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)

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

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 |    1.000000000 |    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 ✔
 1 | 10 |    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 ✔
 2 | 10 |    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 ✔
 3 | 10 |    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 ✔
 4 | 10 |    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 ✔
 5 | 10 |    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 ✔
 6 | 10 |    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 ✔
 7 | 10 |    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 ✔
 8 | 10 |    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 ✔
 9 | 10 |    0.000000000 |    0.000000000 ✔
10 |  1 |    0.000000000 |   -0.000000000 ✔
10 |  2 |    0.000000000 |   -0.000000000 ✔
10 |  3 |    0.000000000 |    0.000000000 ✔
10 |  4 |    0.000000000 |   -0.000000000 ✔
10 |  5 |    0.000000000 |    0.000000000 ✔
10 |  6 |    0.000000000 |    0.000000000 ✔
10 |  7 |    0.000000000 |   -0.000000000 ✔
10 |  8 |    0.000000000 |    0.000000000 ✔
10 |  9 |    0.000000000 |    0.000000000 ✔
10 | 10 |    1.000000000 |    1.000000000 ✔
 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 ✔
 1 | 10 |    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 ✔
 2 | 10 |    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 ✔
 3 | 10 |    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 ✔
 4 | 10 |    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 ✔
 5 | 10 |    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 ✔
 6 | 10 |    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 ✔
 7 | 10 |    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 ✔
 8 | 10 |    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 ✔
 9 | 10 |    0.000000000 |    0.000000000 ✔
10 |  1 |    0.000000000 |   -0.000000000 ✔
10 |  2 |    0.000000000 |   -0.000000000 ✔
10 |  3 |    0.000000000 |    0.000000000 ✔
10 |  4 |    0.000000000 |   -0.000000000 ✔
10 |  5 |    0.000000000 |    0.000000000 ✔
10 |  6 |    0.000000000 |    0.000000000 ✔
10 |  7 |    0.000000000 |   -0.000000000 ✔
10 |  8 |    0.000000000 |    0.000000000 ✔
10 |  9 |    0.000000000 |    0.000000000 ✔
10 | 10 |    1.000000000 |    1.000000000 ✔
 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 ✔
 1 | 10 |    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 ✔
 2 | 10 |    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 ✔
 3 | 10 |    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 ✔
 4 | 10 |    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 ✔
 5 | 10 |    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 ✔
 6 | 10 |    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 ✔
 7 | 10 |    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 ✔
 8 | 10 |    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 ✔
 9 | 10 |    0.000000000 |    0.000000000 ✔
10 |  1 |    0.000000000 |   -0.000000000 ✔
10 |  2 |    0.000000000 |   -0.000000000 ✔
10 |  3 |    0.000000000 |    0.000000000 ✔
10 |  4 |    0.000000000 |   -0.000000000 ✔
10 |  5 |    0.000000000 |    0.000000000 ✔
10 |  6 |    0.000000000 |    0.000000000 ✔
10 |  7 |    0.000000000 |   -0.000000000 ✔
10 |  8 |    0.000000000 |    0.000000000 ✔
10 |  9 |    0.000000000 |    0.000000000 ✔
10 | 10 |    1.000000000 |    1.000000000 ✔
 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 ✔
 1 | 10 |    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 ✔
 2 | 10 |    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 ✔
 3 | 10 |    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 ✔
 4 | 10 |    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 ✔
 5 | 10 |    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 ✔
 6 | 10 |    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 ✔
 7 | 10 |    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 ✔
 8 | 10 |    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 ✔
 9 | 10 |    0.000000000 |    0.000000000 ✔
10 |  1 |    0.000000000 |   -0.000000000 ✔
10 |  2 |    0.000000000 |   -0.000000000 ✔
10 |  3 |    0.000000000 |    0.000000000 ✔
10 |  4 |    0.000000000 |   -0.000000000 ✔
10 |  5 |    0.000000000 |    0.000000000 ✔
10 |  6 |    0.000000000 |    0.000000000 ✔
10 |  7 |    0.000000000 |   -0.000000000 ✔
10 |  8 |    0.000000000 |    0.000000000 ✔
10 |  9 |    0.000000000 |    0.000000000 ✔
10 | 10 |    1.000000000 |    1.000000000 ✔
 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 ✔
 1 | 10 |    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 ✔
 2 | 10 |    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 ✔
 3 | 10 |    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 ✔
 4 | 10 |    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 ✔
 5 | 10 |    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 ✔
 6 | 10 |    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 ✔
 7 | 10 |    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 ✔
 8 | 10 |    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 ✔
 9 | 10 |    0.000000000 |    0.000000000 ✔
10 |  1 |    0.000000000 |    0.000000000 ✔
10 |  2 |    0.000000000 |    0.000000000 ✔
10 |  3 |    0.000000000 |    0.000000000 ✔
10 |  4 |    0.000000000 |    0.000000000 ✔
10 |  5 |    0.000000000 |    0.000000000 ✔
10 |  6 |    0.000000000 |   -0.000000000 ✔
10 |  7 |    0.000000000 |   -0.000000000 ✔
10 |  8 |    0.000000000 |    0.000000000 ✔
10 |  9 |    0.000000000 |    0.000000000 ✔
10 | 10 |    1.000000000 |    1.000000000 ✔
 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 ✔
 1 | 10 |    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 ✔
 2 | 10 |    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 ✔
 3 | 10 |    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 ✔
 4 | 10 |    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 ✔
 5 | 10 |    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 ✔
 6 | 10 |    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 ✔
 7 | 10 |    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 ✔
 8 | 10 |    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 ✔
 9 | 10 |    0.000000000 |    0.000000000 ✔
10 |  1 |    0.000000000 |    0.000000000 ✔
10 |  2 |    0.000000000 |    0.000000000 ✔
10 |  3 |    0.000000000 |    0.000000000 ✔
10 |  4 |    0.000000000 |    0.000000000 ✔
10 |  5 |    0.000000000 |    0.000000000 ✔
10 |  6 |    0.000000000 |   -0.000000000 ✔
10 |  7 |    0.000000000 |   -0.000000000 ✔
10 |  8 |    0.000000000 |    0.000000000 ✔
10 |  9 |    0.000000000 |    0.000000000 ✔
10 | 10 |    1.000000000 |    1.000000000 ✔
 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 ✔
 1 | 10 |    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 ✔
 2 | 10 |    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 ✔
 3 | 10 |    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 ✔
 4 | 10 |    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 ✔
 5 | 10 |    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 ✔
 6 | 10 |    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 ✔
 7 | 10 |    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 ✔
 8 | 10 |    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 ✔
 9 | 10 |    0.000000000 |    0.000000000 ✔
10 |  1 |    0.000000000 |    0.000000000 ✔
10 |  2 |    0.000000000 |    0.000000000 ✔
10 |  3 |    0.000000000 |    0.000000000 ✔
10 |  4 |    0.000000000 |    0.000000000 ✔
10 |  5 |    0.000000000 |    0.000000000 ✔
10 |  6 |    0.000000000 |   -0.000000000 ✔
10 |  7 |    0.000000000 |   -0.000000000 ✔
10 |  8 |    0.000000000 |    0.000000000 ✔
10 |  9 |    0.000000000 |    0.000000000 ✔
10 | 10 |    1.000000000 |    1.000000000 ✔
 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 ✔
 1 | 10 |    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 ✔
 2 | 10 |    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 ✔
 3 | 10 |    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 ✔
 4 | 10 |    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 ✔
 5 | 10 |    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 ✔
 6 | 10 |    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 ✔
 7 | 10 |    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 ✔
 8 | 10 |    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 ✔
 9 | 10 |    0.000000000 |    0.000000000 ✔
10 |  1 |    0.000000000 |    0.000000000 ✔
10 |  2 |    0.000000000 |    0.000000000 ✔
10 |  3 |    0.000000000 |    0.000000000 ✔
10 |  4 |    0.000000000 |    0.000000000 ✔
10 |  5 |    0.000000000 |    0.000000000 ✔
10 |  6 |    0.000000000 |   -0.000000000 ✔
10 |  7 |    0.000000000 |   -0.000000000 ✔
10 |  8 |    0.000000000 |    0.000000000 ✔
10 |  9 |    0.000000000 |    0.000000000 ✔
10 | 10 |    1.000000000 |    1.000000000 ✔

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 |   49.348021579 |   49.348022005 ✔
0.1 | 0.1 | 0.1 |  2 |  197.392081462 |  197.392088022 ✔
0.1 | 0.1 | 0.1 |  3 |  444.132165131 |  444.132198049 ✔
0.1 | 0.1 | 0.1 |  4 |  789.568248176 |  789.568352087 ✔
0.1 | 0.1 | 0.1 |  5 | 1233.700296336 | 1233.700550136 ✔
0.1 | 0.1 | 0.1 |  6 | 1776.528266243 | 1776.528792196 ✔
0.1 | 0.1 | 0.1 |  7 | 2418.052103857 | 2418.053078267 ✔
0.1 | 0.1 | 0.1 |  8 | 3158.271745876 | 3158.273408349 ✔
0.1 | 0.1 | 0.1 |  9 | 3997.187119264 | 3997.189782441 ✔
0.1 | 0.1 | 0.1 | 10 | 4934.798141995 | 4934.802200545 ✔
0.1 | 0.1 | 1.0 |  1 | 4934.802157914 | 4934.802200545 ✔
0.1 | 0.1 | 1.0 |  2 | 19739.208146194 | 19739.208802179 ✔
0.1 | 0.1 | 1.0 |  3 | 44413.216513102 | 44413.219804902 ✔
0.1 | 0.1 | 1.0 |  4 | 78956.824817598 | 78956.835208715 ✔
0.1 | 0.1 | 1.0 |  5 | 123370.029633619 | 123370.055013617 ✔
0.1 | 0.1 | 1.0 |  6 | 177652.826624333 | 177652.879219608 ✔
0.1 | 0.1 | 1.0 |  7 | 241805.210385708 | 241805.307826689 ✔
0.1 | 0.1 | 1.0 |  8 | 315827.174587593 | 315827.340834859 ✔
0.1 | 0.1 | 1.0 |  9 | 399718.711926427 | 399718.978244119 ✔
0.1 | 0.1 | 1.0 | 10 | 493479.814199451 | 493480.220054468 ✔
0.1 | 1.0 | 0.1 |  1 |    4.934802158 |    4.934802201 ✔
0.1 | 1.0 | 0.1 |  2 |   19.739208146 |   19.739208802 ✔
0.1 | 1.0 | 0.1 |  3 |   44.413216513 |   44.413219805 ✔
0.1 | 1.0 | 0.1 |  4 |   78.956824818 |   78.956835209 ✔
0.1 | 1.0 | 0.1 |  5 |  123.370029634 |  123.370055014 ✔
0.1 | 1.0 | 0.1 |  6 |  177.652826624 |  177.652879220 ✔
0.1 | 1.0 | 0.1 |  7 |  241.805210386 |  241.805307827 ✔
0.1 | 1.0 | 0.1 |  8 |  315.827174588 |  315.827340835 ✔
0.1 | 1.0 | 0.1 |  9 |  399.718711926 |  399.718978244 ✔
0.1 | 1.0 | 0.1 | 10 |  493.479814199 |  493.480220054 ✔
0.1 | 1.0 | 1.0 |  1 |  493.480215791 |  493.480220054 ✔
0.1 | 1.0 | 1.0 |  2 | 1973.920814619 | 1973.920880218 ✔
0.1 | 1.0 | 1.0 |  3 | 4441.321651310 | 4441.321980490 ✔
0.1 | 1.0 | 1.0 |  4 | 7895.682481760 | 7895.683520871 ✔
0.1 | 1.0 | 1.0 |  5 | 12337.002963362 | 12337.005501362 ✔
0.1 | 1.0 | 1.0 |  6 | 17765.282662433 | 17765.287921961 ✔
0.1 | 1.0 | 1.0 |  7 | 24180.521038571 | 24180.530782669 ✔
0.1 | 1.0 | 1.0 |  8 | 31582.717458759 | 31582.734083486 ✔
0.1 | 1.0 | 1.0 |  9 | 39971.871192643 | 39971.897824412 ✔
0.1 | 1.0 | 1.0 | 10 | 49347.981419945 | 49348.022005447 ✔
1.0 | 0.1 | 0.1 |  1 |    0.493480216 |    0.493480220 ✔
1.0 | 0.1 | 0.1 |  2 |    1.973920815 |    1.973920880 ✔
1.0 | 0.1 | 0.1 |  3 |    4.441321652 |    4.441321980 ✔
1.0 | 0.1 | 0.1 |  4 |    7.895682481 |    7.895683521 ✔
1.0 | 0.1 | 0.1 |  5 |   12.337002965 |   12.337005501 ✔
1.0 | 0.1 | 0.1 |  6 |   17.765282662 |   17.765287922 ✔
1.0 | 0.1 | 0.1 |  7 |   24.180521036 |   24.180530783 ✔
1.0 | 0.1 | 0.1 |  8 |   31.582717460 |   31.582734083 ✔
1.0 | 0.1 | 0.1 |  9 |   39.971871195 |   39.971897824 ✔
1.0 | 0.1 | 0.1 | 10 |   49.347981418 |   49.348022005 ✔
1.0 | 0.1 | 1.0 |  1 |   49.348021595 |   49.348022005 ✔
1.0 | 0.1 | 1.0 |  2 |  197.392081542 |  197.392088022 ✔
1.0 | 0.1 | 1.0 |  3 |  444.132165194 |  444.132198049 ✔
1.0 | 0.1 | 1.0 |  4 |  789.568248126 |  789.568352087 ✔
1.0 | 0.1 | 1.0 |  5 | 1233.700296503 | 1233.700550136 ✔
1.0 | 0.1 | 1.0 |  6 | 1776.528266171 | 1776.528792196 ✔
1.0 | 0.1 | 1.0 |  7 | 2418.052103606 | 2418.053078267 ✔
1.0 | 0.1 | 1.0 |  8 | 3158.271746002 | 3158.273408349 ✔
1.0 | 0.1 | 1.0 |  9 | 3997.187119519 | 3997.189782441 ✔
1.0 | 0.1 | 1.0 | 10 | 4934.798141783 | 4934.802200545 ✔
1.0 | 1.0 | 0.1 |  1 |    0.049348022 |    0.049348022 ✔
1.0 | 1.0 | 0.1 |  2 |    0.197392082 |    0.197392088 ✔
1.0 | 1.0 | 0.1 |  3 |    0.444132165 |    0.444132198 ✔
1.0 | 1.0 | 0.1 |  4 |    0.789568248 |    0.789568352 ✔
1.0 | 1.0 | 0.1 |  5 |    1.233700297 |    1.233700550 ✔
1.0 | 1.0 | 0.1 |  6 |    1.776528266 |    1.776528792 ✔
1.0 | 1.0 | 0.1 |  7 |    2.418052104 |    2.418053078 ✔
1.0 | 1.0 | 0.1 |  8 |    3.158271746 |    3.158273408 ✔
1.0 | 1.0 | 0.1 |  9 |    3.997187120 |    3.997189782 ✔
1.0 | 1.0 | 0.1 | 10 |    4.934798142 |    4.934802201 ✔
1.0 | 1.0 | 1.0 |  1 |    4.934802159 |    4.934802201 ✔
1.0 | 1.0 | 1.0 |  2 |   19.739208154 |   19.739208802 ✔
1.0 | 1.0 | 1.0 |  3 |   44.413216519 |   44.413219805 ✔
1.0 | 1.0 | 1.0 |  4 |   78.956824813 |   78.956835209 ✔
1.0 | 1.0 | 1.0 |  5 |  123.370029650 |  123.370055014 ✔
1.0 | 1.0 | 1.0 |  6 |  177.652826617 |  177.652879220 ✔
1.0 | 1.0 | 1.0 |  7 |  241.805210361 |  241.805307827 ✔
1.0 | 1.0 | 1.0 |  8 |  315.827174600 |  315.827340835 ✔
1.0 | 1.0 | 1.0 |  9 |  399.718711952 |  399.718978244 ✔
1.0 | 1.0 | 1.0 | 10 |  493.479814178 |  493.480220054 ✔

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 |    0.050000000 |    0.050000000 ✔
0.5 |  1 |    0.250000000 |    0.250000000 ✔
1.0 |  1 |    0.500000000 |    0.500000000 ✔
7.0 |  1 |    3.500000000 |    3.500000000 ✔

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 |    0.002826727 |    0.002826727 ✔
0.5 |  1 |    0.070668185 |    0.070668185 ✔
1.0 |  1 |    0.282672742 |    0.282672742 ✔
7.0 |  1 |   13.850964334 |   13.850964334 ✔

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 |    0.000000000 |    0.000000000 ✔
0.5 |  1 |    0.000000000 |    0.000000000 ✔
1.0 |  1 |    0.000000000 |    0.000000000 ✔
7.0 |  1 |    0.000000000 |    0.000000000 ✔

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 |  986.960431583 |  986.960440109 ✔
0.5 |  1 |   39.478417274 |   39.478417604 ✔
1.0 |  1 |    9.869604319 |    9.869604401 ✔
7.0 |  1 |    0.201420496 |    0.201420498 ✔

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 $p^2$