Question

Consider a collection of N identical harmonic oscillators with negligible (but non-zero) interactions. In a microcanonical ensemble with energy E, the system is on a surface in phase space given by \begin{equation} \sum_{i=1}^{N} \left( \frac{p_i^2}{2m} + \frac{m\omega^2 q_i^2}{2} \right) = E \end{equation} a) Find the volume of phase space enclosed, $\Phi(E)$, as follows: Transform to new variables \begin{align} x_i &= \frac{1}{\sqrt{2m}} p_i & 1 \le i \le N \\ x_i &= \sqrt{\frac{m\omega^2}{2}} q_{i-N} & N+1 \le i \le 2N \end{align} Note that in terms of these variables the constant energy surface is a 2N dimensional sphere. Find its volume. Find the corresponding volume in p-q space. b) Find the entropy S in terms of N and E. c) Find T and express E in terms of N and T

          Consider a collection of N identical harmonic oscillators with negligible (but non-zero) interactions. In a microcanonical ensemble with energy E, the system is on a surface in phase space given by
\begin{equation*}
\sum_{i=1}^{N} \left( \frac{p_i^2}{2m} + \frac{m\omega^2 q_i^2}{2} \right) = E
\end{equation*}
a) Find the volume of phase space enclosed, $\Phi(E)$, as follows:
Transform to new variables
\begin{align*}
x_i &= \frac{1}{\sqrt{2m}} p_i & 1 \le i \le N \\
x_i &= \sqrt{\frac{m\omega^2}{2}} q_{i-N} & N+1 \le i \le 2N
\end{align*}
Note that in terms of these variables the constant energy surface is a 2N dimensional sphere. Find its volume. Find the corresponding volume in p-q space.
b) Find the entropy S in terms of N and E.
c) Find T and express E in terms of N and T

Consider a collection of N identical harmonic oscillators with negligible (but non-zero) interactions. In a microcanonical ensemble with energy E, the system is on a surface in phase space given by

∑i=1^N( (pi^2)/(2m) + (mω^2 qi^2)/(2)) = E

a) Find the volume of phase space enclosed, Φ(E), as follows:
Transform to new variables

xi = (1)/(√(2m)) pi 1 ≤ i ≤ N

xi = √((mω^2)/(2)) qi-N N+1 ≤ i ≤ 2N

Note that in terms of these variables the constant energy surface is a 2N dimensional sphere. Find its volume. Find the corresponding volume in p-q space.
b) Find the entropy S in terms of N and E.
c) Find T and express E in terms of N and T

Added by Barbara B.

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