Temporary HW

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

(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 Failed to parse (unknown function "\math"): {\displaystyle V = 5 \ \mathrm{cm}^3 <\math> piece of an insulating solid having a number of density of <math> 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 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