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Expected Value of the Hamilton operator

  1. Apr 16, 2008 #1
    1. The problem statement, all variables and given/known data

    I have to calculate the expected value of the Hamilton operator (average energy) of a two, non interacting, identical particle system. Thus these particles can be bosons or fermions, but at the moment I just want to look at fermions.

    2. Relevant equations


    The localised wave functions (not eigenstates of the Hamilton operator) are given by:

    \psi_{-}(x_{1}) & = & \exp[-\frac{\beta}{2}(x_{1}-a/2)^{2}]

    \psi_{+}(x_{2}) & = & \exp[-\frac{\beta}{2}(x_{2}+a/2)^{2}][/tex]

    Thus for fermions we need to anti-symmetrise:


    Thus one particle is localised at x = a/2 and one at x = -a/2.

    Thus to get the expected value of the Hamilton operator:


    One actually divides this expression by another function, but I solved that already and not relevant to my problem.

    3. The attempt at a solution

    So far I substituted the Hamilton operator into E(a) to get:


    And then multiply it out to get two terms (both similar, only listing one):


    This is where the problem comes in, calculating that 2nd derivative creates a big mess, even when working with a computer algebra system. Is there a way to rewrite this in a better form?
  2. jcsd
  3. Apr 17, 2008 #2
    Try working in the momentum basis.
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