Problem 1 a)
The roof of the δ-­‐function is at: Since the RHS is > 0, . , where
, stands for the integer part of For plot, let , where and Plot of g(x), assuming C=1. b) Denote:
The , then term in the sum is larger than the rest of the terms. Singling this term out: , which is a 1D-­‐like divergence of the density of states. Problem 2 a) We can just use the solution for 1 a). In this case, there is only one term in the sum b) One can relate and find from there. Alternatively, one can apply the Pauli principle in the following form: The offset c)
, then, increases with as Landau parabola, decreases as while the Fermi energy, counted from the bottom of the . The free energy (per unit volume) is: At , , where In the ultra-­‐quantum limit,
is the last occupied landau level. , and notice that in the grand canonical ensemble is treated as a free parameter till the very end of the calculation. Notice then when calculating the derivative, proportional to
, which is independent of
is kept constant. The coefficient in front of . Therefore, the term proportional to not contribute to magnetization. Dropping this term, one gets: is does Recall that at , Notice the “paramagnetic” sign of magnetization in this limit.