Homework Set 4

Problem 1: Specific heat of insulating solids via the Einstein model of lattice vibrations

(a) Derive an expression for the average energy at temperature $T$ of a single quantum harmonic oscillator having frequency $\omega$.

(b) Assuming unrealistically (as Einstein did) that the normal-mode vibrations of a solid all have the same natural frequency $\omega _{E}$, and using your result in (a), find an expression for the heat capacity of an insulating solid.

(c) Find the high-temperature limit of the heat capacity calculated in (b) and use it to obtain a numerical estimate for the heat capacity of a $V=5\ \mathrm {cm} ^{3}$ piece of an insulating solid having a number of density of $n=6\cdot 10^{28}\ \mathrm {atoms/m} ^{3}$. Would you expect this to be a poor or good estimate for the high-temperature heat capacity of the material?

(d) Find the low-temperature limit of the heat capacity and explain why it is reasonable in terms of the model.

Problem 2: Cosmic microwave background radition

The Universe is currently not in equilibrium. However, in the microwave frequency range it is filled with radiation that is precisely described by a Planck distribution at $T=2.725\pm 0.001$ K which is currently probed, together with spatial inhomogeneities, by the WMAP and PLANCK satellites. The cosmic microwave background (CMB) radiation, whose discovery has earned The Nobel Prize in Physics 1978 is a window back to the decoupling time, about 380 000 years after the Big Bang, when the temperature dropped low enough for the protons and electrons to combine in hydrogen atoms. Light does not travel far in ionized gases. Instead, it accelerates the charges and scatters from them. Hence, before this time, our Universe was very close to an equilibrium soup of electrons, nuclei, and photons. The neutral atoms after this time were transparent enough that almost all of the photons traveled for the next 13 billion years directly into our detectors.

These photons in the meantime gave greatly increased in wavelength. This is due to substantial expansion of the Universe. The initial Planck distribution of photons changed both because of the Doppler shift (a red shift because of the distant gas that emitted the photon appears to be moving away from us) and because the photons are diluted into a larger volume. The Doppler shift both reduced the photon energy and squeezes the overall frequency range of the photons (increasing the number of photons per unit frequency). Thus, one might ask why the current CMB radiation is thermal and why is it at such a low temperature.

(a) If the side of the box $L$ and the wavelengths of the photons in the box are all increased by a factor $f$, what frequency $\omega ^{\prime }$ will result from a photon with initial frequency $\omega$? If the original density of photons is $n(\omega )$, what is the density of photons $n'(\omega ^{\prime })$ after the expansion. Show that Planck form for the number of photons per unit frequency per unit volume:

${\frac {\omega ^{2}}{\pi ^{2}c^{3}(e^{\hbar \omega /k_{B}T}-1)}}$

is preserved, except for a shift in temperature. What is the new temperature $T'$, in terms of the original temperature $T$ and the expansion factor?

(b) How many microwave background photons can you find in a cubic centimeter. How does that compare to the average atomic density $n_{\mathrm {atoms} }=2.5\times 10^{-7}\ \mathrm {atoms/} \mathrm {cm} ^{3}$ in the Universe? NOTE: Cosmologists refer to the current Universe as photon dominated because there are currently many more photons than atoms.

(c) Calculate the internal energy $E$, pressure $P$, and the entropy $S$ for a photon gas in a volume $V$ and temperature $T$. For simplicity, write them in terms of the Stefan-Boltzmann constant $\sigma =\pi ^{2}k_{B}^{4}/60\hbar ^{3}c^{2}$. In this calculation you should ignore the zero point energy in the photon modes, as we did it at the beginning of the course (otherwise, the zero-point energy would make pressure and internal energy infinite, even at zero temperature).

Problem 4: Bose-Einstein condensation in low-dimensional systems

(a) Does Bose condensation occur for a one and two-dimensional ideal Bose gas? If so, find the transition temperature. If not, explain.

(b) As you will find in (a), Bose-Einstein condensation does not occur in ideal one- and twodimensional systems. However, this result holds only if the system is confined by rigid walls. In the following, we want to show that Bose-Einstein condensation can occur if a system is confined by a spatially varying potential.

Let us assume that the confining potential has the form $V(r)\sim r^{n}$. In this case the region accessible to a particle with energy $\varepsilon$ has a radius $\ell \sim \varepsilon ^{1/n}$. Show that the corresponding density of states behaves as:

$g(\varepsilon )\sim \ell ^{d}\varepsilon ^{d/2-1}\sim \varepsilon ^{d/n}\varepsilon ^{d/2-1}\sim \varepsilon ^{\alpha }$

where

$\alpha ={\frac {d}{n}}+{\frac {d}{2}}-1$.

What is the range of values for $n$ for which $T_{c}>0$ in dimensions of space $d=1$ and $d=2$?