Question

At time t = 0 a particle in the potential V(x) = mω²x²/2 is described by the wave function

\(\psi(x, 0) = A\sum\limits_n (1/\sqrt 2)^n\psi_n(x)\),

where \(\psi_n(x)\) are eigenstates of the energy with eigenvalues E_n = (n + 1/2)hω. You are given that \((\psi_n, \psi_{n'})= \delta_{nn'}\).

(a) Find the normalization constant A.

(b) Write an expression for \(\psi\)(x, t) for t > 0.

(c) Show that |\(\psi\)(x, t)|² is a periodic function of time and indicate the longest period \(\tau\).

(d) Find the expectation value of the energy at t = 0.

Answer 1

(a) The normalization condition

gives A = 1/√2, taking A as positive real.

(b) The time-dependent wave function is

(c) The probability density is

Note that the time factor exp [-iω(n - m)t] is a function with period A, the maximum period being 2π/ω.

(d) The expectation value of energy is