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== Problem 3: Bose-Einstein condensation of diluted gases in harmonic traps== | == 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 | [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. | ||
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> | <math> U(\mathbf{r}) = \frac{m}{2} \sum_\alpha \omega^2_\alpha x^2_\alpha </math> | ||
where we allow for the | 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 <math> \hbar \omega \ll k_B T </math>, the the | (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 | ||
Revision as of 12:59, 15 March 2011
Problem 1: Phonons in solids
Problem 2: Cosmic microwave background radition
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 nK.
Close to its minimum, the potential can be expanded to second order, and has the form
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 , the the discreteness of the allowed energies can be largely ignored. Show that in the limit, the number of states and the corresponding density of states are given by
