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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:

    <E>=<p^2>/2m
    =((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

    dextercioby

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    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.
     
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