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Expectation value of x^2 (Q.M.)

  1. Jan 27, 2012 #1
    1. The problem statement, all variables and given/known data
    I am given ψ(x), want to calculate [itex]<x^{2}>[/itex].



    2. Relevant equations
    [itex]\psi(x) = a\exp(ibx-(c/2)(x-d)^2)[/itex]
    [itex]<x^2> = \int\limits_{-∞}^∞ \psi^*x^2\psi \mathrm{d}x[/itex]


    3. The attempt at a solution
    Well, I normalized the wave function and found [itex] a = (\frac{c}{\pi})^{1/4}[/itex].
    So, the integral I have to do becomes:
    [itex]<x^2> = \sqrt{\frac{c}{\pi}} \int\limits_{-∞}^∞ x^2\exp{(-c(x-d)^2)}\mathrm{d}x[/itex].

    Since the function is neither even nor odd, there is no simple trick, and I had a hard time finding the exact integral in my table of integrals.

    Thanks in advance.
     
  2. jcsd
  3. Jan 27, 2012 #2
    I would suggest a change of variables. Try u = x-d

    That should put it in a form where you could split it up then use an integral table.
     
  4. Jan 28, 2012 #3
    then look up a table or formula for integrals of gaussians
     
  5. Jan 28, 2012 #4
    Yeap, after you use y = x -d and split the integral in three: y^2*exp(ay) and d^2*exp(ay) and 2yd*exp(ay) its very easy. The first one gives (π/α)^(3/2) and the second one (π/α)^(1/2). The third one can be found through "derivative integration" if I call it correclty in english?
     
  6. Jan 28, 2012 #5
    Ah, a simple u-substitution works. Thanks for the help everyone.
     
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