Chapter 2 - Answers to selected exercises

1. Neglect the complexities of classical phase space, and consider a system of $N$ distinguishable and noninteracting particles, which may be found in two states of energy, with $\epsilon =0$ and $\epsilon >0$, respectively. Given the total energy $U$ of this system, obtain an expression for the associated number of microscopic states.

$$\Omega \left( U,N\right) =\frac{N!}{\left( \frac{U}{\epsilon }\right) !\left( N-\frac{U}{\epsilon }\right) !}.$$

2. Calculate the number of accessible microscopic states of a system of two localized and independent quantum oscillators, with fundamental frequencies $\omega _{o}$ and $3\omega _{o}$, respectively, and total energy $E=10\hbar \omega _{o}$. $\left( n_{1}=2;n_{2}=2\right)$; $\left( n_{1}=5;n_{2}=1\right)$; $\left( n_{1}=8;n_{2}=0\right)$.

3. Consider a classical one-dimensional system of two noninteracting particles of the same mass $m.$ The motion of the particles is restricted to a region of the $x$ axis between $x=0$ and $x=L>0$. Let $x_{1}$ and $x_{2}$ be the position coordinates of the particles, and $p_{1}$ and $p_{2}$ be the canonically conjugated momenta. The total energy of the system is between $E$ and $E+\delta E$. Draw the projection of phase space in a plane defined by position coordinates. Indicate the region of this plane that is accessible to the system. Draw similar graphs in the plane defined by the momentum coordinates.

4. The position of a one-dimensional harmonic oscillator is given by

$$x=A\cos \left( \omega t+\varphi \right) ,$$

where $A,$ $\omega ,$ and $\varphi$ are positive constants. Obtain $p\left( x\right) dx$, that is, the probability of finding the oscillator with position between $x$ and $x+dx$. Note that it is enough to calculate $dT/T$, where $T$ is a period of oscillation, and $dT$ is an interval of time, within a period, in which the amplitude remains between $x$ and $x+dx$. Draw a graph of $p(x)$ versus $x$.

$$p\left( x\right) =\frac{1}{2\pi A}\left[ 1-\left( \frac{x}{A}\right) ^{2} \right] ^{-1/2}.$$

5. Now consider the classical phase space of an ensemble of identical one-dimensional oscillators with energy between $E$ and $E+\delta E$. Given the energy $E$, we have an ellipse in phase space. So, the accessible region in phase space is a thin elliptical shell bounded by the ellipses associated with energies $E$ and $E+\delta E$, respectively. Obtain an expression for the small area $\delta A$ of this elliptical shell between $x$ and $x+dx$. Show that the probability $p(x)dx$ may also be given by $\delta A/A$, where $A$ is the total area of the elliptical shell. This is one of the few examples where we can check the validity of the ergodic hypothesis and the postulate of equal a priori probabilities. 6. Consider a classical system of $N$ localized and weakly interacting one-dimensional harmonic oscillators, whose Hamiltonian is written as

$$\mathcal{H}=\dsum\limits_{j=1}^{N}\left( \frac{1}{2m}p_{j}^{2}+\frac{1}{2} kx_{j}^{2}\right) ,$$

where $m$ is the mass and $k$ is an elastic constant. Obtain the accessible volume of phase space for $E\leq \mathcal{H}\leq E+\delta E$, with $\delta E<<E$. This classical model for the elastic vibrations of a solid leads to a constant specific heat with temperature (law of Dulong and Petit). The solid of Einstein is a quantum version of this model. The specific heat of Einstein's model decreases with temperature, in qualitative agreement with experimental data.

$$\Omega =\left( \frac{4m}{k}\right) ^{N/2}\delta V_{sp,}$$

where $\delta V_{sp}$ is the volume of a hyperspherical shell (see Appendix) of radius $E$ and thickness $\delta E$.

7. The spin Hamiltonian of a system of $N$ localized magnetic ions is given by

$$\mathcal{H}=D\dsum\limits_{j=1}^{N}S_{j}^{2},$$

where $D>0$ and the spin variable $S_{j}$ may assume the values $\pm 1$ or $0$, for all $j=1,2,3...$. This spin Hamiltonian describes the effects of the electrostatic environment on spin-$1$ ions. An ion in states $\pm 1$ has energy $D>0$, and an ion in state 0 has zero energy. Show that the number of accessible microscopic states of this system with total energy $U$ can be written as

$$\Omega \left( U,N\right) =\frac{N!}{\left( N-\frac{U}{D}\right) !... ...sum\limits_{N_{-}}\left[ \left( \frac{U}{D}-N_{-}\right) !N_{-}!\right] ^{-1},$$

for $N_{-}$ ranging from 0 to $N$, with $N>U/D$ and $N_{-}<U/D$. Thus, we have

$$\Omega \left( U,N\right) =N!2^{U/D}\left[ \left( N-\frac{U}{D}\right) !\left( \frac{U}{D}\right) !\right] ^{-1}.$$

Using Stirling's asymptotic series, show that

$$\frac{1}{N}\ln \Omega \rightarrow \frac{u}{D}\ln 2-\left( 1-\frac{u}{D} \right) \ln \left( 1-\frac{u}{D}\right) -\frac{u}{D}\ln \frac{u}{D},$$

for $N,U\rightarrow \infty$, with $U/N=u$ fixed. This last expression is the entropy per particle in units of Boltzmann's constant, $k_{B}$.

8. In a simplified model of a gas of particles, the system is divided into $V$ cells of unit volume. Find the number of ways to distribute $N$ distinguishable particles (with $0\leq N\leq V$) within $V$ cells, such that each cell may be either empty or filled up by only one particle. How would your answer be modified for indistinguishable particles?

$$\Omega _{d}=\frac{V!}{\left( V-N\right) !}$$

and

$$\Omega _{i}=\frac{V!}{\left( V-N\right) !N!}.$$

9. The atoms of a crystalline solid may occupy either a position of equilibrium, with zero energy, or a displaced position, with energy $\epsilon >0$. To each equilibrium position, there corresponds a unique displaced position. Given the number $N$ of atoms, and the total energy $U$, calculate the number of accessible microscopic states of this system.

$$\Omega \left( U,N\right) =\frac{N!}{\left( \frac{U}{\epsilon }\right) !\left( N-\frac{U}{\epsilon }\right) !}.$$