Schroedinger Simulation by Coding Physics (https://github.com/CodingPhysics)

The time evolution of a quantum particle in one dimension is animated.
The state of this particle is described by a complex wave function that obeys the Schrödinger equation.
This partial differential equation is solved numerically by a finite difference method.

Physical Model

We cosider the motion of a single quantum particle in one dimension for a given potential V(x). The state to this particle is discribed by a complex wave function Ψ(x,t) (probability amplitude). The motion of the particle is determined by the time development operator U(Δt):

\(Ψ(x, t+Δt) = U(Δt) Ψ(x,t)\),
\(U(Δt) = {e^{-iHΔt}}\)

where H is the Hamilton operator of the system:

\(H = -\frac{1}{2} \frac{d^2}{dx^2} + V(x)\)

For a small time step Δt the time development can be approximated by:

\((1 + i H \frac {Δt}{2} Ψ(x,t + Δt) ≈ (1 - i H \frac {Δt}{2}) Ψ(x,t)\) (*)

Algorithm

The continuous wave function is represented by an array of length N:

\({Ψ[j] = Ψ(j Δx)}\), j = 0, ..., N - 1

with \({Ψ[0] = Ψ[N-1] = 0}\) for fixed boundary conditions. The spatial derivative in the Hamilton operator is approximated by:

\(\frac {d^2Ψ(x)}{dx^2}\) ≈ \(\frac {Ψ[j - 1] - 2 Ψ[j] + Ψ[j]}{Δx^2}\)

Therefore, equation (*) can be rewritten as a matrix equation for the vectors Ψ' = Ψ(t + Δt) and Ψ = Ψ(t):

\(T Ψ' = - T* Ψ\) (**)

where T is an tridiagonal matrix with:

\(T[j][j-1] = T[j][j-1] = 1\)

\(T[j][j] = 4 i \frac {Δx^2}{Δt} - 2 Δx^2 V(j Δx) - 2\)

\(T[j][k] = 0\) elsewhere

The matrix equation (**) is solved directly by:

\(a[j] = - \frac {1}{(T[j][j] + a[j-1])}\) with \(a[0] = 0\)

\(b[j] = (-T* Ψ)[j] + a[j-1] b[j-1]\) with \(b[0] = 0\)

\(Ψ'[j] = a[j] (Ψ'[j+1] + b[j])\) with \(Ψ'[N-1] = 0\)