Number States

Consider the quantum mechanical Hamiltonian for a harmonic oscillator with frequency ω:

and define the operators

a) Suppose we define a state |0⟩ to obey

Show that the states

are eigenstates of the number operator, N = a^†a, with eigenvalue n:

b) Show that |n⟩ is also an eigenstate of the Hamiltonian and compute its energy.

Hint: You may assume ⟨n|n⟩ = 1.

c) Using the above operators, evaluate the expectation value ⟨n|q²|n⟩ in terms of E(n), m, and ω.

SOLUTION

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

(1)

where [p, q] = -ih, so

(2)

We may establish the following:

(3)

Apply the number operator a^†a to the state |n⟩ directly:

Since a |0⟩ = 0, we have

(4)

b) We see from (2) that the Hamiltonian is just

We demonstrated in (a) that |n⟩ is an eigenstate of the number operator a^†a so |n⟩ is also an eigenstate of the Hamiltonian with eigenvalues E_n given by

(5)

c) The expectation value ⟨n|q²|n⟩ may be calculated indirectly. Note that

where V(q) is the potential energy. The expectation values of the potential 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 a and a^† to calculate ⟨n|q²|n⟩, so we have

We proceed to find

Thus, the result is the same by both approaches.