|
|
| (83 intermediate revisions by the same user not shown) |
| Line 1: |
Line 1: |
| == Problem 1: Phonons in solids ==
| |
|
| |
|
| == Problem 2: Cosmic microwave background radition ==
| |
|
| |
| == Problem 3: Bose-Einstein condensation of diluted gases in harmonic traps==
| |
|
| |
| [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. Close to its minimum, the potential can be expanded to second order, and has the form
| |
|
| |
| <math> U(\mathbf{r}) = \frac{m}{2} \sum_\alpha \omega^2_\alpha x^2_\alpha </math>
| |
|
| |
| where we allow for the possiblity of anisotropic confinment, with different frequenies along different directions.
| |
|
| |
| (a) We are interested in the limit of wide traps such that <math> \hbar \omega \ll k_B T </math>, the the discretness 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
| |
