Simulating Schrödinger Equation in 1 Dimension

The time dependent schrodingers equation is:

If this is implemented as simple as possible with Euler's method, it will usually be inaccurate and cause the waveform to explode due to integration errors. Simulating the equation will require a more accurate method.

The iterative simulation will approximate $\psi$ as a finite dimensional vector, so it should approximate $\hat{H}$ a matrix.

Assuming the Hamiltonian is constant, integrating gets

The Hamiltonian operator is equal to $\hat{H}=-{\displaystyle \frac{{\hslash }^2}{2m}}{\displaystyle \frac{{\partial }^2}{\partial x^2}}+V(x)$.

The second derivative can be calculated by:

We can approximate this with a finite difference.

So for each position of $-{\displaystyle \frac{{\hslash }^2}{2m}}{\displaystyle \frac{{\partial }^2}{\partial x^2}}$ we can make a matrix below. We assume no repeating boundary conditions (equivelent to a discrete sine transform).

Then for the $V(x)$ part, it is just a value at each point:

So $\hat{H}$ is just the sum of those two matricies

And then each discrete step of the simulation will go through a certain change in time