Please help with the stat-mech question below:

The nal key piece to analyzing quantum statistical mechanics is the density of states, 9(E). This is interpreted so that g[E)dE is the total number of one-particle states within innitesimal energy dE of E. The density of states allows us to approximate the sums over states as integrals. In this problem we will construct the densities of states for a few different systems of interest. A key fact that will help us construct these is the way we change variables in densities. Given two variables a and b, the density in variable a is related to the density in variable b via3 db 9301) (is = 95(5) db = yaw) = 9:30)) E - I11 lecture we derived the density of states for a 3D gas in a cube of side-length L (volume v = L3), h?! ME) = 952va (22)\" E (1) where 9., = 2.9 + 1 is the Spin-degeneracy. Consider instead the density of states for a gas of particles in atland! That is, consider a 2D gas in a bOX of side-length L (area A = L2). (a) Find the density of states for this 2D system. Hint (highlight to resent): [ ] [Noter Fun fact, if we were to do the ID case, we would nd 9(E) DC E'1/2.] Next, lets consider a (3D) gas of ninsstess particles in a cube of side-length L (volume V = L3). Alteri'iatively, we can consider a (3D) gas of particles with mass m in the hyper-retativis tic enemy regime E :9 7311122. In these cases the energy is related to the momentum via E = I35] (5. (b) Find the density of states g(E) for this massless/hyper-relativistic 3D system. Hint (highlight to resent): [ l (c) Using E = tic/A, determine the density of states for this system in the wavelength variable, QAU')' Hint (highlight to resent): [