Question

2. Consider an isolated system consisting of a large number N of very weakly interacting localized particles of spin 1 2. Each particle has a rnagnetic mioment μ which can point parallel or anti-parallel to an applied field H. The energy E of the systern is then E =-(ni-n2):1H, antiparallel to H. (a) Consider the energy range between E and E+δΕ where δΕ < E but is microscopically large so that δΕ μΗ. What is the total number of states Ω(E) lying in this energy range? Be sure to write this in terms of N and E, not ni and n2 since the latter variables are not measurable, and realize that adjacent energy levels are spaced 211H apart. ĮNote that this problem is essentially the same problem as counting the number of "steps the right" out of N steps in the random walk problem from Chapter 1.] (b) Write down an expression for 1n9(E) as a function of E. Simplify this expression by applying Stirling's formula in its simplest form. (c) Assume that the energy E is in a region where S2(E) is appreciable, i.e., that it is not close to the extreme possible values ±ΝμΗ which it can assume. In this case apply a Gaussian approximation to part (a) to obtain a simple expression for Ω(E) as a function of E. [That is, use Eq 1.6.4) of Reif, where you just need to relate the variables of this problem to those in that equation.]

Answer 1