# Blog

Viewing posts by christian

## Two-dimensional collisions

This small Python project is a physical simulation of two-dimensional physics. The animation is carried out using Matplotlib's FuncAnimation method and is implemented by the class Simulation. Each "particle" of the simulation is represented by an instance of the Particle class and depicted as a circle with a fixed radius which undergoes elastic collisions with other particles.

## Packing circles inside a shape

A previous blog post dealt with packing circles into a circle. To fill an arbitrary shape, a slightly different approach is needed. The code is presented in my github repo.

## Packing circles in a circle

The following code attempts to pack a predefined number of smaller circles (of random radii between two given limits) into a larger one.

## The Morse oscillator

The Morse oscillator is a model for a vibrating diatomic molecule that improves on the simple harmonic oscillator model in that the vibrational levels converge with increasing energy and that at some finite energy the molecule dissociates. The potential energy varies with displacement of the internuclear separation from equilibrium, $x = r - r_\mathrm{e}$ as: $$V(x) = D_\mathrm{e}\left[ 1-e^{-ax} \right]^2,$$ where $D_\mathrm{e}$ is the dissociation energy, $a = \sqrt{k_\mathrm{e}/2D_\mathrm{e}}$, and $k_\mathrm{e} = (\mathrm{d}^2V/\mathrm{d}x^2)_\mathrm{e}$ is the bond force constant at the bottom of the potential well.

## The harmonic oscillator wavefunctions

The harmonic oscillator is often used as an approximate model for the behaviour of some quantum systems, for example the vibrations of a diatomic molecule. The SchrÃ¶dinger equation for a particle of mass $m$ moving in one dimension in a potential $V(x) = \frac{1}{2}kx^2$ is $$-\frac{\hbar^2}{2m}\frac{\mathrm{d}^2\psi}{\mathrm{d}x^2} + \frac{1}{2}kx^2\psi = E\psi.$$ With the change of variable, $q = (mk/\hbar^2)^{1/4}x$, this equation becomes $$-\frac{1}{2}\frac{\mathrm{d}^2\psi}{\mathrm{d}q^2} + \frac{1}{2}q^2\psi = \frac{E}{\hbar\omega}\psi,$$ where $\omega = \sqrt{k/m}$. This differential equation has an exact solution in terms of a quantum number $v=0,1,2,\cdots$: $$\psi(q) = N_vH_v(q)\exp(-q^2/2),$$ where $N_v = (\sqrt{\pi}2^vv! )^{-1/2}$ is a normalization constant and $H_v(q)$ is the Hermite polynomial of order $v$, defined by: $$H_v(q) = (-1)^ve^{q^2}\frac{\mathrm{d}^v}{\mathrm{d}q^v}\left(e^{-q^2}\right).$$ The Hermite polynomials obey a useful recursion formula: $$H_{n+1}(q) = 2qH_n(q) - 2nH_{n-1}(q),$$ so given the first two: $H_0 = 1$ and $H_1 = 2q$, we can calculate all the others.