000 |    0.000000000 ✔
0.01 |  2 |  8 |    0.000000000 |    0.000000000 ✔
0.01 |  2 |  9 |    0.000000000 |    0.000000000 ✔
0.01 |  3 |  0 |    0.000000000 |    0.000000000 ✔
0.01 |  3 |  1 |    0.000000000 |    0.000000000 ✔
0.01 |  3 |  2 |    0.000000000 |    0.000000000 ✔
0.01 |  3 |  3 |    1.012654757 |    1.012654759 ✔
0.01 |  3 |  4 |    0.000000000 |    0.000000000 ✔
0.01 |  3 |  5 |    0.000000000 |    0.000000000 ✔
0.01 |  3 |  6 |    0.000000000 |    0.000000000 ✔
0.01 |  3 |  7 |    0.000000000 |    0.000000000 ✔
0.01 |  3 |  8 |    0.000000000 |    0.000000000 ✔
0.01 |  3 |  9 |    0.000000000 |    0.000000000 ✔
0.01 |  4 |  0 |    0.000000000 |    0.000000000 ✔
0.01 |  4 |  1 |    0.000000000 |    0.000000000 ✔
0.01 |  4 |  2 |    0.000000000 |    0.000000000 ✔
0.01 |  4 |  3 |    0.000000000 |    0.000000000 ✔
0.01 |  4 |  4 |    1.015186394 |    1.015186395 ✔
0.01 |  4 |  5 |    0.000000000 |    0.000000000 ✔
0.01 |  4 |  6 |    0.000000000 |    0.000000000 ✔
0.01 |  4 |  7 |    0.000000000 |    0.000000000 ✔
0.01 |  4 |  8 |    0.000000000 |    0.000000000 ✔
0.01 |  4 |  9 |    0.000000000 |    0.000000000 ✔
0.01 |  5 |  0 |    0.000000000 |    0.000000000 ✔
0.01 |  5 |  1 |    0.000000000 |    0.000000000 ✔
0.01 |  5 |  2 |    0.000000000 |    0.000000000 ✔
0.01 |  5 |  3 |    0.000000000 |    0.000000000 ✔
0.01 |  5 |  4 |    0.000000000 |    0.000000000 ✔
0.01 |  5 |  5 |    1.017216766 |    1.017216768 ✔
0.01 |  5 |  6 |    0.000000000 |    0.000000000 ✔
0.01 |  5 |  7 |    0.000000000 |    0.000000000 ✔
0.01 |  5 |  8 |    0.000000000 |    0.000000000 ✔
0.01 |  5 |  9 |    0.000000000 |    0.000000000 ✔
0.01 |  6 |  0 |    0.000000000 |    0.000000000 ✔
0.01 |  6 |  1 |    0.000000000 |    0.000000000 ✔
0.01 |  6 |  2 |    0.000000000 |    0.000000000 ✔
0.01 |  6 |  3 |    0.000000000 |    0.000000000 ✔
0.01 |  6 |  4 |    0.000000000 |    0.000000000 ✔
0.01 |  6 |  5 |    0.000000000 |    0.000000000 ✔
0.01 |  6 |  6 |    1.018912128 |    1.018912129 ✔
0.01 |  6 |  7 |    0.000000000 |    0.000000000 ✔
0.01 |  6 |  8 |    0.000000000 |    0.000000000 ✔
0.01 |  6 |  9 |    0.000000000 |    0.000000000 ✔
0.01 |  7 |  0 |    0.000000000 |    0.000000000 ✔
0.01 |  7 |  1 |    0.000000000 |    0.000000000 ✔
0.01 |  7 |  2 |    0.000000000 |    0.000000000 ✔
0.01 |  7 |  3 |    0.000000000 |    0.000000000 ✔
0.01 |  7 |  4 |    0.000000000 |    0.000000000 ✔
0.01 |  7 |  5 |    0.000000000 |    0.000000000 ✔
0.01 |  7 |  6 |    0.000000000 |    0.000000000 ✔
0.01 |  7 |  7 |    1.020367716 |    1.020367717 ✔
0.01 |  7 |  8 |    0.000000000 |    0.000000000 ✔
0.01 |  7 |  9 |    0.000000000 |    0.000000000 ✔
0.01 |  8 |  0 |    0.000000000 |    0.000000000 ✔
0.01 |  8 |  1 |    0.000000000 |    0.000000000 ✔
0.01 |  8 |  2 |    0.000000000 |    0.000000000 ✔
0.01 |  8 |  3 |    0.000000000 |    0.000000000 ✔
0.01 |  8 |  4 |    0.000000000 |    0.000000000 ✔
0.01 |  8 |  5 |    0.000000000 |    0.000000000 ✔
0.01 |  8 |  6 |    0.000000000 |    0.000000000 ✔
0.01 |  8 |  7 |    0.000000000 |    0.000000000 ✔
0.01 |  8 |  8 |    1.021643176 |    1.021643178 ✔
0.01 |  8 |  9 |    0.000000000 |    0.000000000 ✔
0.01 |  9 |  0 |    0.000000000 |    0.000000000 ✔
0.01 |  9 |  1 |    0.000000000 |    0.000000000 ✔
0.01 |  9 |  2 |    0.000000000 |    0.000000000 ✔
0.01 |  9 |  3 |    0.000000000 |    0.000000000 ✔
0.01 |  9 |  4 |    0.000000000 |    0.000000000 ✔
0.01 |  9 |  5 |    0.000000000 |    0.000000000 ✔
0.01 |  9 |  6 |    0.000000000 |    0.000000000 ✔
0.01 |  9 |  7 |    0.000000000 |    0.000000000 ✔
0.01 |  9 |  8 |    0.000000000 |    0.000000000 ✔
0.01 |  9 |  9 |    1.022778335 |    1.022778336 ✔
0.10 |  0 |  0 |    0.951350770 |    0.951350772 ✔
0.10 |  0 |  1 |    0.000000000 |    0.000000000 ✔
0.10 |  0 |  2 |    0.000000000 |    0.000000000 ✔
0.10 |  0 |  3 |    0.000000000 |    0.000000000 ✔
0.10 |  0 |  4 |    0.000000000 |    0.000000000 ✔
0.10 |  0 |  5 |    0.000000000 |    0.000000000 ✔
0.10 |  0 |  6 |    0.000000000 |    0.000000000 ✔
0.10 |  0 |  7 |    0.000000000 |    0.000000000 ✔
0.10 |  0 |  8 |    0.000000000 |    0.000000000 ✔
0.10 |  0 |  9 |    0.000000000 |    0.000000000 ✔
0.10 |  1 |  0 |    0.000000000 |    0.000000000 ✔
0.10 |  1 |  1 |    1.046485847 |    1.046485848 ✔
0.10 |  1 |  2 |    0.000000000 |    0.000000000 ✔
0.10 |  1 |  3 |    0.000000000 |    0.000000000 ✔
0.10 |  1 |  4 |    0.000000000 |    0.000000000 ✔
0.10 |  1 |  5 |    0.000000000 |    0.000000000 ✔
0.10 |  1 |  6 |    0.000000000 |    0.000000000 ✔
0.10 |  1 |  7 |    0.000000000 |    0.000000000 ✔
0.10 |  1 |  8 |    0.000000000 |    0.000000000 ✔
0.10 |  1 |  9 |    0.000000000 |    0.000000000 ✔
0.10 |  2 |  0 |    0.000000000 |    0.000000000 ✔
0.10 |  2 |  1 |    0.000000000 |    0.000000000 ✔
0.10 |  2 |  2 |    1.098810139 |    1.098810140 ✔
0.10 |  2 |  3 |    0.000000000 |    0.000000000 ✔
0.10 |  2 |  4 |    0.000000000 |    0.000000000 ✔
0.10 |  2 |  5 |    0.000000000 |    0.000000000 ✔
0.10 |  2 |  6 |    0.000000000 |    0.000000000 ✔
0.10 |  2 |  7 |    0.000000000 |    0.000000000 ✔
0.10 |  2 |  8 |    0.000000000 |    0.000000000 ✔
0.10 |  2 |  9 |    0.000000000 |    0.000000000 ✔
0.10 |  3 |  0 |    0.000000000 |    0.000000000 ✔
0.10 |  3 |  1 |    0.000000000 |    0.000000000 ✔
0.10 |  3 |  2 |    0.000000000 |    0.000000000 ✔
0.10 |  3 |  3 |    1.135437144 |    1.135437145 ✔
0.10 |  3 |  4 |    0.000000000 |    0.000000000 ✔
0.10 |  3 |  5 |    0.000000000 |    0.000000000 ✔
0.10 |  3 |  6 |    0.000000000 |    0.000000000 ✔
0.10 |  3 |  7 |    0.000000000 |    0.000000000 ✔
0.10 |  3 |  8 |    0.000000000 |    0.000000000 ✔
0.10 |  3 |  9 |    0.000000000 |    0.000000000 ✔
0.10 |  4 |  0 |    0.000000000 |    0.000000000 ✔
0.10 |  4 |  1 |    0.000000000 |    0.000000000 ✔
0.10 |  4 |  2 |    0.000000000 |    0.000000000 ✔
0.10 |  4 |  3 |    0.000000000 |    0.000000000 ✔
0.10 |  4 |  4 |    1.163823072 |    1.163823074 ✔
0.10 |  4 |  5 |    0.000000000 |    0.000000000 ✔
0.10 |  4 |  6 |    0.000000000 |    0.000000000 ✔
0.10 |  4 |  7 |    0.000000000 |    0.000000000 ✔
0.10 |  4 |  8 |    0.000000000 |    0.000000000 ✔
0.10 |  4 |  9 |    0.000000000 |    0.000000000 ✔
0.10 |  5 |  0 |    0.000000000 |    0.000000000 ✔
0.10 |  5 |  1 |    0.000000000 |    0.000000000 ✔
0.10 |  5 |  2 |    0.000000000 |    0.000000000 ✔
0.10 |  5 |  3 |    0.000000000 |    0.000000000 ✔
0.10 |  5 |  4 |    0.000000000 |    0.000000000 ✔
0.10 |  5 |  5 |    1.187099534 |    1.187099535 ✔
0.10 |  5 |  6 |    0.000000000 |    0.000000000 ✔
0.10 |  5 |  7 |    0.000000000 |    0.000000000 ✔
0.10 |  5 |  8 |    0.000000000 |    0.000000000 ✔
0.10 |  5 |  9 |    0.000000000 |    0.000000000 ✔
0.10 |  6 |  0 |    0.000000000 |    0.000000000 ✔
0.10 |  6 |  1 |    0.000000000 |    0.000000000 ✔
0.10 |  6 |  2 |    0.000000000 |    0.000000000 ✔
0.10 |  6 |  3 |    0.000000000 |    0.000000000 ✔
0.10 |  6 |  4 |    0.000000000 |    0.000000000 ✔
0.10 |  6 |  5 |    0.000000000 |    0.000000000 ✔
0.10 |  6 |  6 |    1.206884526 |    1.206884527 ✔
0.10 |  6 |  7 |    0.000000000 |    0.000000000 ✔
0.10 |  6 |  8 |    0.000000000 |    0.000000000 ✔
0.10 |  6 |  9 |    0.000000000 |    0.000000000 ✔
0.10 |  7 |  0 |    0.000000000 |    0.000000000 ✔
0.10 |  7 |  1 |    0.000000000 |    0.000000000 ✔
0.10 |  7 |  2 |    0.000000000 |    0.000000000 ✔
0.10 |  7 |  3 |    0.000000000 |    0.000000000 ✔
0.10 |  7 |  4 |    0.000000000 |    0.000000000 ✔
0.10 |  7 |  5 |    0.000000000 |    0.000000000 ✔
0.10 |  7 |  6 |    0.000000000 |    0.000000000 ✔
0.10 |  7 |  7 |    1.224125734 |    1.224125734 ✔
0.10 |  7 |  8 |    0.000000000 |    0.000000000 ✔
0.10 |  7 |  9 |    0.000000000 |    0.000000000 ✔
0.10 |  8 |  0 |    0.000000000 |    0.000000000 ✔
0.10 |  8 |  1 |    0.000000000 |    0.000000000 ✔
0.10 |  8 |  2 |    0.000000000 |    0.000000000 ✔
0.10 |  8 |  3 |    0.000000000 |    0.000000000 ✔
0.10 |  8 |  4 |    0.000000000 |    0.000000000 ✔
0.10 |  8 |  5 |    0.000000000 |    0.000000000 ✔
0.10 |  8 |  6 |    0.000000000 |    0.000000000 ✔
0.10 |  8 |  7 |    0.000000000 |    0.000000000 ✔
0.10 |  8 |  8 |    1.239427305 |    1.239427307 ✔
0.10 |  8 |  9 |    0.000000000 |    0.000000000 ✔
0.10 |  9 |  0 |    0.000000000 |    0.000000000 ✔
0.10 |  9 |  1 |    0.000000000 |    0.000000000 ✔
0.10 |  9 |  2 |    0.000000000 |    0.000000000 ✔
0.10 |  9 |  3 |    0.000000000 |    0.000000000 ✔
0.10 |  9 |  4 |    0.000000000 |    0.000000000 ✔
0.10 |  9 |  5 |    0.000000000 |    0.000000000 ✔
0.10 |  9 |  6 |    0.000000000 |    0.000000000 ✔
0.10 |  9 |  7 |    0.000000000 |    0.000000000 ✔
0.10 |  9 |  8 |    0.000000000 |    0.000000000 ✔
0.10 |  9 |  9 |    1.253198720 |    1.253198721 ✔
1.00 |  0 |  0 |    1.000000000 |    1.000000000 ✔
1.00 |  0 |  1 |    0.000000000 |   -0.000000000 ✔
1.00 |  0 |  2 |    0.000000000 |   -0.000000000 ✔
1.00 |  0 |  3 |    0.000000000 |   -0.000000000 ✔
1.00 |  0 |  4 |    0.000000000 |   -0.000000000 ✔
1.00 |  0 |  5 |    0.000000000 |   -0.000000000 ✔
1.00 |  0 |  6 |    0.000000000 |    0.000000000 ✔
1.00 |  0 |  7 |    0.000000000 |   -0.000000000 ✔
1.00 |  0 |  8 |    0.000000000 |    0.000000000 ✔
1.00 |  0 |  9 |    0.000000000 |   -0.000000000 ✔
1.00 |  1 |  0 |    0.000000000 |   -0.000000000 ✔
1.00 |  1 |  1 |    2.000000000 |    2.000000000 ✔
1.00 |  1 |  2 |    0.000000000 |    0.000000000 ✔
1.00 |  1 |  3 |    0.000000000 |    0.000000000 ✔
1.00 |  1 |  4 |    0.000000000 |    0.000000000 ✔
1.00 |  1 |  5 |    0.000000000 |    0.000000000 ✔
1.00 |  1 |  6 |    0.000000000 |   -0.000000000 ✔
1.00 |  1 |  7 |    0.000000000 |    0.000000000 ✔
1.00 |  1 |  8 |    0.000000000 |   -0.000000000 ✔
1.00 |  1 |  9 |    0.000000000 |    0.000000000 ✔
1.00 |  2 |  0 |    0.000000000 |   -0.000000000 ✔
1.00 |  2 |  1 |    0.000000000 |    0.000000000 ✔
1.00 |  2 |  2 |    3.000000000 |    3.000000000 ✔
1.00 |  2 |  3 |    0.000000000 |   -0.000000000 ✔
1.00 |  2 |  4 |    0.000000000 |    0.000000000 ✔
1.00 |  2 |  5 |    0.000000000 |   -0.000000000 ✔
1.00 |  2 |  6 |    0.000000000 |   -0.000000000 ✔
1.00 |  2 |  7 |    0.000000000 |   -0.000000000 ✔
1.00 |  2 |  8 |    0.000000000 |    0.000000000 ✔
1.00 |  2 |  9 |    0.000000000 |   -0.000000000 ✔
1.00 |  3 |  0 |    0.000000000 |   -0.000000000 ✔
1.00 |  3 |  1 |    0.000000000 |    0.000000000 ✔
1.00 |  3 |  2 |    0.000000000 |   -0.000000000 ✔
1.00 |  3 |  3 |    4.000000000 |    4.000000000 ✔
1.00 |  3 |  4 |    0.000000000 |   -0.000000000 ✔
1.00 |  3 |  5 |    0.000000000 |    0.000000000 ✔
1.00 |  3 |  6 |    0.000000000 |    0.000000000 ✔
1.00 |  3 |  7 |    0.000000000 |    0.000000000 ✔
1.00 |  3 |  8 |    0.000000000 |   -0.000000000 ✔
1.00 |  3 |  9 |    0.000000000 |   -0.000000000 ✔
1.00 |  4 |  0 |    0.000000000 |   -0.000000000 ✔
1.00 |  4 |  1 |    0.000000000 |    0.000000000 ✔
1.00 |  4 |  2 |    0.000000000 |    0.000000000 ✔
1.00 |  4 |  3 |    0.000000000 |   -0.000000000 ✔
1.00 |  4 |  4 |    5.000000000 |    5.000000000 ✔
1.00 |  4 |  5 |    0.000000000 |   -0.000000000 ✔
1.00 |  4 |  6 |    0.000000000 |   -0.000000000 ✔
1.00 |  4 |  7 |    0.000000000 |   -0.000000000 ✔
1.00 |  4 |  8 |    0.000000000 |    0.000000000 ✔
1.00 |  4 |  9 |    0.000000000 |    0.000000000 ✔
1.00 |  5 |  0 |    0.000000000 |   -0.000000000 ✔
1.00 |  5 |  1 |    0.000000000 |    0.000000000 ✔
1.00 |  5 |  2 |    0.000000000 |   -0.000000000 ✔
1.00 |  5 |  3 |    0.000000000 |    0.000000000 ✔
1.00 |  5 |  4 |    0.000000000 |   -0.000000000 ✔
1.00 |  5 |  5 |    6.000000000 |    6.000000000 ✔
1.00 |  5 |  6 |    0.000000000 |    0.000000000 ✔
1.00 |  5 |  7 |    0.000000000 |   -0.000000000 ✔
1.00 |  5 |  8 |    0.000000000 |   -0.000000000 ✔
1.00 |  5 |  9 |    0.000000000 |    0.000000000 ✔
1.00 |  6 |  0 |    0.000000000 |    0.000000000 ✔
1.00 |  6 |  1 |    0.000000000 |   -0.000000000 ✔
1.00 |  6 |  2 |    0.000000000 |   -0.000000000 ✔
1.00 |  6 |  3 |    0.000000000 |    0.000000000 ✔
1.00 |  6 |  4 |    0.000000000 |   -0.000000000 ✔
1.00 |  6 |  5 |    0.000000000 |    0.000000000 ✔
1.00 |  6 |  6 |    7.000000000 |    7.000000000 ✔
1.00 |  6 |  7 |    0.000000000 |    0.000000000 ✔
1.00 |  6 |  8 |    0.000000000 |    0.000000000 ✔
1.00 |  6 |  9 |    0.000000000 |    0.000000000 ✔
1.00 |  7 |  0 |    0.000000000 |   -0.000000000 ✔
1.00 |  7 |  1 |    0.000000000 |    0.000000000 ✔
1.00 |  7 |  2 |    0.000000000 |   -0.000000000 ✔
1.00 |  7 |  3 |    0.000000000 |    0.000000000 ✔
1.00 |  7 |  4 |    0.000000000 |   -0.000000000 ✔
1.00 |  7 |  5 |    0.000000000 |   -0.000000000 ✔
1.00 |  7 |  6 |    0.000000000 |    0.000000000 ✔
1.00 |  7 |  7 |    8.000000000 |    8.000000000 ✔
1.00 |  7 |  8 |    0.000000000 |    0.000000000 ✔
1.00 |  7 |  9 |    0.000000000 |   -0.000000000 ✔
1.00 |  8 |  0 |    0.000000000 |    0.000000000 ✔
1.00 |  8 |  1 |    0.000000000 |   -0.000000000 ✔
1.00 |  8 |  2 |    0.000000000 |    0.000000000 ✔
1.00 |  8 |  3 |    0.000000000 |   -0.000000000 ✔
1.00 |  8 |  4 |    0.000000000 |    0.000000000 ✔
1.00 |  8 |  5 |    0.000000000 |   -0.000000000 ✔
1.00 |  8 |  6 |    0.000000000 |    0.000000000 ✔
1.00 |  8 |  7 |    0.000000000 |    0.000000000 ✔
1.00 |  8 |  8 |    9.000000000 |    9.000000000 ✔
1.00 |  8 |  9 |    0.000000000 |   -0.000000000 ✔
1.00 |  9 |  0 |    0.000000000 |   -0.000000000 ✔
1.00 |  9 |  1 |    0.000000000 |    0.000000000 ✔
1.00 |  9 |  2 |    0.000000000 |   -0.000000000 ✔
1.00 |  9 |  3 |    0.000000000 |   -0.000000000 ✔
1.00 |  9 |  4 |    0.000000000 |    0.000000000 ✔
1.00 |  9 |  5 |    0.000000000 |    0.000000000 ✔
1.00 |  9 |  6 |    0.000000000 |    0.000000000 ✔
1.00 |  9 |  7 |    0.000000000 |   -0.000000000 ✔
1.00 |  9 |  8 |    0.000000000 |   -0.000000000 ✔
1.00 |  9 |  9 |   10.000000000 |   10.000000000 ✔

Normalization & 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 |    1.000000000 |    1.000000000 ✔
 0 |  1 |    0.000000000 |   -0.000000000 ✔
 0 |  2 |    0.000000000 |   -0.000000000 ✔
 0 |  3 |    0.000000000 |    0.000000000 ✔
 0 |  4 |    0.000000000 |    0.000000000 ✔
 0 |  5 |    0.000000000 |   -0.000000000 ✔
 0 |  6 |    0.000000000 |   -0.000000000 ✔
 0 |  7 |    0.000000000 |   -0.000000000 ✔
 0 |  8 |    0.000000000 |   -0.000000000 ✔
 0 |  9 |    0.000000000 |    0.000000000 ✔
 1 |  0 |    0.000000000 |   -0.000000000 ✔
 1 |  1 |    1.000000000 |    1.000000000 ✔
 1 |  2 |    0.000000000 |   -0.000000000 ✔
 1 |  3 |    0.000000000 |    0.000000000 ✔
 1 |  4 |    0.000000000 |    0.000000000 ✔
 1 |  5 |    0.000000000 |   -0.000000000 ✔
 1 |  6 |    0.000000000 |   -0.000000000 ✔
 1 |  7 |    0.000000000 |   -0.000000000 ✔
 1 |  8 |    0.000000000 |   -0.000000000 ✔
 1 |  9 |    0.000000000 |    0.000000000 ✔
 2 |  0 |    0.000000000 |   -0.000000000 ✔
 2 |  1 |    0.000000000 |   -0.000000000 ✔
 2 |  2 |    1.000000000 |    1.000000000 ✔
 2 |  3 |    0.000000000 |    0.000000000 ✔
 2 |  4 |    0.000000000 |    0.000000000 ✔
 2 |  5 |    0.000000000 |    0.000000000 ✔
 2 |  6 |    0.000000000 |   -0.000000000 ✔
 2 |  7 |    0.000000000 |   -0.000000000 ✔
 2 |  8 |    0.000000000 |   -0.000000000 ✔
 2 |  9 |    0.000000000 |   -0.000000000 ✔
 3 |  0 |    0.000000000 |    0.000000000 ✔
 3 |  1 |    0.000000000 |    0.000000000 ✔
 3 |  2 |    0.000000000 |    0.000000000 ✔
 3 |  3 |    1.000000000 |    1.000000000 ✔
 3 |  4 |    0.000000000 |    0.000000000 ✔
 3 |  5 |    0.000000000 |   -0.000000000 ✔
 3 |  6 |    0.000000000 |    0.000000000 ✔
 3 |  7 |    0.000000000 |   -0.000000000 ✔
 3 |  8 |    0.000000000 |   -0.000000000 ✔
 3 |  9 |    0.000000000 |   -0.000000000 ✔
 4 |  0 |    0.000000000 |    0.000000000 ✔
 4 |  1 |    0.000000000 |    0.000000000 ✔
 4 |  2 |    0.000000000 |    0.000000000 ✔
 4 |  3 |    0.000000000 |    0.000000000 ✔
 4 |  4 |    1.000000000 |    1.000000000 ✔
 4 |  5 |    0.000000000 |    0.000000000 ✔
 4 |  6 |    0.000000000 |   -0.000000000 ✔
 4 |  7 |    0.000000000 |    0.000000000 ✔
 4 |  8 |    0.000000000 |    0.000000000 ✔
 4 |  9 |    0.000000000 |   -0.000000000 ✔
 5 |  0 |    0.000000000 |   -0.000000000 ✔
 5 |  1 |    0.000000000 |   -0.000000000 ✔
 5 |  2 |    0.000000000 |    0.000000000 ✔
 5 |  3 |    0.000000000 |   -0.000000000 ✔
 5 |  4 |    0.000000000 |    0.000000000 ✔
 5 |  5 |    1.000000000 |    1.000000000 ✔
 5 |  6 |    0.000000000 |   -0.000000000 ✔
 5 |  7 |    0.000000000 |    0.000000000 ✔
 5 |  8 |    0.000000000 |    0.000000000 ✔
 5 |  9 |    0.000000000 |   -0.000000000 ✔
 6 |  0 |    0.000000000 |   -0.000000000 ✔
 6 |  1 |    0.000000000 |   -0.000000000 ✔
 6 |  2 |    0.000000000 |   -0.000000000 ✔
 6 |  3 |    0.000000000 |    0.000000000 ✔
 6 |  4 |    0.000000000 |   -0.000000000 ✔
 6 |  5 |    0.000000000 |   -0.000000000 ✔
 6 |  6 |    1.000000000 |    1.000000000 ✔
 6 |  7 |    0.000000000 |    0.000000000 ✔
 6 |  8 |    0.000000000 |   -0.000000000 ✔
 6 |  9 |    0.000000000 |   -0.000000000 ✔
 7 |  0 |    0.000000000 |   -0.000000000 ✔
 7 |  1 |    0.000000000 |   -0.000000000 ✔
 7 |  2 |    0.000000000 |   -0.000000000 ✔
 7 |  3 |    0.000000000 |   -0.000000000 ✔
 7 |  4 |    0.000000000 |    0.000000000 ✔
 7 |  5 |    0.000000000 |    0.000000000 ✔
 7 |  6 |    0.000000000 |    0.000000000 ✔
 7 |  7 |    1.000000000 |    1.000000000 ✔
 7 |  8 |    0.000000000 |   -0.000000000 ✔
 7 |  9 |    0.000000000 |    0.000000000 ✔
 8 |  0 |    0.000000000 |   -0.000000000 ✔
 8 |  1 |    0.000000000 |   -0.000000000 ✔
 8 |  2 |    0.000000000 |   -0.000000000 ✔
 8 |  3 |    0.000000000 |   -0.000000000 ✔
 8 |  4 |    0.000000000 |    0.000000000 ✔
 8 |  5 |    0.000000000 |    0.000000000 ✔
 8 |  6 |    0.000000000 |   -0.000000000 ✔
 8 |  7 |    0.000000000 |   -0.000000000 ✔
 8 |  8 |    1.000000000 |    1.000000000 ✔
 8 |  9 |    0.000000000 |   -0.000000000 ✔
 9 |  0 |    0.000000000 |    0.000000000 ✔
 9 |  1 |    0.000000000 |    0.000000000 ✔
 9 |  2 |    0.000000000 |   -0.000000000 ✔
 9 |  3 |    0.000000000 |   -0.000000000 ✔
 9 |  4 |    0.000000000 |   -0.000000000 ✔
 9 |  5 |    0.000000000 |   -0.000000000 ✔
 9 |  6 |    0.000000000 |   -0.000000000 ✔
 9 |  7 |    0.000000000 |    0.000000000 ✔
 9 |  8 |    0.000000000 |   -0.000000000 ✔
 9 |  9 |    1.000000000 |    1.000000000 ✔

Eigenvalues

\[ \begin{aligned} E_n &= \int \psi^\ast_n(r) \hat{H} \psi_n(r) \mathrm{d}r \\ &= \int \psi^\ast_n(r) \left[ \hat{V} + \hat{T} \right] \psi(r) \mathrm{d}r \\ &= \int \psi^\ast_n(r) \left[ V(r) - \frac{\hbar^2}{2m} \frac{\mathrm{d}^{2}}{\mathrm{d} r^{2}} \right] \psi(r) \mathrm{d}r \\ &\simeq \int \psi^\ast_n(r) \left[ V(r)\psi(r) -\frac{\hbar^2}{2m} \frac{\psi(r+\Delta r) - 2\psi(r) + \psi(r-\Delta r)}{\Delta r^{2}} \right] \mathrm{d}r. \end{aligned}\]

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 |   -0.094849824 |   -0.094849824 ✔
0.1 |  1 |   -0.084957924 |   -0.084957924 ✔
0.1 |  2 |   -0.075610627 |   -0.075610627 ✔
0.1 |  3 |   -0.066807932 |   -0.066807933 ✔
0.1 |  4 |   -0.058549841 |   -0.058549842 ✔
0.1 |  5 |   -0.050836353 |   -0.050836354 ✔
0.1 |  6 |   -0.043667468 |   -0.043667475 ✔
0.1 |  7 |   -0.037043186 |   -0.037043173 ✔
0.1 |  8 |   -0.030963506 |   -0.030963499 ✔
0.1 |  9 |   -0.025428430 |   -0.025428323 ✔
0.2 |  0 |   -0.092756429 |   -0.092756429 ✔
0.2 |  1 |   -0.079086190 |   -0.079086191 ✔
0.2 |  2 |   -0.066505158 |   -0.066505159 ✔
0.2 |  3 |   -0.055013332 |   -0.055013333 ✔
0.2 |  4 |   -0.044610711 |   -0.044610713 ✔
0.2 |  5 |   -0.035297297 |   -0.035297299 ✔
0.2 |  6 |   -0.027073089 |   -0.027073091 ✔
0.2 |  7 |   -0.019938086 |   -0.019938089 ✔
0.2 |  8 |   -0.013892290 |   -0.013892292 ✔
0.2 |  9 |   -0.008935699 |   -0.008935703 ✔
0.3 |  0 |   -0.091165949 |   -0.091165949 ✔
0.3 |  1 |   -0.074723205 |   -0.074723205 ✔
0.3 |  2 |   -0.059914269 |   -0.059914270 ✔
0.3 |  3 |   -0.046739142 |   -0.046739144 ✔
0.3 |  4 |   -0.035197824 |   -0.035197826 ✔
0.3 |  5 |   -0.025290316 |   -0.025290318 ✔
0.3 |  6 |   -0.017016616 |   -0.017016618 ✔
0.3 |  7 |   -0.010376725 |   -0.010376726 ✔
0.3 |  8 |   -0.005370643 |   -0.005370644 ✔
0.3 |  9 |   -0.001998370 |   -0.001998371 ✔
0.1 |  0 |   -0.094849824 |   -0.094849824 ✔
0.1 |  1 |   -0.084957924 |   -0.084957924 ✔
0.1 |  2 |   -0.075610627 |   -0.075610627 ✔
0.1 |  3 |   -0.066807932 |   -0.066807933 ✔
0.1 |  4 |   -0.058549841 |   -0.058549842 ✔
0.1 |  5 |   -0.050836353 |   -0.050836354 ✔
0.1 |  6 |   -0.043667468 |   -0.043667475 ✔
0.1 |  7 |   -0.037043186 |   -0.037043173 ✔
0.1 |  8 |   -0.030963506 |   -0.030963499 ✔
0.1 |  9 |   -0.025428430 |   -0.025428323 ✔

Recurrence Relation between $E_{n+1}$ and $E_n$

\[\begin{equation} \left\{ \, \begin{aligned} 0 < \Delta E && 0 \leq n \leq n_\mathrm{max} \\ \Delta E < 0 && \mathrm{otherwise} \end{aligned} \right. \end{equation}\]

\[\Delta E = E_{n+1} - E_n\]

\[n_\mathrm{max} = \left\lfloor\frac{2 D_{\mathrm{e}}-h \nu_0}{h \nu_0}\right\rfloor\]

 n  Eₙ          ΔE
 0  -0.094850  +0.009892  0 < ΔE  ✔
 1  -0.084958  +0.009347  0 < ΔE  ✔
 2  -0.075611  +0.008803  0 < ΔE  ✔
 3  -0.066808  +0.008258  0 < ΔE  ✔
 4  -0.058550  +0.007713  0 < ΔE  ✔
 5  -0.050836  +0.007169  0 < ΔE  ✔
 6  -0.043667  +0.006624  0 < ΔE  ✔
 7  -0.037043  +0.006080  0 < ΔE  ✔
 8  -0.030964  +0.005535  0 < ΔE  ✔
 9  -0.025428  +0.004990  0 < ΔE  ✔
10  -0.020438  +0.004446  0 < ΔE  ✔
11  -0.015992  +0.003901  0 < ΔE  ✔
12  -0.012091  +0.003357  0 < ΔE  ✔
13  -0.008734  +0.002812  0 < ΔE  ✔
14  -0.005922  +0.002267  0 < ΔE  ✔
15  -0.003655  +0.001723  0 < ΔE  ✔
16  -0.001932  +0.001178  0 < ΔE  ✔
17  -0.000754  +0.000634  0 < ΔE  ✔
18  -0.000120  +0.000089  0 < ΔE  ✔
-----------------------------  nₘₐₓ(MP) = 18
19  -0.000031  -0.000456  ΔE < 0  ✔
20  -0.000486  -0.001000  ΔE < 0  ✔
21  -0.001487  -0.001545  ΔE < 0  ✔
22  -0.003031  -0.002089  ΔE < 0  ✔
23  -0.005121  -0.002634  ΔE < 0  ✔