Quantum Statistics

a. Show that the density of states of a 2-D Fermi gas in a rectangular solid of dimension L2 equals N(E) = b. Show that the chemical potential of a Fermi gas in 2 dimensions is given by for n electrons per unit area. Use an integral table to solve the integral and watch your limits of integration. c. Show that for all small T, μ becomes independent of T. (See attachment for proper display of equations, symbols)

Quantum statistics

Consider a 2-level system with an energy splitting between the upper and lower level. Using Boltzmann statistics, show that the heat capacity of the 2-level system equals: (see attachment for equation) What happens in the limit of (see attachment for equation)

hydrogen atom

For a hydrogen atom in the ground state, calculate the probability of finding the electron in the "classically forbidden region."

Fermions in harmonic oscillator potential

Two identical, non-interacting spin-1/2 fermions are placed in the 1-D harmonic potential V(x) = (1/2)mω2x2, Where m is the mass of the fermion and ω is its angular frequency. a. Find the energies of the ground and first excited states of this two-fermion system. Express the eigenstates corresponding to these two energy levels in terms of harmonic oscillator wave functions and the singlet and triplet spin states. b. Calculate the square of the separation of the two fermions, <(x1-x2)2>=<(x12+x22-2x1x2)> for the lowest energy state of the two-fermion system. c. Repeat the calculations for the first excited states.

Raman effect

I am having a hard time finding out how to work with the Raman effect using the Kramers-Heisenburg formula. I am calculating differential cross sections for scattering and I dont know how to do this or what approximations to use with the Raman effect. If anyone knows anything on this topic, please respond ASAP.