Thermodynamics, Statistical Physics, and Quantum Mechanics

QUANTUM MECHANICS 261

Since a we have

b) We see from (S.5.16.2) that the Hamiltonian is just

We demonstrated in (a) that is an eigenstate of the number operator

so is also an eigenstate of the Hamiltonian with eigenvalues

given by

where is the potential energy. The expectation values of the poten-
tial and kinetic energies are equal for the quantum oscillator, as for time
averages in the classical oscillator. Therefore, they are half of the total
energy:

In this problem, however, you are explicitly asked to use the operators
and to calculate so we have

c) The expectation value may be calculated indirectly. Note that