Given a Gaussian
المؤلف:
Sidney B. Cahn, Gerald D. Mahan And Boris E. Nadgorny
المصدر:
A GUIDE TO PHYSICS PROBLEMS
الجزء والصفحة:
part 2 , p 56
14-8-2016
1321
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 ѱn(x) with eigenvalue En. 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 an:
(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 Hѱn = En ѱn. The probability Pn of energy En is |an|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.
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