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Normalize the wave function and more Please help

  1. Oct 18, 2009 #1
    normalize the wave function and more!! Please help!!!

    1. The problem statement, all variables and given/known data

    i) Normalize the wave function

    ii) Calculate <x>

    iii) Calculate [tex]<x^{2}>[/tex]

    iv) What would happen if a < 0?



    2. Relevant equations

    [tex]\psi\left(x\right) = N\left(1+i\right)exp\left(-a|x|\right)[/tex], for -inf < x < inf and a > 0

    3. The attempt at a solution

    It would take ages for me to work out the latex for all my steps (I'm new to latex!!) so I'll do what i need to and hope someone can help!!

    First:

    [tex]^{inf}_{-inf}\int 2N^{2}e^{-2a|x|}dx[/tex]
    [tex]=N^{2}\left[-e^{-2a|x|}\right]^{inf}_{-inf}[/tex]

    so: [tex]\frac{-N^{2}}{a}=1[/tex]
    so: [tex]N=\sqrt{\frac{-1}{a}}[/tex]

    therefore:
    [tex]\psi\left(x\right) = \sqrt{\frac{-1}{a}}\left(1+i\right)exp\left(-a|x|\right)[/tex]


    ii+iii) for the expectation values, I got both equalling zero

    iv) and if a < 0, you'd get exponential growth as x approaches infinity (+ and -)

    Is this right??!

    I get so confused when the limits are infinity!!
     
  2. jcsd
  3. Oct 18, 2009 #2
    Re: normalize the wave function and more!! Please help!!!

    I think you lost a 2 when doing your initial normalization.

    You should get an expectation value for position of zero. Think about physically why that is the case.

    The expectation value of x^2 should not be zero. Check that integral.

    Your completely correct on the a<0 regieme. The function would increase to infinity as x goes to infinity and you could not normalize the function. The solution becomes non-physical.
     
  4. Oct 18, 2009 #3

    nicksauce

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    Re: normalize the wave function and more!! Please help!!!

    Your integration to normalize the WF looks wrong. You shouldn't get a negative answer when you're integrating a positive definite function. I'd try it again more carefully if I were you.
     
  5. Oct 18, 2009 #4
    Re: normalize the wave function and more!! Please help!!!

    Also, it is difficult to integrate an absolute value. You need to split the integral into two regions. One where the Abs(x) = x and one where Abs(x) = -x
     
  6. Oct 20, 2009 #5
    Re: normalize the wave function and more!! Please help!!!

    Ok! I redid the integration for the wave function and got [tex]N=\sqrt{\frac{a}{2}}[/tex] now, which looks alot better and my expectation value of x is still zero so that's good too!! My problem now is that when i try to calculate the expectation value of x^2, (using integration by parts) I just keep getting deeper and deeper into it!!

    [tex]<x^{2}>=^{inf}_{-inf}\int\psi^{*}x^{2}\psi\: dx[/tex]
    [tex]=\sqrt{\frac{a}{2}}\left(1-i\right)e^{-a|x|}x^{2}\sqrt{\frac{a}{2}}\left(1-i\right)e^{-a|x|}\: dx[/tex]
    [tex]=^{inf}_{-inf}\int ax^{2}e^{-2a|x|}[/tex]
    [tex]=^{0}_{-inf}\int ax^{2}e^{2ax}+\;^{inf}_{0}\int ax^{2}e^{-2ax}[/tex]
    [tex]=\left[\frac{x^{2}e^{2ax}}{2}\right]^{0}_{-inf}-\:^{0}_{-inf}\int \frac{e^{2ax}}{x}\:dx+\left[\frac{-x^{2}e^{-2ax}}{2}\right]^{inf}_{0}-\:^{inf}_{0}\int \frac{-e^{-2ax}}{x}\:dx[/tex]

    =......

    so far I have carried on another 2 integrals and the denominator in each new integral has an increased power of x (this one is x, next x^2, then x^3)! I can see no end!!!

    Have I made yet another mistake?!
     
  7. Oct 22, 2009 #6
    Re: normalize the wave function and more!! Please help!!!

    I don't know how where i was going with that intgral!! I started again from scratch following your advice and every thing seemed to fall into place itself so thanks to both of you for your help!:smile:
     
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