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| == Problem 1: Specific heat of the Einstein model of lattice vibrations in solids ==
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| (a) Derive an expression for the average energy at temperature <math> T </math> of a single quantum harmonic oscillator having frequency <math> \omega </math>.
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| (b) Assuming unrealistically (as Einstein did) that the normal-mode vibrations of a solid all have the same natural frequency <math> \omega_E </math>, and using your result in (a),
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| find an expression for the heat capacity of an insulating solid.
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| (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 <math> 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 </math>. Would you expect this to be a poor or good estimate for the high-temperature
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| heat capacity of the material?
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| (d) Find the low-temperature limit of the heat capacity and explain why it is reasonable in terms of the model.
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| == Problem 2: Cosmic microwave background radition ==
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| == Problem 3: Bose-Einstein condensation of diluted gases in harmonic traps==
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| [http://nobelprize.org/nobel_prizes/physics/laureates/2001/press.html 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 <math> \sim 10 </math> nK.
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| Close to its minimum, the potential can be expanded to second order, and has the form
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| <math> U(\mathbf{r}) = \frac{m}{2} \sum_\alpha \omega^2_\alpha x^2_\alpha </math>
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| where we allow for the possibility of anisotropic confinement, with different frequencies along different directions.
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| (a) We are interested in the limit of wide traps such that <math> \hbar \omega \ll k_B T </math>, the the discreteness of the allowed energies can be largely ignored. Show that in the limit, the number of states <math> N(E) </math> and the corresponding density of states <math> g(E) </math> are given by
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