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Homework Help: Quantum Mechanics question

  1. Dec 11, 2007 #1
    By the way, this isn't a homework problem, it's self study.

    1. The problem statement, all variables and given/known data
    We're given that we have a wave function in an infinite potential well, with the well having 0 PE from 0 to a. The wave function would be described by

    [tex]{\Phi}_n={\sqrt{\frac{2}{a}}Sin({\frac{n{\pi}x}{a})}[/tex]. It was given that the function is originally in it's second exicited state, meaning it's original state can be described by [tex]{\Phi}_3={\sqrt{\frac{2}{a}}Sin({\frac{3{\pi}x}{a})}[/tex]

    The box is suddenly expanded, so that it's rightmost boundary is now 2a. This means the new basis vectors are of the form
    [tex]|{\Theta}_n>={\sqrt{\frac{1}{a}}} Sin{\frac{n{\pi}x}{2a}})[/tex]

    What the book asks is for me to find the probability of this particle being found in states n=1 and 2. I figured the probability of finding it in any given state would be [tex]<{\Theta}_n|{\Phi_3}>[/tex]
    =[tex]{\int}_0^{2a}({\sqrt{\frac{1}{a}}} Sin({\frac{n{\pi}x}{2a}}))({\sqrt{\frac{2}{a}}Sin({\frac{3{\pi}x}{a})})dx[/tex]

    I had no problems up to here. It's the next step in their analysis that threw me off.

    [tex]={\frac{{\sqrt{2}}}{a}{\int}_0^aSin({\frac{n{\pi}x}{2a}})Sin({\frac{3{\pi}x}{a})}dx[/tex]

    Looks like they simplified the equation in the exact same way that I did. The problem is that the upper limit of integration has changed from 2a to a. I've been looking at this problem for a while now and I cannot figure out why the limit of integration changes. Why is it that the upper limit of integration changes from 2a to a when all they did was simplify a few terms? Shouldn't the limit of integration still be 2a? When I do use 2a, all of the probabilities vanish, so obviously I'm wrong and the book is right. I just cannot figure out why that is the case. Why does the upper limit become a instead of 2a?
     
  2. jcsd
  3. Dec 11, 2007 #2
    Shouldn't it be 0 to a from the beginning since that is the area the particle is confined to immediately after the walls have been expanded?
     
  4. Dec 11, 2007 #3

    Galileo

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    The integration is from 0 to 2a, but your initial wavefunction is zero for |x|>a
     
  5. Dec 11, 2007 #4
    Oh, thanks, I didn't forgot about that. Thank you.
     
  6. Dec 11, 2007 #5
    Wait, in that case, shouldn't the equations be:

    =[tex]{\int}_0^{a}({\sqrt{\frac{1}{a}}} Sin({\frac{n{\pi}x}{2a}}))({\sqrt{\frac{2}{a}}Sin({\frac{3{\pi}x}{a})})dx[/tex] ?

    Because the book does in fact write this integral out with 2a as the upper limit of integration.
     
  7. Dec 11, 2007 #6

    nrqed

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    then that's a typo in the book.
     
  8. Dec 11, 2007 #7
    Oh okay, thanks.
     
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