Chapter 10 - Answers to selected exercises

1. Consider a system of ideal bosons of zero spin ($\gamma =1$), within a container of volume $V$. (a) Show that the entropy above the condensation temperature $T_{o}$ is given by

$$S=k_{B}\frac{V}{\lambda ^{3}}\left[ \frac{5}{2}g_{5/2}\left( z\right) -\frac{ \mu }{k_{B}T}g_{3/2}\left( z\right) \right] ,$$

where

$$\lambda =\frac{h}{\sqrt{2\pi mk_{B}T}}$$

and

$$g_{\alpha }\left( z\right) =\dsum\limits_{n=1}^{\infty }\frac{z^{n}}{ n^{\alpha }}.$$

(b) Given $N$ and $V$, what is the expression of the entropy below $T_{o}$? What is the entropy associated with the particles of the condensate? (c) From the expression for the entropy, show that the specific heat at constant volume above $T_{o}$ is given by

$$c_{V}=\frac{3}{4}k_{B}\left[ 5\frac{g_{5/2}\left( z\right) }{g_{3... ...( z\right) }-3\frac{g_{3/2}\left( z\right) }{g_{1/2}\left( z\right) }\right] .$$

(d) Below $T_{o}$, show that the specific heat at constant volume is given by

$$c_{V}=\frac{15}{4}k_{B}\frac{v}{\lambda ^{3}}g_{5/2}\left( 1\right) .$$

(e) Given the specific volume $v$, sketch a graph of $c_{V}/k_{B}$ versus $\lambda ^{-2}$ (that is, in terms of the temperature in convenient units). Obtain the asymptotic expressions of the specific heat for $T\rightarrow 0$ and $T\rightarrow \infty$. Obtain the value of the specific heat at $T=T_{o}$.

2. Consider again the same problem for an ideal gas of two-dimensional bosons confined to a surface of area $A$. What are the changes in the expressions of item (a)? Show that there is no Bose-Einstein condensation in two dimensions (that is, show that in this case the Bose-Einstein temperature vanishes).

3. Consider an ideal gas of bosons with internal degrees of freedom. Suppose that, besides the ground state with zero energy ( $\epsilon _{o}=0$), we have to take into account the first excited state, with internal energy $\epsilon _{1}>0$. In other words, assume that the energy spectrum is given by

$$\epsilon _{j}=\epsilon _{\vec{k},\sigma }=\frac{\hbar ^{2}k^{2}}{2m} +\epsilon _{1}\sigma ,$$

where $\sigma =0,1$. Obtain an expression for the Bose-Einstein condensation temperature as a function of the parameter $\epsilon _{1}$ .

4. Consider a gas of non-interacting bosons associated with the energy spectrum

$$\epsilon =\hbar c\left\vert \vec{k}\right\vert ,$$

where $\hbar$ and $c$ are constants, and $\vec{k}$ is a wave vector. Calculate the pressure of this gas at zero chemical potential. What is the pressure of radiation of a gas of photons?

*5. The Hamiltonian of a gas of photons within an empty cavity of volume $V$ is given by the expression

$$\mathcal{H}=\dsum\limits_{\vec{k},j}\hbar \omega _{\vec{k},j}\left( a_{\vec{k },j}^{\dagger }a_{\vec{k},j}+\frac{1}{2}\right) ,$$

where $j=1,2$ indicates the polarization, $\vec{k}$ is the wave vector,

$$\omega _{\vec{k},j}=ck,$$

$c$ is the velocity of light, and $k=\left\vert \vec{k}\right\vert$. (a) Use the formalism of the occupation numbers (second quantization) to obtain the canonical partition function associated with this system. (b) Show that the internal energy is given by

$$U=\sigma VT^{n}.$$

Obtain the value of the constants $n$ and $\sigma$. (c) Consider the Sun as a black body at temperature $T\approx 5800\,\mathrm{K }$. The solar diameter and the distance between the Sun and the Earth are of the order of $10^{9}\,\mathrm{m}$ and $10^{11\,}\mathrm{m}$, respectively. Obtain the intensity of the total radiation that reaches the surface of Earth. What is the value of the pressure of this radiation?