Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Obtain an expression for the expectation value in terms of energy

  1. Nov 5, 2011 #1
    1. The problem statement, all variables and given/known data
    The ground state wave function for a particle of mass m moving with energy E in a one-dimensional harmonic oscillator potential with classical frequency omega is:

    u(subscript 0) (x)= N(subscript 0) exp((-alpha^2)(x^2)/2) and alpha=sqrt (m *omega/h-bar)

    where N(subscript 0) is some normalisation constant

    a) for the ground state, show explicitly that the quantum mechanical expectation values <x> and <p> are both zero
    b) if the uncertainties delta x and delta p are given by:

    (delta x)^2=<x^2>-<x>^2
    (delta p)^2=<p^2>-<p>^2

    obtain an expression for the expectation value of E in terms of the uncertainties
    c)If ΔxΔp = c for c a constant, deduce a value for c by minimising the ground state energy. What significance
    does the value of c have in light of the uncertainty principle ?

    3. The attempt at a solution

    I haven't tried part a yet, but for part b:

    =((delta p)^2 - <p>^2)^2)/2m

    I don't know if the above is correct and I have no idea how to get it in terms of delta x. Please help
    Last edited: Nov 5, 2011
  2. jcsd
  3. Nov 5, 2011 #2


    User Avatar
    Science Advisor
    Homework Helper

    The expectation value (average over an ensemble) for the Hamiltonian is

    [tex] \langle H\rangle =\frac{1}{2m} \langle p^2\rangle + C \langle x^2\rangle [/tex]

    , wherer C=C(m,ω)>0.
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook