Please help with the question a, b, d, e below:

The hint is "It's not just half of them! Think about the combinatorics. If we know we have N? total up-spins and we know the first spin is one of the up-spins, what does that leave us?"

Let's return to our paramagnet system. Consider a two-state paramagnet (our usual setup) with N spins and consider the macrostate with N, of the N spins aligned with the magnetic field B (and therefore we have NJ = N - N, spins anti-aligned with the magnetic field). The energy and number of spin orientations were related by N U U = -HB(N+ - N); N U N1 = 2 B' NI = 2 + 2 B For the multiplicity you found something equivalent to the following form (which will be a useful form for us), -N+ - NJ n ( U ) ~ If we have the energy U (and thus the number of up-spins N, ) fixed then the fundamental assumption of statistical mechanics says the microstate probability distribution is P(01, 02, . . . , ON) = n ( U ) where of = 11 gives the spin-state of the Mi-th spin (with +1 for aligned, -1 for anti-aligned). In Problem Set 5 we also found the relationship between energy and temperature for this system, U(T) = -NMB tanh (MB KBT Suppose we know that our system has N spins with N, spins being aligned with the magnetic field (spin-up). Let's see what we can learn about the value of the first spin of (or equivalently any particular one of the N spins).(a) How many microstates in the microcanonical ensemble associated with N. have the first spin as spin-up? Given your answer and the fundamental assumption of statistical mechanics, what is the probability Pi(1, U) of finding the first spin in a spin-up state? You may answer in terms of N and Nt. Hint (highlight to reveal): [It's not just half of them! Think about the combinatorics. If we know we have N, total up-spins and we know the first spin is one of the up-spins, what does that leave us?] (b) Given your answer to (a), the relationship between N and energy, and the relationship between energy and temperature, show that etuB/KBT PI (1, T ) = . 2 cosh HB KBT (c) Extra Part (Not for Credit) Follow similar steps to find Pi(), T) and show that our results are normalized as we should expect, PI(1, T) + PI(+, T). (d) What is the expectation value (01) (T) for the first spin? Hint (highlight to reveal): [Recall that given a probability distribution P(X) for an observable X, we have (X) = _x XP(X).] (e) What happens to U(T), Pi(1, T), and (01)(7) in the two limits T -+ 0 and T -+ co? Plot the non-dimensionalized energy-per-spin u = U/NMB, Pi(1), and (1) as functions of the non- dimensionalized temperature T = KBT/HB