Problem 1
The Hamiltonian for an electron spin degree of freedom in the external magnetic field is given by:
where is the Bohr magneton and is the vector of the Pauli matrices:
(a) In the quantum canonical ensemble, evaluate the density matrix if is along the z axis.
(b) Repeat the calculation from (a) assuming that points along the x axis.
(c) Calculate the average energy in each of the above cases.
Problem 2
Consider a single one-dimensional quantum harmonic oscillator described by the Hamiltonian:
,
where .
(a) Find the partition function in quantum canonical ensemble at temperature .
(b) Using result from (a), calculate the averge energy .
(c) Write down the formal expression for the canonical density operator in terms of the eigenstates of the Hamiltonian and the corresponding energy levels .
(d) Using result in (c), write down the density matrix in a coordinate representation .
(e) In the coordinate representation, calculate explicitly in the high temperature limit .
HINT: One approach is to utilize the following result
which you can apply to the Boltzmann operator:
while neglecting terms of order and higher since is very small in the high temperature limit.
(f) At low temperatures, is dominated by low-energy states. Use the ground state wave function only, evaluate the limiting behavior of as .