Homework Set 2

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Problem 1: Canonical partition function for single spin in magnetic field

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: Canonical partition function of two-interacting qubits

Problem 3: Canonical partition function and density matrix of 1D harmonic oscillator

Consider a single one-dimensional quantum harmonic oscillator described by the Hamiltonian:

,

where .

(a) Find the partition function in the quantum canonical ensemble at temperature .

(b) Using the 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 the result in (c), write down the density matrix in the 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 .