- #1

Homo Novus

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## Homework Statement

For a Hamiltonian of the form [itex]H=\sum_{j=1}^m a_j p_j^2 + \sum_{j=1}^n b_j q_j^2 + H'(q_{n+1}, \dots, q_{sN}, p_{m+1}, \dots, p_{sN})[/itex] (for a system with n coordinates and m momenta, s degrees of freedom and N particles), show that [itex]\overline{a_jp_j^2}=\frac{kT}{2}[/itex], for j=1,...,m and [itex]\overline{b_jq_j^2}=\frac{kT}{2}[/itex], for j=1,...,n.

## Homework Equations

[itex]P(q,p)=\frac{exp[-\beta H(q,p)]}{Z_N}[/itex]

[itex]Z_N=\frac{1}{N!h^{sN}}\int \int exp[-\beta H(q,p)]\,d^{sN}q\,d^{sN}p[/itex]

[itex]P_x(x_1,...,x_N)dx_1...dx_n = P_y(y_1(x_1,...,x_N),...,y_N(x_1,...,x_N)) \left|J\left(\frac{y_1,...,y_N}{x_1,...,x_N}\right)\right| dx_1...dx_N[/itex]

Note that [itex]\beta=\frac{1}{kT}[/itex].

## The Attempt at a Solution

I honestly don't really know where to go with this... I've tried plugging stuff in, but I'm really at a loss. Any help would be greatly appreciated. Thanks so much in advance!