
\sect{Equipartition theorem}

%Baruch's  A27.

This is an MCE version of A23: An equipartition type relation
is obtained in the following way:

Consider N particles with coordinates $\vec q_i$, and conjugate
momenta $\vec p_i$ (with $i = 1,...,N$), and subject to a
Hamiltonian ${\cal H}({\vec p_i},{\vec q_i})$.
\begin{itemize}
\item [(a)]
Using the classical micro canonical ensemble (MCE)
show that the entropy $S$ is invariant under the
rescaling $\vec q_i\rightarrow \lambda {\vec q_i}$ and ${\vec
p_i}\rightarrow {\vec p_i}/\lambda $ of a pair of conjugate
variables, i.e. $S[{\cal H}_{\lambda}]$ is independent of
$\lambda$, where ${\cal H}_{\lambda}$ is the Hamiltonian obtained
after the above rescaling.
\item[(b)] Now assume a Hamiltonian of the form
${\cal H}=\sum_i\frac{({\vec p_i})^2}{2m}+V(\{{\vec q_i}\})$.
Use the result that $S[{\cal H}_{\lambda}]$ is independent of
$\lambda$ to prove the virial relation
\[\left\langle \frac{({\vec p_1})^2}{m}\right\rangle =
\left\langle \frac{\partial V}{\partial {\vec q}_1}\cdot {\vec
q}_1\right\rangle\] where the brackets denote MCE averages.
Hint: $S$ can also be expressed with the accumulated number of states
$\Sigma(E)$.
\item[(c)] Show that classical equipartition,
$\langle x_i\frac{\partial {\cal H}}{\partial
x_j}\rangle=\delta_{ij}k_BT$, also yields the result (b). Note that this
form may fail for quantum systems.
\item[(d)] Quantum mechanical version: Write down the expression for
the entropy in the quantum case. Show that it is also invariant
under the rescalings $\vec q_i\rightarrow \lambda {\vec q_i}$ and
${\vec p_i}\rightarrow {\vec p_i}/\lambda $  where $\vec p_i$ and
$\vec q_i$ are now quantum mechanical operators. (Hint: Use
Schr\"{o}dinger's equation and $\vec p_i=-i\hbar
\partial/\partial \vec q_i$.) Show that the result in (b) is valid
also in the quantum case.

\end{itemize}

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