Cheat sheets/Statistical mechanics

Statistical mechanics is a branch of mathematical physics that studies, using probability theory, the average behaviour of a mechanical system where the state of the system is uncertain. A common use of statistical mechanics is in explaining the thermodynamic behaviour of large systems.

Thermal wavelength

 * $$\lambda = \frac{h}{\sqrt{2\pi m k_B T}} = h \sqrt{\frac{\beta}{2\pi m}}= \hbar \sqrt{\frac{2\pi\beta}{m}}$$
 * $${1 \over \lambda} = \frac{\sqrt{2\pi m k_B T}}{h} $$

Partition function
Canonical: (in 3d)
 * $$Z = \int \prod_{i=1}^N \frac{d^3 q_i d^3 p_i}{h^3} e^{-\beta \frac{p_i^2}{2m}} e^{-\beta U(q_i)} = \int \prod_{i=1}^N \frac{d^3 q_i d^3 p_i}{h^3} e^{-\beta H} $$