Exercises

1. Consider a particle of mass $m$ confined within a cubic box of dimensions $L_x=L_y=L_z$ . According to elementary quantum mechanics, the possible energy levels of this particle are given by
   $$E = \frac{\hbar^{ 2} \pi^{ 2}}{2 m} \left(\frac{n_x^{ 2}}{L_x^{ 2}}+ \frac{n_y^{ 2}}{L_y^{ 2}}+\frac{n_z^{ 2}}{L_z^{ 2}}\right),$$
   where $n_x$ , $n_y$ , and $n_z$ are positive integers. (See Section C.10.)
   1. Suppose that the particle is in a given state specified by particular values of the three quantum numbers, $n_x$ , $n_y$ , $n_z$ . By considering how the energy of this state must change when the length, $L_x$ , of the box parallel to the $x$ -axis is very slowly changed by a small amount $dL_x$ , show that the force exerted by a particle in this state on a wall perpendicular to the $x$ -axis is given by $F_x= -\partial E/\partial L_x$ .
   2. Explicitly calculate the force per unit area (or pressure) acting on this wall. By averaging over all possible states, find an expression for the mean pressure on this wall. (Hint: exploit the fact that $\overline{n_x^{ 2}}=\overline{n_y^{ 2}}=\overline{n_z^{ 2}}$ must all be equal, by symmetry.) Show that this mean pressure can be written
      $$\overline{p} = \frac{2}{3}\frac{\overline{E}} {V},$$
      where $\overline{E}$ is the mean energy of the particle, and $V=L_x L_y L_z$ the volume of the box.

2. The state of a system with $f$ degrees of freedom at time $t$ is specified by its generalized coordinates, $q_1, \cdots, q_f$ , and conjugate momenta, $p_1,\cdots, p_f$ . These evolve according to Hamilton's equations (see Section B.9):
   $$\dot{q}_i = \frac{\partial H}{\partial p_i},$$
   $$\dot{p}_i = -\frac{\partial H}{\partial q_i}.$$
   Here, $H(q_1, \cdots, q_f, p_1,\cdots, p_f, t)$ is the Hamiltonian of the system. Consider a statistical ensemble of such systems. Let $\rho(q_1, \cdots, q_f, p_1,\cdots, p_f, t)$ be the number density of systems in phase-space. In other words, let $\rho(q_1, \cdots, q_f, p_1,\cdots, p_f, t) dq_1 dq_2\cdots dq_f dp_1 dp_2\cdots dp_f$ be the number of states with $q_1$ lying between $q_1$ and $q_1+dq_1$ , $p_1$ lying between $p_1$ and $p_1+dp_1$ , et cetera, at time $t$ .
   1. Show that $\rho$ evolves in time according to Liouville's theorem:
      $$\frac{\partial \rho}{\partial t} +\sum_{i=1,f} \left(\dot{q}_i ... ...o} {\partial q_i} + \dot{p}_i \frac{\partial \rho}{\partial p_i}\right) = 0.$$
      [Hint: Consider how the the flux of systems into a small volume of phase-space causes the number of systems in the volume to change in time.]

   2. By definition,
      $$N = \int_{-\infty}^{\infty} \rho dq_1\cdots dq_f dp_1\cdots dp_f$$
      is the total number of systems in the ensemble. The integral is over all of phase-space. Show that Liouville's theorem conserves the total number of systems (i.e., $d N/d t=0$ ). You may assume that $\rho$ becomes negligibly small if any of its arguments (i.e., $q_1, \cdots, q_f$ and $p_1,\cdots, p_f$ ) becomes very large. This is equivalent to assuming that all of the systems are localized to some region of phase-space.

   3. Suppose that $H$ has no explicit time dependence (i.e., $\partial H/\partial t=0$ ). Show that the ensemble-averaged energy,
      $$\overline{H} = \int_{-\infty}^{\infty} H \rho dq_1\cdots dq_f dp_1\cdots dp_f,$$
      is a constant of the motion.

   4. Show that if $H$ is also not an explicit function of the coordinate $q_j$ then the ensemble average of the conjugate momentum,
      $$\overline{p_j} = \int_{-\infty}^{\infty} p_j \rho dq_1\cdots dq_f dp_1\cdots dp_f,$$
      is a constant of the motion.

3. Consider a system consisting of very many particles. Suppose that an observation of a macroscopic variable, $x$ , can result in any one of a great many closely-spaced values, $x_r$ . Let the (approximately constant) spacing between adjacent values be $\delta x$ . The probability of occurrence of the value $x_r$ is denoted $P_r$ . The probabilities are assumed to be properly normalized, so that
   $$\sum_r P_r \simeq \int_{-\infty}^\infty \frac{P_r(x_r)}{\delta x} dx_r=1,$$
   where the summation is over all possible values. Suppose that we know the mean and the variance of $x$ , so that
   $$\overline{x} = \sum_r x_r P_r$$
   and
   $$\overline{({\mit\Delta} x)^{ 2}} = \sum_r (x_r-\overline{x})^{ 2} P_r$$
   are both fixed. According to the $H$ -theorem, the system will naturally evolve towards a final equilibrium state in which the quantity
   $$H = \sum_r P_r \ln P_r$$
   is minimized. Used the method of Lagrange multipliers to minimixe $H$ with respect to the $P_r$ , subject to the constraints that the probabilities remain properly normalized, and that the mean and variance of $x$ remain constant. (See Section B.6.) Show that the most general form for the $P_r$ which can achieve this goal is
   $$P_r (x_r)\simeq \frac{\delta x}{ \sqrt{2\pi \overline{({\mit \De... ...\frac{(x_r-\overline{x})^{ 2}}{2 \overline{({\mit \Delta} x)^{ 2}}}\right].$$
   This result demonstrates that the system will naturally evolve towards a final equilibrium state in which all of its macroscopic variables have Gaussian probability distributions, which is in accordance with the central limit theorem. (See Section 2.10.)