Given a Gaussian

A particle of mass m is coupled to a simple harmonic oscillator in one dimension. The oscillator has frequency ω and distance constant x²₀ = h/mω. At time t = 0 the particle’s wave function ψ (x, t) is given by

(i)

The constant σ is unrelated to any other parameters. What is the probability that a measurement of energy at t = 0 finds the value of E₀ = hω/2?

SOLUTION

Denote the eigenfunctions of the harmonic oscillator as ѱ_n(x) with eigenvalue E_n. They are a complete set of states, and we can expand any function in this set. In particular, we expand our function Ψ(x, 0) in terms of coefficients a_n:

(1)

(2)

The expectation value of the energy is the integral of the Hamiltonian H for the harmonic oscillator:

(3)

where we used the fact that Hѱ_n = E_n ѱ_n. The probability P_n of energy E_n is |a_n|². So the probability of E₀ is given by

(4)

where

(5)

and finally

(6)

It is easy to show that this quantity is less than unity for any value of σ ≠ x₀ and is unity if σ = x₀.