Solving Time-Dependent Schrödinger Equations in Python

Introduction

Based on this article by Mathcube

Any quantum mechanical system evolves according to the TDSE given below

$$\frac{\partial}{\partial t} \left| \psi (t) \right> = - i H (t) \left| \psi (t) \right>$$

To simulate this emulation, we need to solve this differential equation. We can use the Euler Approximation (1^st Order approximation) by discretizing the time domain. The differential equation is then approximated in the forward direction by

$$\frac{\left| \psi \right>^{(n+1)} - \left| \psi \right>^{(n)}}{\Delta t} = -i H \left| \psi \right>^{(n)}$$

where

$$\left| \psi \right>^{(n)} = \left| \psi (n \Delta t) \right>$$

This gives us the Explicit form of the wave-function at the next time step.

$$\left| \psi \right>^{(n+1)} = (1 - i H \Delta t) \left| \psi \right>^{(n)}$$

Similarly for the backward approximation

$$\frac{\left| \psi \right>^{(n+1)} - \left| \psi \right>^{(n)}}{\Delta t} = -i H \left| \psi \right>^{(n+1)}$$

we get the implicit form

$$(1 + i H \Delta t) \left| \psi \right>^{(n+1)} = \left| \psi \right>^{(n)}$$

But in these approximation, we lose the normalization of the wave-function.

Crank–Nicolson Scheme

To combat the issue of loss of normalization, we can take the average of the implicit and explicit forms given above to get

$$\left( 1 + \frac{i\Delta t}{2} H \right) \left| \psi \right>^{(n+1)} = \left( 1 - \frac{i\Delta t}{2} H \right) \left| \psi \right>^{(n)}$$

which defines a unitary transformation

$$U(t) = \left( 1 + \frac{i\Delta t}{2} H(t) \right)^{-1} \cdot \left( 1 - \frac{i\Delta t}{2} H(t) \right)$$

This can then be used to approximate the evolution of the system.

Results

The Crank-Nicolson scheme was implemented in python and the evolution of a gaussian wave-packet in harmonic potential simulated.

Notes

The python script outputs a series of ordered JPEG images, which were later stitched using ffmpeg.

ffmpeg -f image2 -framerate 12 -i %03d.jpg Solution.gif