Simulation of The 1D Time-Dependent Schrodinger Equation

Описание: This is a simulation that uses the one dimensional Schrodinger equation to evolve a Gaussian wave packet. I simulated it for 3 potentials: the infinite square well, a single barrier, and a step potential.

The method of simulation is the Crank-Nicolson scheme. Normally in finite difference methods one can get away with using a centered spatial 2nd derivative + forward 1st time derivative at each time step, but naively applying this to Schrodinger's equation doesn't work (see (1) below). This is because of the special condition that the wave function must be normalized at each time step. To get around this, the Crank-Nicolson scheme approximates every derivative as an average of the current derivative (t = t) and the derivative at the next time step
(t = t+dt) (see (2) below). There are other methods for arriving at the same result of the Crank-Nicolson scheme, namely approximating the time-evolution operator exp(-itH/hbar) with Cayley's form (see (3) below).

If you're interested in the code email me at [email protected]

1) https://www.researchgate.net/publicat...

2)
https://artmenlope.github.io/solving-...

3)
https://static.uni-graz.at/fileadmin/...