(i) A number $N$ of non-interacting particles move in one dimension in a potential $V(x, t)$. Write down the Hamiltonian and Hamilton's equations for one particle.

At time $t$, the number density of particles in phase space is $f(x, p, t)$. Write down the time derivative of $f$ along a particle's trajectory. By equating the rate of change of the number of particles in a fixed domain $V$ in phase space to the flux into $V$ across its boundary, deduce that $f$ is a constant along any particle's trajectory.

(ii) Suppose that $V(x)=\frac{1}{2} m \omega^{2} x^{2}$, and particles are injected in such a manner that the phase space density is a constant $f_{1}$ at any point of phase space corresponding to a particle energy being smaller than $E_{1}$ and zero elsewhere. How many particles are present?

Suppose now that the potential is very slowly altered to the square well form

$$V(x)=\left\{\begin{array}{cc} 0, & -L<x<L \\ \infty & \text { elsewhere } \end{array}\right.$$

Show that the greatest particle energy is now

$$E_{2}=\frac{\pi^{2}}{8} \frac{E_{1}^{2}}{m L^{2} \omega^{2}} .$$