For a gas of particles confined inside a two-dimensional box, the density of states is constant, independent of (see Problem 7.28). Investigate the behavior of a gas of noninteracting bosons in a two-dimensional box. You should find that the chemical potential remains significantly less than zero as long as is significantly greater than zero, and hence that there is no abrupt condensation of particles into the ground state. Explain how you know that this is the case, and describe what does happen to this system as the temperature decreases. What property must have in order for there to be an abrupt Bose-Einstein condensation?
Consider two single-particle states, A and B, in a system of fermions, where and that is, level A lies below by the same amount that level B lies above . Prove that the probability of level B being occupied is the same as the probability of level A being unoccupied. In other words, the Fermi-Dirac distribution is "symmetrical" about the point where .
Consider a gas of identical spin-0 bosons confined by an isotropic three-dimensional harmonic oscillator potential. (In the rubidium experiment discussed above, the confining potential was actually harmonic, though not isotropic.) The energy levels in this potential are , where is any nonnegative integer and is the classical oscillation frequency. The degeneracy of level .
(a) Find a formula for the density of states, , for an atom confined by this potential. (You may assume .)
(b) Find a formula for the condensation temperature of this system, in terms of the oscillation frequency .
(c) This potential effectively confines particles inside a volume of roughly the cube of the oscillation amplitude. The oscillation amplitude, in turn, can be estimated by setting the particle's total energy (of order ) equal to the potential energy of the "spring." Making these associations, and neglecting all factors of 2 and and so on, show that your answer to part (b) is roughly equivalent to the formula derived in the text for the condensation temperature of bosons confined inside a box with rigid walls.
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