Make this result plausible physically. In writing about experiments on the scattering of a particles in helium Rutherford said, "On account of the impossibility of distinguishing between the scattered alpha particles and the projected He nuclei, the results are subject to a certain ambiguity." Explain how an awareness of quantum statistics could have removed the ambiguity. What determines whether a gas obeys Bose or Fermi distributions? 19. How can the ordered state of the He II explain its lack of resistance to heat conduction? 20. What examples of a Fermi gas are there other than an electron gas and a gas of He 3 atoms? 21. In the ideal gas equations we use the rest mass of particles. Should we ever use the relativistic mass instead? Consider the effect of temperature and the nature of the particle. 22. Give a plausibility argument for the relation, (11-57), between the Fermi energy eF and 23.

In the Fermi distribution we obtain the result that at the Fermi energy gF the average number of particles per quantum state is exactly one-half. This is definitely not the same as saying that 50% of the particles are at energies above the Fermi energy and 50% below. Explain. 24. Justify the assumption that conduction electrons behave approximately as a system of free noninteracting particles. Is there a connection between Vo , the depth of the potential well for conduction electrons in a metal, and electron diffraction experiments of the Davisson-Germer type? Can we determine V0 from such experiments? 24. Justify the assumption that conduction electrons behave approximately as a system of free noninteracting particles. Is there a connection between Vo , the depth of the potential well for conduction electrons in a metal, and electron diffraction experiments of the Davisson-Germer type? Can we determine V0 from such experiments?

26. Explain physically the effect of letting h 0 in expressions for the density of states, such as (11-49). Explain physically the effect of letting h -4.0 in equations involving the quantum degeneracy term, such as (11-53). PROBLEMS 1. The equilibrium state is one of maximum entropy S in thermodynamics and one of maximum probability P in statistics. Assuming then that S is a function of P, show that we should expect S = k In P, where k is a universal constant. This relation is sometimes called the Boltzmann postulate. (Hint: Consider the effect on S and P of combining two systems.) d a o a 1 3 w s (cid:9) (cid:9) 2. The Maxwell distribution can be developed by looking at elastic collisions between two particles. If initially these particles have energies f1 and g2, and finally g3 and e4, then

N T ^ If all possible states are equally probable, the number of collisions per second P is proportional to the number of particles in each initial state, i.e. + = (cid:9) a)+(e2 +5) S C T S T A T S M U T N A U Q I I P1,2 = CP(g1)P(e2) where Red is the probability of a state being occupied, and C is a constant. Similarly P3 , 4 = CP(^3)P(e4). In equilibrium, for each collision (1,2) -+ (3,4) there must be a collision (3,4) (1,2). Thus P 1,2 = P3 , 4. (a) Show that P(g1) = e-gilkT solves this equation. (b) Use similar reasoning to derive the Fermi distribution. Here, however, the initial states must be filled and the final states must be empty, and the number of collisions becomes P1,2 = CP(GP(e2)[1 P(e3)][1 P(4)] Then show that the equation P1,2 = P3,4 can be solved by 1 P(6L) P(^`) J