Simulating Schrödinger Equation in 1 Dimension

The time dependent schrodingers equation is:

∂ψ(x,t)∂t=−iℏH^ψ(x,t)

A naive implementation using Eulers method as simple as possible with Euler's method will usually cause the waveform to explode due to integration errors.

From quantization ψ is an infinite dimensional vector, but we will approximate it as a finite dimensional vector. H^ is now a finite matrix.

∂ψ(x,t)∂t=−iℏH^ψ(x,t)

Assuming the Hamiltonian isn't time dependent, the differential equation has the solution.

ψ(x,t+Δt)=e−iℏH^Δtψ(x,t)

We define the Hamiltonian operator to be a non-relativistic particle with some potential.

H^=−ℏ22m∂2∂x2+V(x)
.

Since we approximated ψ as a finite dimensional vector, we should approximate the second derivative using the finite difference method.

limh→0⁡df(x+h)dx−df(x)dxh
=limh→0⁡f(x+h)−f(x)h−f(x)−f(x−h)hh
=limh→0⁡(f(x+h)−f(x))−(f(x)−f(x−h))h2
=limh→0⁡f(x+h)−2f(x)+f(x−h)h2
≈f(x+Δx)−2f(x)+f(x−Δx)Δx2

So for each position of −ℏ22m∂2∂x2 we can make a matrix below. We assume no repeating boundary conditions. The matrix is equivelent to the Laplacian of a path graph.

−ℏ22m1Δx2[−111−21⋱1−211−1]

The potential term V(x) is simply the value at each point:

[V(0)V(Δx)⋱V((n−1)Δx)V(nΔx)]

H^ is the sum of those two matricies

H^=−ℏ22m1Δx2[−111−21⋱1−211−1]+[V(0)V(Δx)⋱V((n−1)Δx)V(nΔx)]

Each discrete step of the simulation will go through a certain change in time using the matrix exponent of the Hamiltonian operator.

ψt+1=e−iℏH^Δtψt