The Schrödinger equation describes the energy and time-evolution of a particle or system of particles, and is one of the fundamental building blocks of modern physics. In it’s general form, the (time-independent) Schrödinger equation for a one-dimensional harmonic oscillator reads thus:¹

\begin{equation} \label{eq:sch} \frac{-\hbar^2}{2m} \frac{\partial^2}{\partial z^2}\psi(z) + \frac{mz^2}{2} \psi(z) = E\psi(z) \end{equation}

There are relatively few situations in which the Schrödinger equation can be solved analytically, and numerical methods and approximations are one way around that analytical limitation. To demonstrate how this is possible and how a numerical solution works, what better way than to solve a system which can be solved analytically and comparing the results.

The general Runge-Kutta algorithm is one of a few algorithms for solving first order ordinary differential equations. Below is a specific implementation for solving equations of motion and other second order ordinary differential equations (ODEs) for Physics simulations, amongst other things.

I’m getting ready for starting a course in computational physics, and so, ignoring the fact I’m meant to be revising for an exam this Monday, I thought I’d prepare for the more exciting of the two. I’ve always wanted to code this little physics model ever since I saw it on one of those JavaScript snippet websites back when dial-up was fast and people downloaded mp3s one at a time. Those were the days.

Just a quick one here. It’s pushing a month since I last checked in, and not for want of desire. I’ve not been entirely negligent though: I’ve been busy putting together a few typed up solutions to both foundational and pretty challenging/rare problems in statistical physics. My motivation is two-fold: a) to get extremely familiar with the material, what with exams just around the corner and b) for a few of the problems, the solutions are nearly impossible to do or to find help online for if you can’t do them.