Infinite Potential Well (Particle in a Box)

The infinite potential well (particle in a box) is the simplest model for quantum mechanical system.

Definitions

Antique.InfinitePotentialWell — Type

Model

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:

IPW = InfinitePotentialWell(L=1.0, m=1.0, ℏ=1.0)

$L$ is the length of the box, $m$ is the mass of particle and $\hbar$ is the reduced Planck constant (Dirac's constant).

References

D. J. Griffiths, D. F. Schroeter, Introduction to Quantum Mechanics Third Edition (Cambridge University Press, 2018) p.31, 2.2 THE INFINITE SQUARE WELL

Proofs

source

Potential

Antique.V — Method

V(model::InfinitePotentialWell; x)

\[V(x) = \left\{ \begin{array}{ll} \infty & x \lt 0, L \lt x \\ 0 & 0 \leq x \leq L \end{array} \right.\]

source

Eigenvalues

Antique.E — Method

E(model::InfinitePotentialWell; n::Int=1)

\[E_n = \frac{\hbar^2 n^2 \pi^2}{2 m L^2}\]

source

Eigenfunctions

Antique.ψ — Method

ψ(model::InfinitePotentialWell, x; n::Int=1)

\[\psi_n(x) = \sqrt{\frac{2}{L}} \sin \frac{n\pi x}{L}\]

source

Install Antique.jl for the first use and run using Antique before each use. The energy E(), wavefunction ψ() and potential V() will be exported. In this system, the model is generated by InfinitePotentialWell and several parameters L, m and ℏ are set as optional arguments.

using Antique
IPW = InfinitePotentialWell(L=1.0, m=1.0, ℏ=1.0)

Parameters:

julia> IPW.L1.0
julia> IPW.m1.0
julia> IPW.ℏ1.0

Eigenvalues:

julia> E(IPW, n=1)4.934802200544679
julia> E(IPW, n=2)19.739208802178716

Wave functions:

using CairoMakie

# settings
f = Figure()
ax = Axis(f[1,1], xlabel=L"$x$", ylabel=L"$\psi(x)$")

# plot
w1 = lines!(ax, 0..1, x -> ψ(IPW, x, n=1))
w2 = lines!(ax, 0..1, x -> ψ(IPW, x, n=2))
w3 = lines!(ax, 0..1, x -> ψ(IPW, x, n=3))
w4 = lines!(ax, 0..1, x -> ψ(IPW, x, n=4))
w5 = lines!(ax, 0..1, x -> ψ(IPW, x, n=5))

# legend
axislegend(ax, [w1, w2, w3, w4, w5], [L"n=1", L"n=2", L"n=3", L"n=4", L"n=5"], 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=(-0.5,1.5,0,140))
# hidespines!(ax)
# hidedecorations!(ax)

# plot
for n in 1:5
  # energy
  lines!(ax, [0,IPW.L], fill(E(IPW,n=n),2), color=:black, linewidth=2)
  # wave function
  lines!(ax, 0..IPW.L, x -> E(IPW,n=n) + 5*ψ(IPW,x,n=n), linewidth=2)
end

#potential
lines!(ax, [0,0,IPW.L,IPW.L], [140,0,0,140], color=:black, linewidth=2)

f

Testing

Unit testing and Integration testing were done using numerical integration (QuadGK.jl). The test script is here.

Normalization & Orthogonality of $\psi_n(x)$

\[\int_{0}^{L} \psi_i^\ast(x) \psi_j(x) ~\mathrm{d}x = \delta_{ij}\]

 i |  j |        analytical |         numerical 
-- | -- | ----------------- | ----------------- 
 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 ✔
 1 | 10 |    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 ✔
 2 | 10 |    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 ✔
 3 | 10 |    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 ✔
 4 | 10 |    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 ✔
 5 | 10 |    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 ✔
 6 | 10 |    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 ✔
 7 | 10 |    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 ✔
 8 | 10 |    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 ✔
 9 | 10 |    0.000000000000 |    0.000000000000 ✔
10 |  1 |    0.000000000000 |    0.000000000000 ✔
10 |  2 |    0.000000000000 |    0.000000000000 ✔
10 |  3 |    0.000000000000 |    0.000000000000 ✔
10 |  4 |    0.000000000000 |    0.000000000000 ✔
10 |  5 |    0.000000000000 |    0.000000000000 ✔
10 |  6 |    0.000000000000 |   -0.000000000000 ✔
10 |  7 |    0.000000000000 |   -0.000000000000 ✔
10 |  8 |    0.000000000000 |    0.000000000000 ✔
10 |  9 |    0.000000000000 |    0.000000000000 ✔
10 | 10 |    1.000000000000 |    1.000000000000 ✔

Eigenvalues

\[ \begin{aligned} E_n &= \int_0^L \psi^\ast_n(x) \hat{H} \psi_n(x) ~\mathrm{d}x \\ &= \int_0^L \psi^\ast_n(x) \left[ \hat{V} + \hat{T} \right] \psi(x) ~\mathrm{d}x \\ &= \int_0^L \psi^\ast_n(x) \left[ 0 - \frac{\hbar^2}{2m} \frac{\mathrm{d}^{2}}{\mathrm{d} x^{2}} \right] \psi(x) ~\mathrm{d}x \\ &\simeq \int_0^L \psi^\ast_n(x) \left[ -\frac{\hbar^2}{2m} \frac{\psi(x+\Delta x) - 2\psi(x) + \psi(x-\Delta x)}{\Delta x^{2}} \right] ~\mathrm{d}x. \end{aligned}\]

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:

\[\begin{aligned} \psi(x+\Delta x) &= \psi(x) + \frac{\mathrm{d} \psi(x)}{\mathrm{d} x} \Delta x + \frac{1}{2!} \frac{\mathrm{d}^{2} \psi(x)}{\mathrm{d} x^{2}} \Delta x^{2} + \frac{1}{3!} \frac{\mathrm{d}^{3} \psi(x)}{\mathrm{d} x^{3}} \Delta x^{3} + O\left(\Delta x^{4}\right), \\ \psi(x-\Delta x) &= \psi(x) - \frac{\mathrm{d} \psi(x)}{\mathrm{d} x} \Delta x + \frac{1}{2!} \frac{\mathrm{d}^{2} \psi(x)}{\mathrm{d} x^{2}} \Delta x^{2} - \frac{1}{3!} \frac{\mathrm{d}^{3} \psi(x)}{\mathrm{d} x^{3}} \Delta x^{3} + O\left(\Delta x^{4}\right). \end{aligned}\]

  L |   m |   ℏ |  n |        analytical |         numerical 
--- | --- | --- | -- | ----------------- | ----------------- 
0.1 | 0.1 | 0.1 |  1 |   49.348021579139 |   49.348022005447 ✔
0.1 | 0.1 | 0.1 |  2 |  197.392081461942 |  197.392088021787 ✔
0.1 | 0.1 | 0.1 |  3 |  444.132165131018 |  444.132198049021 ✔
0.1 | 0.1 | 0.1 |  4 |  789.568248175979 |  789.568352087149 ✔
0.1 | 0.1 | 0.1 |  5 | 1233.700296336187 | 1233.700550136170 ✔
0.1 | 0.1 | 0.1 |  6 | 1776.528266243334 | 1776.528792196084 ✔
0.1 | 0.1 | 0.1 |  7 | 2418.052103857080 | 2418.053078266893 ✔
0.1 | 0.1 | 0.1 |  8 | 3158.271745875927 | 3158.273408348594 ✔
0.1 | 0.1 | 0.1 |  9 | 3997.187119264267 | 3997.189782441189 ✔
0.1 | 0.1 | 0.1 | 10 | 4934.798141994514 | 4934.802200544678 ✔
0.1 | 0.1 | 1.0 |  1 | 4934.802157913905 | 4934.802200544678 ✔
0.1 | 0.1 | 1.0 |  2 | 19739.208146194214 | 19739.208802178713 ✔
0.1 | 0.1 | 1.0 |  3 | 44413.216513101754 | 44413.219804902103 ✔
0.1 | 0.1 | 1.0 |  4 | 78956.824817597895 | 78956.835208714852 ✔
0.1 | 0.1 | 1.0 |  5 | 123370.029633618717 | 123370.055013616948 ✔
0.1 | 0.1 | 1.0 |  6 | 177652.826624333364 | 177652.879219608410 ✔
0.1 | 0.1 | 1.0 |  7 | 241805.210385707964 | 241805.307826689212 ✔
0.1 | 0.1 | 1.0 |  8 | 315827.174587592541 | 315827.340834859409 ✔
0.1 | 0.1 | 1.0 |  9 | 399718.711926426622 | 399718.978244118916 ✔
0.1 | 0.1 | 1.0 | 10 | 493479.814199451241 | 493480.220054467791 ✔
0.1 | 1.0 | 0.1 |  1 |    4.934802157914 |    4.934802200545 ✔
0.1 | 1.0 | 0.1 |  2 |   19.739208146194 |   19.739208802179 ✔
0.1 | 1.0 | 0.1 |  3 |   44.413216513102 |   44.413219804902 ✔
0.1 | 1.0 | 0.1 |  4 |   78.956824817598 |   78.956835208715 ✔
0.1 | 1.0 | 0.1 |  5 |  123.370029633619 |  123.370055013617 ✔
0.1 | 1.0 | 0.1 |  6 |  177.652826624333 |  177.652879219608 ✔
0.1 | 1.0 | 0.1 |  7 |  241.805210385708 |  241.805307826689 ✔
0.1 | 1.0 | 0.1 |  8 |  315.827174587593 |  315.827340834859 ✔
0.1 | 1.0 | 0.1 |  9 |  399.718711926427 |  399.718978244119 ✔
0.1 | 1.0 | 0.1 | 10 |  493.479814199451 |  493.480220054468 ✔
0.1 | 1.0 | 1.0 |  1 |  493.480215791390 |  493.480220054468 ✔
0.1 | 1.0 | 1.0 |  2 | 1973.920814619422 | 1973.920880217871 ✔
0.1 | 1.0 | 1.0 |  3 | 4441.321651310176 | 4441.321980490210 ✔
0.1 | 1.0 | 1.0 |  4 | 7895.682481759791 | 7895.683520871485 ✔
0.1 | 1.0 | 1.0 |  5 | 12337.002963361872 | 12337.005501361695 ✔
0.1 | 1.0 | 1.0 |  6 | 17765.282662433339 | 17765.287921960840 ✔
0.1 | 1.0 | 1.0 |  7 | 24180.521038570794 | 24180.530782668924 ✔
0.1 | 1.0 | 1.0 |  8 | 31582.717458759253 | 31582.734083485939 ✔
0.1 | 1.0 | 1.0 |  9 | 39971.871192642662 | 39971.897824411892 ✔
0.1 | 1.0 | 1.0 | 10 | 49347.981419945128 | 49348.022005446779 ✔
1.0 | 0.1 | 0.1 |  1 |    0.493480215948 |    0.493480220054 ✔
1.0 | 0.1 | 0.1 |  2 |    1.973920815419 |    1.973920880218 ✔
1.0 | 0.1 | 0.1 |  3 |    4.441321651944 |    4.441321980490 ✔
1.0 | 0.1 | 0.1 |  4 |    7.895682481265 |    7.895683520871 ✔
1.0 | 0.1 | 0.1 |  5 |   12.337002965030 |   12.337005501362 ✔
1.0 | 0.1 | 0.1 |  6 |   17.765282661715 |   17.765287921961 ✔
1.0 | 0.1 | 0.1 |  7 |   24.180521036064 |   24.180530782669 ✔
1.0 | 0.1 | 0.1 |  8 |   31.582717460023 |   31.582734083486 ✔
1.0 | 0.1 | 0.1 |  9 |   39.971871195191 |   39.971897824412 ✔
1.0 | 0.1 | 0.1 | 10 |   49.347981417827 |   49.348022005447 ✔
1.0 | 0.1 | 1.0 |  1 |   49.348021594816 |   49.348022005447 ✔
1.0 | 0.1 | 1.0 |  2 |  197.392081541864 |  197.392088021787 ✔
1.0 | 0.1 | 1.0 |  3 |  444.132165194438 |  444.132198049021 ✔
1.0 | 0.1 | 1.0 |  4 |  789.568248126463 |  789.568352087149 ✔
1.0 | 0.1 | 1.0 |  5 | 1233.700296503016 | 1233.700550136170 ✔
1.0 | 0.1 | 1.0 |  6 | 1776.528266171473 | 1776.528792196084 ✔
1.0 | 0.1 | 1.0 |  7 | 2418.052103606433 | 2418.053078266892 ✔
1.0 | 0.1 | 1.0 |  8 | 3158.271746002275 | 3158.273408348594 ✔
1.0 | 0.1 | 1.0 |  9 | 3997.187119519121 | 3997.189782441190 ✔
1.0 | 0.1 | 1.0 | 10 | 4934.798141782662 | 4934.802200544679 ✔
1.0 | 1.0 | 0.1 |  1 |    0.049348021595 |    0.049348022005 ✔
1.0 | 1.0 | 0.1 |  2 |    0.197392081542 |    0.197392088022 ✔
1.0 | 1.0 | 0.1 |  3 |    0.444132165194 |    0.444132198049 ✔
1.0 | 1.0 | 0.1 |  4 |    0.789568248126 |    0.789568352087 ✔
1.0 | 1.0 | 0.1 |  5 |    1.233700296503 |    1.233700550136 ✔
1.0 | 1.0 | 0.1 |  6 |    1.776528266171 |    1.776528792196 ✔
1.0 | 1.0 | 0.1 |  7 |    2.418052103606 |    2.418053078267 ✔
1.0 | 1.0 | 0.1 |  8 |    3.158271746002 |    3.158273408349 ✔
1.0 | 1.0 | 0.1 |  9 |    3.997187119519 |    3.997189782441 ✔
1.0 | 1.0 | 0.1 | 10 |    4.934798141783 |    4.934802200545 ✔
1.0 | 1.0 | 1.0 |  1 |    4.934802159482 |    4.934802200545 ✔
1.0 | 1.0 | 1.0 |  2 |   19.739208154186 |   19.739208802179 ✔
1.0 | 1.0 | 1.0 |  3 |   44.413216519444 |   44.413219804902 ✔
1.0 | 1.0 | 1.0 |  4 |   78.956824812646 |   78.956835208715 ✔
1.0 | 1.0 | 1.0 |  5 |  123.370029650302 |  123.370055013617 ✔
1.0 | 1.0 | 1.0 |  6 |  177.652826617147 |  177.652879219608 ✔
1.0 | 1.0 | 1.0 |  7 |  241.805210360643 |  241.805307826689 ✔
1.0 | 1.0 | 1.0 |  8 |  315.827174600228 |  315.827340834859 ✔
1.0 | 1.0 | 1.0 |  9 |  399.718711951912 |  399.718978244119 ✔
1.0 | 1.0 | 1.0 | 10 |  493.479814178266 |  493.480220054468 ✔

Expected Value of $x$

\[\langle x \rangle_{n=1} = \int_{0}^{L} \psi_1^\ast(x) \hat{x} \psi_1(x) ~\mathrm{d}x = \frac{2(2a)^2}{\pi^3} \left( \frac{\pi^3}{6} - \frac{\pi}{4} \right)\]

for only $n=1$.

Reference:

LibreTexts PHYSICS, 6.4: Expectation Values, Observables, and Uncertainty

  L |  n |        analytical |         numerical 
--- | -- | ----------------- | ----------------- 
0.1 |  1 |    0.050000000000 |    0.050000000000 ✔
0.5 |  1 |    0.250000000000 |    0.250000000000 ✔
1.0 |  1 |    0.500000000000 |    0.500000000000 ✔
7.0 |  1 |    3.500000000000 |    3.500000000000 ✔

Expected Value of $x^2$

\[\langle x^2 \rangle_{n=1} = \int_{0}^{L} \psi_1^\ast(x) \hat{x}^2 \psi_1(x) ~\mathrm{d}x = \frac{2(2a)^2}{\pi^3} \left( \frac{\pi^3}{6} - \frac{\pi}{4} \right)\]

Reference:

LibreTexts PHYSICS, 6.4: Expectation Values, Observables, and Uncertainty

  L |  n |        analytical |         numerical 
--- | -- | ----------------- | ----------------- 
0.1 |  1 |    0.002826727415 |    0.002826727415 ✔
0.5 |  1 |    0.070668185378 |    0.070668185378 ✔
1.0 |  1 |    0.282672741512 |    0.282672741512 ✔
7.0 |  1 |   13.850964334096 |   13.850964334096 ✔

Expected Value of $p$

\[\langle p \rangle_{n=1} = \int_{0}^{L} \psi_1^\ast(x) \hat{p} \psi_1(x) ~\mathrm{d}x = 0\]

Reference:

LibreTexts PHYSICS, 6.4: Expectation Values, Observables, and Uncertainty

\[ \begin{aligned} \langle p \rangle_{n=1} &= \int_0^L \psi^\ast_n(x) \hat{p} \psi_n(x) ~\mathrm{d}x \\ &= \int_0^L \psi^\ast_n(x) \left[ -i\hbar\frac{\mathrm{d}}{\mathrm{d} x} \right] \psi(x) ~\mathrm{d}x \\ &\simeq \int_0^L \psi^\ast_n(x) \left[ -i\hbar \frac{\psi(x+\Delta x) - \psi(x-\Delta x)}{2\Delta x} \right] ~\mathrm{d}x. \end{aligned}\]

Where, the difference formula for the 2nd-order derivative:

\[\begin{aligned} % 2\frac{\mathrm{d} \psi(x)}{\mathrm{d}x} \Delta x % + O\left(\Delta x^{3}\right) % &= % \psi(x+\Delta x) % - \psi(x-\Delta x) % \\ % 2\frac{\mathrm{d} \psi(x)}{\mathrm{d}x} \Delta x % &= % \psi(x+\Delta x) % - \psi(x-\Delta x) % - O\left(\Delta x^{3}\right) % \\ % \frac{\mathrm{d} \psi(x)}{\mathrm{d}x} % &= % \frac{\psi(x+\Delta x)- \psi(x-\Delta x)}{2\Delta x} % - \frac{O\left(\Delta x^{3}\right)}{2\Delta x} % \\ \frac{\mathrm{d} \psi(x)}{\mathrm{d}x} &= \frac{\psi(x+\Delta x)- \psi(x-\Delta x)}{2\Delta x} + O\left(\Delta x^{2}\right), \end{aligned}\]

are given by the sum of 2 Taylor series:

\[\begin{aligned} \psi(x+\Delta x) &= \psi(x) + \frac{\mathrm{d} \psi(x)}{\mathrm{d}x} \Delta x + \frac{1}{2!} \frac{\mathrm{d}^{2} \psi(x)}{\mathrm{d}x^{2}} \Delta x^{2} + O\left(\Delta x^{3}\right), \\ \psi(x-\Delta x) &= \psi(x) - \frac{\mathrm{d} \psi(x)}{\mathrm{d}x} \Delta x + \frac{1}{2!} \frac{\mathrm{d}^{2} \psi(x)}{\mathrm{d}x^{2}} \Delta x^{2} + O\left(\Delta x^{3}\right). \end{aligned}\]

  L |  n |        analytical |         numerical 
--- | -- | ----------------- | ----------------- 
0.1 |  1 |    0.000000000003 |    0.000000000000 ✔
0.5 |  1 |    0.000000000000 |    0.000000000000 ✔
1.0 |  1 |    0.000000000000 |    0.000000000000 ✔
7.0 |  1 |    0.000000000000 |    0.000000000000 ✔

Expected Value of $p^2$

\[\langle p^2 \rangle = \int_{0}^{L} \psi_1^\ast(x) \hat{p}^2 \psi_1(x) ~\mathrm{d}x = \frac{\pi^2\hbar^2}{L^2}\]

Reference:

LibreTexts PHYSICS, 6.4: Expectation Values, Observables, and Uncertainty

\[ \begin{aligned} \langle p^2 \rangle &= \int_0^L \psi^\ast_n(x) \hat{p} \psi_n(x) ~\mathrm{d}x \\ &= \int_0^L \psi^\ast_n(x) \left[ -\hbar^2\frac{\mathrm{d}^2}{{\mathrm{d}x}^2} \right] \psi(x) ~\mathrm{d}x \\ &\simeq \int_0^L \psi^\ast_n(x) \left[ -\hbar^2 \frac{\psi(x+\Delta x) - 2\psi(x) + \psi(x-\Delta x)}{\Delta x^{2}} \right] ~\mathrm{d}x. \end{aligned}\]

Where, the difference formula for the 2nd-order derivative:

are given by the sum of 2 Taylor series:

  L |  n |        analytical |         numerical 
--- | -- | ----------------- | ----------------- 
0.1 |  1 |  986.960431582781 |  986.960440108936 ✔
0.5 |  1 |   39.478417274195 |   39.478417604357 ✔
1.0 |  1 |    9.869604318963 |    9.869604401089 ✔
7.0 |  1 |    0.201420496383 |    0.201420497981 ✔

Infinite Potential Well (Particle in a Box)

Definitions

Potential

Eigenvalues

Eigenfunctions

Usage & Examples

Testing

Normalization & Orthogonality of $\psi_n(x)$

Eigenvalues

Expected Value of $x$

Expected Value of $x^2$

Expected Value of $p$

Expected Value of $p^2$