Statistical mechanics

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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)

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