Problems are listed in approximate order of difficulty. A single dot (•) indicates straigh...

Problems are listed in approximate order of difficulty. A single dot (•) indicates straightforward problems involving just one main concept and sometimes requiring no more than substitution of numbers in the appropriate formula. Two dots (••) identify problems that are slightly more challenging and usually involve more than one concept. Three dots (•••) indicate problems that are distinctly more challenging, either because they are intrinsically difficult or involve lengthy calculations. Needless to say, these distinctions are hard to draw and are only approximate.

••• Consider a quantum ideal gas of N spinless particles in a container of volume V = a3. If the temperature is sufficiently high and the density is sufficiently low, the gas is in the classical nondegenerate regime, in which the probability that any particular point in k space is occupied is much less than one, corresponding to Fig. 15.12(a). In this high-temperature, low- density limit, the properties of the gas are the same regardless of whether the particles of the gas are fermions or bosons. (a) Show that this classical regime occurs when the number density (number per volume) n obeys n ≫ (3kBTm)3/2/6 π2ℏ3, where m is the mass of a gas molecule. [Hints: From the equipartition theorem, the average kinetic energy of a single particle is , and so a typical value of k = |k| for a particle is given roughly by . Also, recall that the density of points in k space is 1 point per (π/a3.] (b) For a gas of helium atoms at atmospheric pressure, below what temperature (roughly) does the gas become degenerate? (c) For a conduction electron gas in a metal, how high a temperature would be necessary for the electrons to be in the classical regime? Is such a temperature possible? Assume 1 conduction electron per atom, and a volume per atom of about (0.3 nm)3.