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Expectation Value

  1. Oct 8, 2006 #1
    OK, here is the problem:

    An electron is in a 1-D box of length L. Its wavefunction is a linear combination of the ground and first-excited stationary states (and here it is):

    [tex]\phi(x,t) = \sqrt\frac{2}{L}[sin (\frac {\pi x} {L})e^{-i \omega_1 t} + sin\frac {2 \pi x}{L} e^{-i \omega_2 t}][/tex]

    where [tex]E_1 = \hbar\omega_1 = \frac{\pi^2 \hbar^2}{2mL^2}[/tex], and [tex]E_2 = \hbar\omega_2 = \frac{4 \pi^2 \hbar^2}{2mL^2}[/tex]

    First, I need to find the expectation value <x> for the wavefunction.

    It seems to me that I need to multiply the wavefunction by its complex conjugate, put x in the middle, and integrate from -infinity to infinity. But attempting to do this yields some nasty integrals, like [tex]\int x sin^2 (\frac{\pi x}{L})[/tex] and [tex]\int sin(\frac{\pi x}{L})sin(\frac{2 \pi x}{L})e^{it(\omega_2-\omega_1)}[/tex] ; am I just doing this completely wrong or do I need to plow through the integrals?
     
    Last edited: Oct 8, 2006
  2. jcsd
  3. Oct 8, 2006 #2
    Well, Integrals might be nasty... But you are wrong about your integration limits, they're [0,L] and not [-inf,inf].
     
  4. Oct 8, 2006 #3
    Thanks, I was wondering about that too. But I'm doing it correctly otherwise? I don't even know how to begin that second integral. Guh.
     
  5. Oct 8, 2006 #4
    Well, you should use the formulae that convert a sine multiplication to a sum of cosines... Sorry, Integrals in QM are ugly... And it only gets worse...
     
  6. Oct 9, 2006 #5

    jtbell

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    Staff: Mentor

    Whenever I have an integral that isn't a basic polynomial or trig function, and I can't recognize a simple substitution that will solve it, I pull out my CRC Handbook of Mathematical Tables and look it up. I tell my students that doing this is OK, so long as they give me a reference: the name of the book or the address of the Web site, and the number that it uses to identify the integral.
     
  7. Oct 9, 2006 #6
    Yeah, I think I figured them out, thanks to you guys :)
     
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