Temporary HW

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 microwave background radiation 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 microwave background

Problem 3: Bose-Einstein condensation of diluted gases in harmonic traps

The Nobel Prize in Physics 2001 has been awarded for "the achievement of Bose-Einstein condensation in dilute gases of alkali atoms, and for early fundamental studies of the properties of the condensates". Such experimental realizations of BEC rely on trapping cold atoms in a potential where laser cooling and evaporative cooling bring bosons to a temperature of the order of $\sim 10$ nK.

Close to its minimum, the potential can be expanded to second order, and has the form

$U(\mathbf {r} )={\frac {m}{2}}\sum _{\alpha }\omega _{\alpha }^{2}x_{\alpha }^{2}$

where we allow for the possibility of anisotropic confinement, with different frequencies along different directions.

(a) We are interested in the limit of wide traps such that $\hbar \omega \ll k_{B}T$, the the discreteness of the allowed energies can be largely ignored. Show that in the limit, the number of states $N(E)$ and the corresponding density of states $g(E)$ are given by