Chapter 8- Answers to selected exercises

1. Obtain the explicit forms of the ground state and of the first excited state for a system of two free bosons, with zero spin, confined to a one-dimensional region of length $L$. Repeat this problem for two fermions of spin $1/2$. The single-particle states are given by $\varphi \left( x\right) =A\sin \left( kx\right)$, where $k=\sqrt{2mE}/\hbar$ and $kL=n\pi$, with $n=1,2,3...$. Note that $\varphi \left( x\right)$ vanishes at $x=0$ and $x=L$ . The ground state is given by $C\sin \left( \pi x_{1}/L\right) \sin \left( \pi x_{2}/L\right)$, where $C$ is a normalization constant. The first excited state is given by

$$D\left[ \sin \left( \pi x_{1}/L\right) \sin \left( 2\pi x_{2}/L\right) +\sin \left( 2\pi x_{1}/L\right) \sin \left( \pi x_{2}/L\right) \right] .$$

2. Show that the entropy of an ideal quantum gas may be written as

$$S=-k_{B}\dsum\limits_{j}\left\{ f_{j}\ln f_{j}\pm \left( 1\mp f_{j}\right) \ln \left( 1\mp f_{j}\right) \right\} ,$$

where the upper (lower) sign refers to fermions (bosons), and

$$f_{j}=\left\langle n_{j}\right\rangle =\frac{1}{\exp \left[ \beta \left( \epsilon _{j}-\mu \right) \right] \pm 1}$$

is the Fermi-Dirac (Bose-Einstein) distribution. Show that we can still use these equations to obtain the usual results for the classical ideal gas.

3. Show that the equation of state

$$pV=\frac{2}{3}U$$

holds for both free bosons and fermions (and also in the classical case). Show that an ideal ultrarelativistic gas, given by the energy spectrum $\epsilon =c\hbar k$, still obeys the same equation of state. For fermions, we have

$$\Phi =-pV=-\frac{1}{\beta }\dsum\limits_{j}\ln \left[ 1+z\exp \left( -\beta \epsilon _{j}\right) \right] .$$

In the thermodynamic limit, for free fermions, we can write

$$\dsum\limits_{j}\ln \left[ 1+z\exp \left( -\beta \epsilon _{j}\ri... ...}dk\ln \left[ 1+z\exp \left( -\frac{\beta \hbar ^{2}k^{2}}{2m}\right) \right]$$

$$=\frac{V\gamma }{\left( 2\pi \right) ^{3}}\left\{ \left. \frac{4\... ...-1}\exp \left( \frac{\beta \hbar ^{2}k^{2}}{2m}\right) +1\right] ^{-1}\right\}$$

$$\sim \frac{2\beta }{3}\dsum\limits_{j}\frac{\epsilon _{j}}{z^{-1}\exp \left( \beta \epsilon _{j}\right) +1}=\frac{2\beta }{3}U,$$

from which we prove that $pV=2U/3$. The same manipulations can be carried out for bosons, but we should pay special attention to the $k=0$ state. For the classical gas, this result is trivial.

4. Consider a quantum ideal gas inside a cubic vessel of side $L$, and suppose that the orbitals of the particles are associated with wave functions that vanish at the surfaces of the cube. Find the density of states in $\vec{k}$ space. In the thermodynamic limit, show that we have the same expressions as calculated with periodic boundary conditions.

5. An ideal gas of $N$ atoms of mass $m$ is confined to a vessel of volume $V$, at a given temperature $T$. Calculate the classical limit of the chemical potential of this gas. Now consider a ``two-dimensional'' gas of $N_{A}$ free particles adsorbed on a surface of area $A$. The energy of an adsorbed particle is given by

$$\epsilon _{A}=\frac{1}{2m}\vec{p}^{2}-\epsilon _{o},$$

where $\vec{p}$ is the (two-dimensional) momentum, and $\epsilon _{o}>0$ is the binding energy that keeps the particle stuck to the surface. In the classical limit, calculate the chemical potential $\mu _{A}$ of the adsorbed gas. The condition of equilibrium between the adsorbed particles and the particles of the three-dimensional gas can be expressed in terms of the respective chemical potentials. Use this condition to find the surface density of adsorbed particles as a function of temperature and pressure $p$ of the surrounding gas. In the thermodynamic limit, the chemical potential of the three-dimensional gas is given by

$$\mu =k_{B}T\ln \left[ \frac{8\pi ^{3}}{\gamma }\frac{N}{V}\left( \frac{\beta \hbar ^{2}}{2\pi m}\right) ^{3/2}\right] .$$

For the adsorbed gas, we have the classical limit

$$\ln \Xi =\dsum\limits_{j}z\exp \left( -\beta \epsilon _{j}\right)... ...i k\exp \left[ - \frac{\beta \hbar ^{2}k^{2}}{2m}+\beta \epsilon _{o}\right] ,$$

which yields the chemical potential

$$\mu _{A}=k_{B}T\ln \left[ \frac{4\pi ^{2}}{\gamma }\frac{N_{A}}{A}\left( \frac{\beta \hbar ^{2}}{2\pi m}\right) \right] -\epsilon _{o}.$$

From the physical requirement of equilibrium, $\mu =\mu _{A}$, we obtain $N_{A}/A$ in terms of temperature and pressure.

6. Obtain an expression for the entropy per particle, in terms of temperature and density, for a classical ideal monatomic gas of $N$ particles of spin $S$ adsorbed on a surface of area $A$. Obtain the expected values of $\mathcal{H}$, $\mathcal{H}^{2}$, $\mathcal{H}^{3}$, where $\mathcal{H}$ is the Hamiltonian of the system. What are the expressions of the second and third moments of the Hamiltonian with respect to its average value?

7. Consider a homogeneous mixture of two ideal monatomic gases, at temperature $T$, inside a container of volume $V$. Suppose that there are $N_{A}$ particles of gas $A$ and $N_{B}$ particles of gas $B$. Write an expression for the grand partition function associated with this system (it should depend on $T$, $V$, and the chemical potentials $\mu _{A}$ and $\mu _{B}$). In the classical limit, obtain expressions for the canonical partition function, the Helmholtz free energy $F$, and the pressure $p$ of the gas. Show that $p=p_{A}+p_{B}$ (Dalton's law), where $p_{A}$ and $p_{B}$ are the partial pressures of $A$ and $B$, respectively.

8. Under some conditions, the amplitudes of vibration of a diatomic molecule may be very large, with a certain degree of anharmonicity. In this case, the vibrational energy levels are given by the approximate expression

$$\epsilon _{n}=\left( n+\frac{1}{2}\right) \hbar \omega -x\left( n+\frac{1}{2} \right) ^{2}\hbar \omega ,$$

where $x$ is the parameter of anharmonicity. To first order in $x$, obtain an expression for the vibrational specific heat of this system. Note that

$$\ln Z_{v}=\ln S\left( \alpha \right) +x\frac{\alpha }{S\left( \al... ...ht) }\frac{d^{2}}{d\alpha ^{2}}S\left( \alpha \right) +O\left( x^{2}\right) ,$$

where

$$S\left( \alpha \right) =\dsum\limits_{n=0}^{\infty }\exp \left[ -... ...t( n+\frac{1}{2}\right) \right] =\left[ 2\sinh \frac{\alpha }{2}\right] ^{-1},$$

with $\alpha =\beta \hbar \omega$. We then have

$$u=-\hbar \omega \frac{\partial }{\partial \alpha }\ln Z_{v} c=k_{B}\left( \frac{\hbar \omega }{k_{B}T}\right) ^{2}\frac{ \partial ^{2}}{\partial \alpha ^{2}}\ln Z_{v},$$

from which we calculate the first-order correction to the specific heat,

$$\Delta c=k_{B}\left( \frac{\hbar \omega }{k_{B}T}\right) ^{2}x\fr... ...t( \alpha \right) }\frac{d^{2}}{ d\alpha ^{2}}S\left( \alpha \right) \right] .$$

9. The potential energy between atoms of a hydrogen molecule may be described by the Morse potential,

$$V\left( r\right) =V_{o}\left\{ \exp \left[ -\frac{2\left( r-r_{o}\right) }{a} \right] -2\exp \left[ -\frac{r-r_{o}}{a}\right] \right\} ,$$

where $V_{o}=7\times 10^{-19}\,\mathrm{J}$, $r_{o}=8\times 10^{-11}\,\mathrm{ m}$, and $a=5\times 10^{-11\,}\mathrm{m}$. Sketch a graph of $V\left( r\right)$ versus $r$. Calculate the characteristic temperatures of vibration and rotation to compare with experimental data (see table on Section 8.4). Note that

$$\frac{dV}{dr}=0 r=r_{o}$$

and

$$\left( \frac{d^{2}V}{dr^{2}}\right) _{r_{o}}=\frac{2V_{o}}{a^{2}}>0,$$

which shows the existence of a minimum at $r=r_{o}$. The characteristic temperature of rotation is given by

$$\Theta _{r}=\frac{\hbar ^{2}}{2k_{B}I}=\frac{\hbar ^{2}}{2k_{B}Mr_{o}^{2}},$$

where $I$ is the moment of inertia and $M$ is the reduced mass of the hydrogen molecule. From the frequency of vibrations,

$$\omega =\left( \frac{2V_{o}}{a^{2}M}\right) ^{1/2},$$

we write the characteristic temperature of vibrations,

$$\Theta _{v}=\frac{\hbar }{k_{B}}\left( \frac{2V_{o}}{a^{2}M}\right) ^{1/2}.$$

Now it is easy to check the numerical predictions.