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Quantum Mechanics question

  1. Feb 7, 2007 #1
    1. The problem statement, all variables and given/known data

    Assume that a particle in a one-dimensional box is in its first excited state. Calculate the expectation values [x], [p], and [E], and the uncertainties
    delta(x), delta(p), and delta(E). Verify that delta(x)*delta(p)>=h_bar/2.

    2. Relevant equations

    Psi=sqrt(2/a) cos(pi*x/a) e^(-i*E*t/h_bar)

    [x]=Int(Psi_star x Psi, -a/2, a/2)

    [p]=Int(Psi_star (-i*h_bar*d/dx) Psi, -a/2, a/2)

    [E]=Int(Psi_star (i*h_bar*d/dt) Psi, -a/2, a/2)

    3. The attempt at a solution

    After evaluating the above integrals, I get:

    [x]=0

    [p]=0

    [E]=h_bar*pi^2*n^2 / 2*m*a^2

    I am trying to calculate the quantities delta(x), delta(p), and delta(E) but I am having trouble doing that. Can you please suggest some hints on how to proceed. Thank you.
     
  2. jcsd
  3. Feb 7, 2007 #2
    What are the definitions of delta(x) and delta(p) in terms of expectation values?
     
  4. Feb 7, 2007 #3
    This is a trivial question, but would I be able to approximate it using the relations:

    delta(x)=h_bar/sqrt(2m(V0 - E))
     
  5. Feb 7, 2007 #4
    That's a new relation to me. Just look at the definition of the variance and follow the prescription.
     
  6. Feb 7, 2007 #5
    Yep. You need to calculate six expectation values in order to calculate deltax, deltap and deltaE, and only three of them are <x>, <p> and <E>.
     
  7. Feb 8, 2007 #6

    dextercioby

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    Homework Helper

    Do you know a reason why [itex] \Delta E [/itex] is zero for this problem ? Besides the actual computation of it, which can be avoided by knowing this reason.
     
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