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Expectation Value for Gaussian Wave Packet.

  1. Oct 2, 2011 #1
    1. The problem statement, all variables and given/known data

    Calculate <x> for the Gaussian wave packet [itex]\psi(x)=Ne-(x-x0)/2k2[/itex]

    2. Relevant equations

    [itex]\left\langle x \right\rangle = \int dx x|\psi(x)|2[/itex]

    3. The attempt at a solution

    So I've been reviewing for the up-coming midterm and I've had the painful realization that I'm basically killing myself with basic issues; in this case solving the integral.

    [itex]\left\langle x \right\rangle = \int dx x|\psi(x)|2
    = \int dx x|Ne-(x-x0)/2k2|2
    =N2\int dx x|e-(x-x0)/2k2|2[/itex]

    It's at this point that I know there a u substitution to be made (at least I'm pretty sure that's the best way to solve the integral), but I'm not exactly sure for which value? Is there a better way to solve this integral?

    Thanks for any pointers!
     
  2. jcsd
  3. Oct 2, 2011 #2

    G01

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    Let [itex]u=x-x_o[/itex].

    That should give you two terms that both proportional to standard Gaussian integrals. Consider the symmetry of the integrand and the solution shouldn't be hard to come by.
     
  4. Oct 3, 2011 #3
    Ah, ok. Since the other expectation values are next on the block, and I can screw up any given integral is the following mathematically "legal"?

    [itex]=N2\intdxx|e-(x-x0)/2k2|2[/itex]
    [itex]=N2\intdux|e-u/2k2|\intdu|e-u/2k2|[/itex]
    [itex]=N2x\intdu|e-u/2k2|\intdu|e-u/2k2|[/itex]
     
  5. Oct 3, 2011 #4
    Am I way off or does this come out to be <x>=x0?
     
  6. Oct 3, 2011 #5

    G01

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    That's the answer.

    Your Latex is not formatted correctly, so just to make sure that we are on the same page, this is the type of integral we are considering:

    [tex]<x^n>=\int dx x^n e^{\frac{-2(x-x_o)^2}{2k^2}}[/tex]

    Correct? Quote my post to see how I coded the above equation.
     
    Last edited: Oct 3, 2011
  7. Oct 3, 2011 #6
    Yep, that's the one! (sorry about that, Latex and I have a sordid history of being hit or miss)
     
  8. Oct 3, 2011 #7

    G01

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    Oops, I actually forgot a square in the numerator. Check that integral again and make sure it's still right.
     
  9. Oct 3, 2011 #8
    There should also be a negative sign, using numerical syntax:

    int(x^n*exp[(-(x-x0)^2)/(k^2))]dx).
     
  10. Oct 3, 2011 #9

    G01

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    Ooops. Yes, of course! Typo#2 corrected.


    Now, do you still need advice on finding the other expectation values?
     
  11. Oct 3, 2011 #10
    The basic technique is still the same correct? <x^2>=<psi|x^2|psi> or <p>=<psi|p|psi>
     
  12. Oct 3, 2011 #11

    G01

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    Yes, the basic idea is the same. Try them out. If you get stuck, feel free to come back and ask for some advice.
     
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