Harmonic Oscillator
The harmonic oscillator is the most frequently used model in quantum physics.
Definitions
This model is described with the time-independent Schrödinger equation
\[ \hat{H} \psi(x) = E \psi(x),\]
and the Hamiltonian
\[ \hat{H} = - \frac{\hbar^2}{2m} \frac{\mathrm{d}^2}{\mathrm{d}x ^2} + V(x).\]
Parameters are specified with the following struct.
Parameters
Antique.HarmonicOscillator — TypeHarmonicOscillator(k=1.0, m=1.0, ℏ=1.0)
$k$ is the force constant, $m$ is the mass of particle and $\hbar$ is the reduced Planck constant (Dirac's constant).
Potential
Antique.V — MethodV(model::HarmonicOscillator, x)
\[V(x) = \frac{1}{2} k x^2 = \frac{1}{2} m \omega^2 x^2 = \frac{1}{2} \hbar \omega \xi^2,\]
where $\omega = \sqrt{k/m}$ is the angular frequency and $\xi = \sqrt{\frac{m\omega}{\hbar}}x$.
Eigen Values
Antique.E — MethodE(model::HarmonicOscillator; n=0)
\[E_n = \hbar \omega \left( n + \frac{1}{2} \right),\]
where $\omega = \sqrt{k/m}$ is the angular frequency.
Eigen Functions
Antique.ψ — Methodψ(model::HarmonicOscillator, x; n=0)
\[\psi_n(x) = A_n H_n(\xi) \exp{\left( -\frac{\xi^2}{2} \right)},\]
where $\omega = \sqrt{k/m}$, $\xi = \sqrt{\frac{m\omega}{\hbar}}x$, $A_n = \sqrt{\frac{1}{n! 2^n} \sqrt{\frac{m\omega}{\pi\hbar}}}$, $H_n(x) = (-1)^n \mathrm{e}^{x^2} \frac{\mathrm{d}^n}{\mathrm{d}x^n} \mathrm{e}^{-x^2}$ are defined.
Hermite Polynomials
Antique.H — MethodH(model::HarmonicOscillator, x; n=0)
Rodrigues' formula & closed-form:
\[\begin{aligned} H_{n}(x) &:= (-1)^n \mathrm{e}^{x^2} \frac{\mathrm{d}^n}{\mathrm{d}x^n} \mathrm{e}^{-x^2} \\ &= n! \sum_{m=0}^{\lfloor n/2 \rfloor} \frac{(-1)^m}{m! (n-2m)!}(2 x)^{n-2m}. \end{aligned}\]
Examples:
\[\begin{aligned} H_{0}(x) &= 1, \\ H_{1}(x) &= 2 x, \\ H_{2}(x) &= -2 + 4 x^{2}, \\ H_{3}(x) &= -12 x + 8 x^{3}, \\ H_{4}(x) &= 12 - 48 x^{2} + 16 x^{4}, \\ H_{5}(x) &= 120 x - 160 x^{3} + 32 x^{5}, \\ H_{6}(x) &= -120 + 720 x^{2} - 480 x^{4} + 64 x^{6}, \\ H_{7}(x) &= -1680 x + 3360 x^{3} - 1344 x^{5} + 128 x^{7}, \\ H_{8}(x) &= 1680 - 13440 x^{2} + 13440 x^{4} - 3584 x^{6} + 256 x^{8}, \\ H_{9}(x) &= 30240 x - 80640 x^{3} + 48384 x^{5} - 9216 x^{7} + 512 x^{9}, \\ &\vdots \end{aligned}\]
Reference
- DLMF 18.5.18
- cpprefjp
- The Digital Library of Mathematical Functions (DLMF) 18.3 Table1, 18.5 Table1, 18.5.13, 18.5.18
- L. D. Landau, E. M. Lifshitz, Quantum Mechanics (Pergamon Press, 1965) p.595 (a.4), (a.6)
- L. I. Schiff, Quantum Mechanics (McGraw-Hill Book Company, 1968) p.71 (13.12)
- A. Messiah, Quanfum Mechanics (Dover Publications, 1999) p.491 (B.59)
- W. Greiner, Quantum Mechanics: An Introduction Third Edition (Springer, 1994) p.152 (7.22)
- D. J. Griffiths, Introduction to Quantum Mechanics (Prentice Hall, 1995) p.41 Table 2.1, p.43 (2.70)
- D. A. McQuarrie, J. D. Simon, Physical Chemistry: A Molecular Approach (University Science Books, 1997) p.170 Table 5.2
- P. W. Atkins, J. De Paula, Atkins' Physical Chemistry, 8th edition (W. H. Freeman, 2008) p.293 Table 9.1
- J. J. Sakurai, J. Napolitano, Modern Quantum Mechanics Third Edition (Cambridge University Press, 2021) p.524 (B.29)
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 HarmonicOscillator and several parameters k, m and ℏ are set as optional arguments.
using Antique
HO = HarmonicOscillator(k=1.0, m=1.0, ℏ=1.0)Parameters:
julia> HO.k1.0julia> HO.m1.0julia> HO.ℏ1.0
Eigen values:
julia> E(HO, n=0)0.5julia> E(HO, n=1)1.5
Potential energy curve:
using CairoMakie
f = Figure()
ax = Axis(f[1,1], xlabel=L"$x$", ylabel=L"$V(x)$")
lines!(ax, -5..5, x -> V(HO, x))
f
Wave functions:
using CairoMakie
# setting
f = Figure()
ax = Axis(f[1,1], xlabel=L"$x$", ylabel=L"$\psi(x)$")
# plot
w0 = lines!(ax, -5..5, x -> ψ(HO, x, n=0))
w1 = lines!(ax, -5..5, x -> ψ(HO, x, n=1))
w2 = lines!(ax, -5..5, x -> ψ(HO, x, n=2))
w3 = lines!(ax, -5..5, x -> ψ(HO, x, n=3))
w4 = lines!(ax, -5..5, x -> ψ(HO, 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, Wave functions:
using CairoMakie
# settings
f = Figure()
ax = Axis(f[1,1], xlabel=L"$x$", ylabel=L"$V(x),~E_n,~\psi_n(x) \times 5 + E_n$", aspect=1, limits=(-5,5,0,5.2))
# hidespines!(ax)
# hidedecorations!(ax)
for n in 0:4
# classical turning point
xE = sqrt(2*HO.k*E(HO, n=n))
# energy
lines!(ax, [-xE,xE], fill(E(HO,n=n),2), color=:black, linewidth=2)
hlines!(ax, E(HO, n=n), color=:black, linewidth=1, linestyle=:dash)
# wave function
lines!(ax, -5..5, x -> E(HO,n=n) + 0.5*ψ(HO,x,n=n), linewidth=2)
end
#potential
lines!(ax, -5..5, x -> V(HO, x), color=:black, linewidth=2)
f
Testing
Unit testing and Integration testing were done using computer algebra system (Symbolics.jl) and numerical integration (QuadGK.jl). The test script is here.
Hermite Polynomials $H_n(x)$
\[ \begin{aligned} H_{n}(x) &:= (-1)^n \mathrm{e}^{x^2} \frac{\mathrm{d}^n}{\mathrm{d}x^n} \mathrm{e}^{-x^2} \\ &= n! \sum_{m=0}^{\lfloor n/2 \rfloor} \frac{(-1)^m}{m! (n-2m)!}(2 x)^{n-2m}. \end{aligned}\]
$n=0:$ ✔
\[\begin{aligned} H_{0}(x) = e^{ - x^{2}} e^{x^{2}} &= 1 \\ &= 1 \end{aligned}\]
$n=1:$ ✔
\[\begin{aligned} H_{1}(x) = - e^{x^{2}} \frac{\mathrm{d} e^{ - x^{2}}}{\mathrm{d}x} &= 2 x \\ &= 2 x \end{aligned}\]
$n=2:$ ✔
\[\begin{aligned} H_{2}(x) = e^{x^{2}} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d} e^{ - x^{2}}}{\mathrm{d}x} &= -2 + 4 x^{2} \\ &= -2 + 4 x^{2} \end{aligned}\]
$n=3:$ ✔
\[\begin{aligned} H_{3}(x) = - \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d} e^{ - x^{2}}}{\mathrm{d}x} e^{x^{2}} &= - 12 x + 8 x^{3} \\ &= - 12 x + 8 x^{3} \end{aligned}\]
$n=4:$ ✔
\[\begin{aligned} H_{4}(x) = e^{x^{2}} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d} e^{ - x^{2}}}{\mathrm{d}x} &= 12 - 48 x^{2} + 16 x^{4} \\ &= 12 - 48 x^{2} + 16 x^{4} \end{aligned}\]
$n=5:$ ✔
\[\begin{aligned} H_{5}(x) = - \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d} e^{ - x^{2}}}{\mathrm{d}x} e^{x^{2}} &= 120 x - 160 x^{3} + 32 x^{5} \\ &= 120 x - 160 x^{3} + 32 x^{5} \end{aligned}\]
$n=6:$ ✔
\[\begin{aligned} H_{6}(x) = e^{x^{2}} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d} e^{ - x^{2}}}{\mathrm{d}x} &= -120 + 720 x^{2} - 480 x^{4} + 64 x^{6} \\ &= -120 + 720 x^{2} - 480 x^{4} + 64 x^{6} \end{aligned}\]
$n=7:$ ✔
\[\begin{aligned} H_{7}(x) = - e^{x^{2}} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d} e^{ - x^{2}}}{\mathrm{d}x} &= - 1680 x + 3360 x^{3} - 1344 x^{5} + 128 x^{7} \\ &= - 1680 x + 3360 x^{3} - 1344 x^{5} + 128 x^{7} \end{aligned}\]
$n=8:$ ✔
\[\begin{aligned} H_{8}(x) = e^{x^{2}} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d} e^{ - x^{2}}}{\mathrm{d}x} &= 1680 - 13440 x^{2} + 13440 x^{4} - 3584 x^{6} + 256 x^{8} \\ &= 1680 - 13440 x^{2} + 13440 x^{4} - 3584 x^{6} + 256 x^{8} \end{aligned}\]
$n=9:$ ✔
\[\begin{aligned} H_{9}(x) = - e^{x^{2}} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d} e^{ - x^{2}}}{\mathrm{d}x} &= 30240 x - 80640 x^{3} + 48384 x^{5} - 9216 x^{7} + 512 x^{9} \\ &= 30240 x - 80640 x^{3} + 48384 x^{5} - 9216 x^{7} + 512 x^{9} \end{aligned}\]
Normalization & Orthogonality of $H_n(x)$
\[\int_{-\infty}^\infty H_j(x) H_i(x) \mathrm{e}^{-x^2} \mathrm{d}x = \sqrt{\pi} 2^j j! \delta_{ij}\]
i | j | analytical | numerical
-- | -- | ----------------- | -----------------
0 | 0 | 1.772453850906 | 1.772453850906 ✔
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 | 0.000000000000 ✔
0 | 8 | 0.000000000000 | -0.000000000001 ✔
0 | 9 | 0.000000000000 | 0.000000000000 ✔
1 | 0 | 0.000000000000 | 0.000000000000 ✔
1 | 1 | 3.544907701811 | 3.544907701811 ✔
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 | -0.000000000000 ✔
1 | 8 | 0.000000000000 | -0.000000000000 ✔
1 | 9 | 0.000000000000 | 0.000000000014 ✔
2 | 0 | 0.000000000000 | 0.000000000000 ✔
2 | 1 | 0.000000000000 | 0.000000000000 ✔
2 | 2 | 14.179630807244 | 14.179630807244 ✔
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 | -0.000000000011 ✔
2 | 9 | 0.000000000000 | -0.000000000002 ✔
3 | 0 | 0.000000000000 | 0.000000000000 ✔
3 | 1 | 0.000000000000 | -0.000000000000 ✔
3 | 2 | 0.000000000000 | -0.000000000000 ✔
3 | 3 | 85.077784843465 | 85.077784843465 ✔
3 | 4 | 0.000000000000 | -0.000000000000 ✔
3 | 5 | 0.000000000000 | 0.000000000000 ✔
3 | 6 | 0.000000000000 | -0.000000000000 ✔
3 | 7 | 0.000000000000 | -0.000000000000 ✔
3 | 8 | 0.000000000000 | -0.000000000000 ✔
3 | 9 | 0.000000000000 | 0.000000000139 ✔
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 | 680.622278747718 | 680.622278747718 ✔
4 | 5 | 0.000000000000 | -0.000000000000 ✔
4 | 6 | 0.000000000000 | 0.000000000002 ✔
4 | 7 | 0.000000000000 | 0.000000000000 ✔
4 | 8 | 0.000000000000 | -0.000000000063 ✔
4 | 9 | 0.000000000000 | 0.000000000000 ✔
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 | 6806.222787477181 | 6806.222787477180 ✔
5 | 6 | 0.000000000000 | 0.000000000000 ✔
5 | 7 | 0.000000000000 | 0.000000000009 ✔
5 | 8 | 0.000000000000 | 0.000000000000 ✔
5 | 9 | 0.000000000000 | 0.000000001339 ✔
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 | 0.000000000002 ✔
6 | 5 | 0.000000000000 | 0.000000000000 ✔
6 | 6 | 81674.673449726179 | 81674.673449726135 ✔
6 | 7 | 0.000000000000 | 0.000000000004 ✔
6 | 8 | 0.000000000000 | 0.000000000397 ✔
6 | 9 | 0.000000000000 | -0.000000000087 ✔
7 | 0 | 0.000000000000 | 0.000000000000 ✔
7 | 1 | 0.000000000000 | -0.000000000000 ✔
7 | 2 | 0.000000000000 | 0.000000000000 ✔
7 | 3 | 0.000000000000 | -0.000000000000 ✔
7 | 4 | 0.000000000000 | 0.000000000000 ✔
7 | 5 | 0.000000000000 | 0.000000000009 ✔
7 | 6 | 0.000000000000 | 0.000000000004 ✔
7 | 7 | 1143445.428296166472 | 1143445.428296166705 ✔
7 | 8 | 0.000000000000 | -0.000000000007 ✔
7 | 9 | 0.000000000000 | 0.000000011649 ✔
8 | 0 | 0.000000000000 | -0.000000000001 ✔
8 | 1 | 0.000000000000 | -0.000000000000 ✔
8 | 2 | 0.000000000000 | -0.000000000011 ✔
8 | 3 | 0.000000000000 | -0.000000000000 ✔
8 | 4 | 0.000000000000 | -0.000000000063 ✔
8 | 5 | 0.000000000000 | 0.000000000000 ✔
8 | 6 | 0.000000000000 | 0.000000000397 ✔
8 | 7 | 0.000000000000 | -0.000000000007 ✔
8 | 8 | 18295126.852738663554 | 18295126.852738667279 ✔
8 | 9 | 0.000000000000 | 0.000000001630 ✔
9 | 0 | 0.000000000000 | 0.000000000000 ✔
9 | 1 | 0.000000000000 | 0.000000000014 ✔
9 | 2 | 0.000000000000 | -0.000000000002 ✔
9 | 3 | 0.000000000000 | 0.000000000139 ✔
9 | 4 | 0.000000000000 | 0.000000000000 ✔
9 | 5 | 0.000000000000 | 0.000000001339 ✔
9 | 6 | 0.000000000000 | -0.000000000087 ✔
9 | 7 | 0.000000000000 | 0.000000011649 ✔
9 | 8 | 0.000000000000 | 0.000000001630 ✔
9 | 9 | 329312283.349295914173 | 329312283.349295675755 ✔Normalization & Orthogonality of $\psi_n(x)$
\[\int \psi_i^\ast(x) \psi_j(x) \mathrm{d}x = \delta_{ij}\]
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 | 0.000000000000 ✔
0 | 8 | 0.000000000000 | -0.000000000000 ✔
0 | 9 | 0.000000000000 | 0.000000000000 ✔
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 | -0.000000000000 ✔
1 | 8 | 0.000000000000 | 0.000000000000 ✔
1 | 9 | 0.000000000000 | 0.000000000000 ✔
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 | -0.000000000000 ✔
2 | 9 | 0.000000000000 | 0.000000000000 ✔
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 | 0.000000000000 ✔
3 | 8 | 0.000000000000 | -0.000000000000 ✔
3 | 9 | 0.000000000000 | 0.000000000000 ✔
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 | -0.000000000000 ✔
4 | 9 | 0.000000000000 | -0.000000000000 ✔
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 | 0.000000000000 ✔
5 | 8 | 0.000000000000 | 0.000000000000 ✔
5 | 9 | 0.000000000000 | 0.000000000000 ✔
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 | 0.000000000000 ✔
6 | 9 | 0.000000000000 | 0.000000000000 ✔
7 | 0 | 0.000000000000 | 0.000000000000 ✔
7 | 1 | 0.000000000000 | -0.000000000000 ✔
7 | 2 | 0.000000000000 | 0.000000000000 ✔
7 | 3 | 0.000000000000 | 0.000000000000 ✔
7 | 4 | 0.000000000000 | 0.000000000000 ✔
7 | 5 | 0.000000000000 | 0.000000000000 ✔
7 | 6 | 0.000000000000 | -0.000000000000 ✔
7 | 7 | 1.000000000000 | 1.000000000000 ✔
7 | 8 | 0.000000000000 | 0.000000000000 ✔
7 | 9 | 0.000000000000 | 0.000000000000 ✔
8 | 0 | 0.000000000000 | -0.000000000000 ✔
8 | 1 | 0.000000000000 | 0.000000000000 ✔
8 | 2 | 0.000000000000 | -0.000000000000 ✔
8 | 3 | 0.000000000000 | -0.000000000000 ✔
8 | 4 | 0.000000000000 | -0.000000000000 ✔
8 | 5 | 0.000000000000 | 0.000000000000 ✔
8 | 6 | 0.000000000000 | 0.000000000000 ✔
8 | 7 | 0.000000000000 | 0.000000000000 ✔
8 | 8 | 1.000000000000 | 1.000000000000 ✔
8 | 9 | 0.000000000000 | -0.000000000000 ✔
9 | 0 | 0.000000000000 | 0.000000000000 ✔
9 | 1 | 0.000000000000 | 0.000000000000 ✔
9 | 2 | 0.000000000000 | 0.000000000000 ✔
9 | 3 | 0.000000000000 | 0.000000000000 ✔
9 | 4 | 0.000000000000 | -0.000000000000 ✔
9 | 5 | 0.000000000000 | 0.000000000000 ✔
9 | 6 | 0.000000000000 | 0.000000000000 ✔
9 | 7 | 0.000000000000 | 0.000000000000 ✔
9 | 8 | 0.000000000000 | -0.000000000000 ✔
9 | 9 | 1.000000000000 | 1.000000000000 ✔Virial Theorem
The virial theorem $\langle T \rangle = \langle V \rangle$ and the definition of Hamiltonian $\langle H \rangle = \langle T \rangle + \langle V \rangle$ derive $\langle H \rangle = 2 \langle V \rangle = 2 \langle T \rangle$.
\[2 \int \psi_n^\ast(x) V(x) \psi_n(x) \mathrm{d}x = E_n\]
k | n | analytical | numerical
--- | -- | ----------------- | -----------------
0.1 | 0 | 0.500000000000 | 0.500000000000 ✔
0.1 | 1 | 1.500000000000 | 1.500000000000 ✔
0.1 | 2 | 2.500000000000 | 2.500000000000 ✔
0.1 | 3 | 3.500000000000 | 3.500000000000 ✔
0.1 | 4 | 4.500000000000 | 4.500000000000 ✔
0.1 | 5 | 5.500000000000 | 5.500000000000 ✔
0.1 | 6 | 6.500000000000 | 6.500000000000 ✔
0.1 | 7 | 7.500000000000 | 7.500000000000 ✔
0.1 | 8 | 8.500000000000 | 8.500000000000 ✔
0.1 | 9 | 9.500000000000 | 9.500000000000 ✔
0.5 | 0 | 0.500000000000 | 0.500000000000 ✔
0.5 | 1 | 1.500000000000 | 1.500000000000 ✔
0.5 | 2 | 2.500000000000 | 2.500000000000 ✔
0.5 | 3 | 3.500000000000 | 3.500000000000 ✔
0.5 | 4 | 4.500000000000 | 4.500000000000 ✔
0.5 | 5 | 5.500000000000 | 5.500000000000 ✔
0.5 | 6 | 6.500000000000 | 6.500000000000 ✔
0.5 | 7 | 7.500000000000 | 7.500000000000 ✔
0.5 | 8 | 8.500000000000 | 8.500000000000 ✔
0.5 | 9 | 9.500000000000 | 9.500000000000 ✔
1.0 | 0 | 0.500000000000 | 0.500000000000 ✔
1.0 | 1 | 1.500000000000 | 1.500000000000 ✔
1.0 | 2 | 2.500000000000 | 2.500000000000 ✔
1.0 | 3 | 3.500000000000 | 3.500000000000 ✔
1.0 | 4 | 4.500000000000 | 4.500000000000 ✔
1.0 | 5 | 5.500000000000 | 5.500000000000 ✔
1.0 | 6 | 6.500000000000 | 6.500000000000 ✔
1.0 | 7 | 7.500000000000 | 7.500000000000 ✔
1.0 | 8 | 8.500000000000 | 8.500000000000 ✔
1.0 | 9 | 9.500000000000 | 9.500000000000 ✔
5.0 | 0 | 0.500000000000 | 0.500000000000 ✔
5.0 | 1 | 1.500000000000 | 1.500000000000 ✔
5.0 | 2 | 2.500000000000 | 2.500000000000 ✔
5.0 | 3 | 3.500000000000 | 3.500000000000 ✔
5.0 | 4 | 4.500000000000 | 4.500000000000 ✔
5.0 | 5 | 5.500000000000 | 5.500000000000 ✔
5.0 | 6 | 6.500000000000 | 6.500000000000 ✔
5.0 | 7 | 7.500000000000 | 7.500000000000 ✔
5.0 | 8 | 8.500000000000 | 8.500000000000 ✔
5.0 | 9 | 9.500000000000 | 9.500000000000 ✔Eigen Values
\[ \begin{aligned} E_n &= \int \psi^\ast_n(x) \hat{H} \psi_n(x) \mathrm{d}x \\ &= \int \psi^\ast_n(x) \left[ \hat{V} + \hat{T} \right] \psi(x) \mathrm{d}x \\ &= \int \psi^\ast_n(x) \left[ V(x) - \frac{\hbar^2}{2m} \frac{\mathrm{d}^{2}}{\mathrm{d} x^{2}} \right] \psi(x) \mathrm{d}x \\ &\simeq \int \psi^\ast_n(x) \left[ V(x)\psi(x) -\frac{\hbar^2}{2m} \frac{\psi(x+\Delta x) - 2\psi(x) + \psi(x-\Delta x)}{\Delta x^{2}} \right] \mathrm{d}x. \end{aligned}\]
Where, the difference formula for the 2nd-order derivative:
\[\begin{aligned} % 2\psi(x) % + \frac{\mathrm{d}^{2} \psi(x)}{\mathrm{d} x^{2}} \Delta x^{2} % + O\left(\Delta x^{4}\right) % &= % \psi(x+\Delta x) % + \psi(x-\Delta x) % \\ % \frac{\mathrm{d}^{2} \psi(x)}{\mathrm{d} x^{2}} \Delta x^{2} % &= % \psi(x+\Delta x) % - 2\psi(x) % + \psi(x-\Delta x) % - O\left(\Delta x^{4}\right) % \\ % \frac{\mathrm{d}^{2} \psi(x)}{\mathrm{d} x^{2}} % &= % \frac{\psi(x+\Delta x) - 2\psi(x) + \psi(x-\Delta x)}{\Delta x^{2}} % - \frac{O\left(\Delta x^{4}\right)}{\Delta x^{2}} % \\ \frac{\mathrm{d}^{2} \psi(x)}{\mathrm{d} x^{2}} &= \frac{\psi(x+\Delta x) - 2\psi(x) + \psi(x-\Delta x)}{\Delta x^{2}} + O\left(\Delta x^{2}\right) \end{aligned}\]
are given by the sum of 2 Taylor series:
\[\begin{aligned} \psi(x+\Delta x) &= \psi(x) + \frac{\mathrm{d} \psi(x)}{\mathrm{d} x} \Delta x + \frac{1}{2!} \frac{\mathrm{d}^{2} \psi(x)}{\mathrm{d} x^{2}} \Delta x^{2} + \frac{1}{3!} \frac{\mathrm{d}^{3} \psi(x)}{\mathrm{d} x^{3}} \Delta x^{3} + O\left(\Delta x^{4}\right), \\ \psi(x-\Delta x) &= \psi(x) - \frac{\mathrm{d} \psi(x)}{\mathrm{d} x} \Delta x + \frac{1}{2!} \frac{\mathrm{d}^{2} \psi(x)}{\mathrm{d} x^{2}} \Delta x^{2} - \frac{1}{3!} \frac{\mathrm{d}^{3} \psi(x)}{\mathrm{d} x^{3}} \Delta x^{3} + O\left(\Delta x^{4}\right). \end{aligned}\]
k | n | analytical | numerical
--- | -- | ----------------- | -----------------
0.1 | 0 | 0.158113883008 | 0.158113879883 ✔
0.1 | 1 | 0.474341649025 | 0.474341633410 ✔
0.1 | 2 | 0.790569415042 | 0.790569374409 ✔
0.1 | 3 | 1.106797181059 | 1.106797102928 ✔
0.1 | 4 | 1.423024947076 | 1.423024818987 ✔
0.1 | 5 | 1.739252713093 | 1.739252522506 ✔
0.1 | 6 | 2.055480479109 | 2.055480213500 ✔
0.1 | 7 | 2.371708245126 | 2.371707891950 ✔
0.1 | 8 | 2.687936011143 | 2.687935558100 ✔
0.1 | 9 | 3.004163777160 | 3.004163211450 ✔
0.5 | 0 | 0.353553390593 | 0.353553374944 ✔
0.5 | 1 | 1.060660171780 | 1.060660093649 ✔
0.5 | 2 | 1.767766952966 | 1.767766749878 ✔
0.5 | 3 | 2.474873734153 | 2.474873343556 ✔
0.5 | 4 | 3.181980515339 | 3.181979874817 ✔
0.5 | 5 | 3.889087296526 | 3.889086343463 ✔
0.5 | 6 | 4.596194077713 | 4.596192749665 ✔
0.5 | 7 | 5.303300858899 | 5.303299093519 ✔
0.5 | 8 | 6.010407640086 | 6.010405374197 ✔
0.5 | 9 | 6.717514421272 | 6.717511593266 ✔
1.0 | 0 | 0.500000000000 | 0.499999968773 ✔
1.0 | 1 | 1.500000000000 | 1.499999843774 ✔
1.0 | 2 | 2.500000000000 | 2.499999593764 ✔
1.0 | 3 | 3.500000000000 | 3.499999218732 ✔
1.0 | 4 | 4.500000000000 | 4.499998718747 ✔
1.0 | 5 | 5.500000000000 | 5.499998093755 ✔
1.0 | 6 | 6.500000000000 | 6.499997343602 ✔
1.0 | 7 | 7.500000000000 | 7.499996468887 ✔
1.0 | 8 | 8.500000000000 | 8.499995468843 ✔
1.0 | 9 | 9.500000000000 | 9.499994343445 ✔
5.0 | 0 | 1.118033988750 | 1.118033832523 ✔
5.0 | 1 | 3.354101966250 | 3.354101184969 ✔
5.0 | 2 | 5.590169943749 | 5.590167912524 ✔
5.0 | 3 | 7.826237921249 | 7.826234014984 ✔
5.0 | 4 | 10.062305898749 | 10.062299492494 ✔
5.0 | 5 | 12.298373876249 | 12.298364344997 ✔
5.0 | 6 | 14.534441853749 | 14.534428572309 ✔
5.0 | 7 | 16.770509831248 | 16.770492175222 ✔
5.0 | 8 | 19.006577808748 | 19.006555152416 ✔
5.0 | 9 | 21.242645786248 | 21.242617504750 ✔