Consider an isolated system consisting of a large number N of very weakly intera
ID: 1917706 • Letter: C
Question
Consider an isolated system consisting of a large number N of very weakly interacting localized particles of spinExplanation / Answer
naturally, when the spin points antiparallel, its energy is mH and when it points parallel, its energy is - mH. Thus, the ground state has energy -NmH, since in this state all the N number of spins are parallel. The state next to the ground state has (N-1) number of parallel spins and 1 antiparallel spin and hence has energy -(N-1)mH+mH=-(N-2)mH. Thus, the difference in energy between ground state and the next state is 2mH, which is the difference in energy between any two macrostates. Therefore, in energy range del E, there are (del E)/2mH number of energy gaps or (del E)/2mH + 1 number of energy levels, which is approximately equal to del E/2mH since since del E >> mH and hence we can neglect 1. Now, total energy is E = (-n1 + n2)mH and total number of spins is n1 + n2 = N. Solving, we get n1=(1/2)(N-(E/mH)) and n2=(1/2)(N+(E/mH)). Now, for a given state, when n1 spins are parallel and n2 spins are antiparallel, the state can occur in N!/n1!n2! ways, which is the degeneracy of each state. Thus, the total number of available states is number of states in range del E multiplied by the degeneracy of each state. Therefore W(E) = (N!/n1!n2!)(del E/2mH). Substituting the expressions for n1 and n2, the result matches with the answer given in the book.