Thermodynamics, Statistical Physics, and Quantum Mechanics

260 SOLUTIONS

Up to an arbitrary phase, we see that

Starting with the vacuum ket we can write an energy eigenket

g) The energyspectrum is where takes all positive integer values
and zero. From (S.5.15.9) and the fact that the norm of the eigenvectors is
positive (actually, 1), we see that cannot be negative, and so no negative
eigenvalues are possible.

5.16 Number States (Stony Brook)

a) In this problem it is important to use only the information given. We
may write the Hamiltonian as

We may establish the following:

Apply the number operator to the state directly:

where (see Problem5.15), so