Time-Dependent Harmonic Oscillator II

Consider a simple harmonic oscillator in one dimension. Introduce the raising and lowering operators, a^† and a, respectively. The Hamiltonian H and wave function Ψ at t = 0 are

(i)

(ii)

where |n⟩ denotes the eigenfunction of energy E_n = hω(n +1/2).

a) What is wave function Ψ(t) at positive times?

b) What is the expectation value for the energy?

c) The position x can be represented in operators by x = X₀(a + a^†).

where  is a constant. Derive an expression for the expectation of the time-dependent position

(iii)

You may need operator expressions such as  and

SOLUTION

a) The time dependence of the wave function is

(1)

b) The expectation value for the energy is

(2)

which is independent of time.

c) To find the average value of the position operator, we first need to show that

(3)

(4)

(5)

Then

(6)

The expectation value of the position operator oscillates in time.