A blog of Python-related topics and code.

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

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 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, $a = \sqrt{k_\mathrm{r}/2D_\mathrm{e}}$, 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 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.

The following code simulates (very approximately) the growth of a polycrystal from a number of seeds. Atoms are added to the crystal lattice of each of the resulting grains until no more will fit, creating realistic-looking boundaries where two grains meet.