This post recounts my attempt to learn about the numerical solution of the time-dependent Schrödinger equation (TDSE). It arose from my interest in the dynamics of electrons in solids, rather than the static properties usually considered in electronic structure calculations – see Post 1 in this series. Solving the 1-electron TDSE is about the most basic thing you can do in this direction, and it is interesting enough in its own right. It’s also at the heart of time-dependent density functional theory.
So I began to learn about the Crank-Nicolson method, which seems to be the most popular method for solving the partial differential equations of electron dynamics, and then wrote a simple 1-dimensional code, TDSE1, to see if I could make it work in problems that I knew about. I had looked at quite a few problems in R1 which involved wave-packet dynamics – problems in which the Hamiltonian is constant in time. A lot of progress can be made analytically, or semi-analytically, on such problems, so I used them to test the accuracy of the fully numerical TDSE1 code.
However, the problems for which a numerical method is usually mandatory are those in which the Hamiltonian varies with time – ie the dynamics of driven systems. In fact, I found I could use wavepacket dynamics methods to treat one problem of this kind: a 1-dimensional model with an electric field which is uniform in space but oscillatory in time (an extreme simplification of the process of ionization in a strong infra-red laser field). For this problem, too, my simple code TDSE1 performed very well.
Finally, in the course of this project I came across (or possibly remembered) the idea that propagation in imaginary time projects out of an initial wave-packet the bound states of the system in order of their energy (lowest first). I modified TDSE1 to implement this algorithm in the code ITP and tested it on some simple bound state problems. Again, it works pretty well.
Solving the TDSE .pdf
