(i) Compute
(ii) Compute Tr [
].
(iii) Compute
. Using the identity
check your answer to (i).
(iv) Confirm that the expectation
where is the Fermi function.
(v) Repeat the above procedure assuming that
and
are boson operators. What are the restrictions on the values of
and what happens when one of these energies
becomes zero?
(vi) Calculate