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probset10 - Physics 5524 Statistical Mechanics Problem Set...

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Physics 5524 Statistical Mechanics Problem Set 10 Due: Wednesday, Apr. 7 10.1 In 1995 Bose-Einstein condensation was realized experimentally for dilute gases of alkali atoms (Li, Na, K, Rb, Cs). Unlike He 4 , these gases are sufficiently dilute to be well described by the ideal Bose gas model. In these experiments, atoms are trapped using inhomogeneous magnetic fields which couple to the atoms through the Zeeman coupling to the unpaired s -electron in the out- ermost shell. The resulting “magnetic trap” can be modeled as a three-dimensional har- monic potential V ( r ) = 1 2 2 0 r 2 for which the single-particle energy levels are given by E n x ,n y ,n z = ¯ 0 ( n x + n y + n z + 3 / 2). In what follows it is convenient to subtract off the zero-point energy so that the ground state has energy E = 0 rather than 3 2 ¯ 0 . The energy levels are then E n x ,n y ,n z = ¯ 0 ( n x + n y + n z ) . The total number of bosons N for such a harmonically confined gas is given by the usual sum over Bose occupation factors, N = n x ,n y ,n z n ( E n x ,n y ,n
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