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Wavefunction in a delta potential well

  1. Jan 12, 2017 #1
    1. The problem statement, all variables and given/known data
    Using the equations given, show that the wave function for a particle in the periodic delta function potential can be written in the form

    ##\psi (x) = C[\sin(kx) + e^{-iKa}\sin k(a-x)], \quad 0 \leq x \leq a##

    2. Relevant equations
    Given equations:

    ##\psi (x) =A\sin(kx) + B\cos(kx), \quad 0<x<a##
    ##A\sin(ka) = [e^{iKa} - \cos(ka)]B##

    Note that ##k## and ##K## are different constants.
    3. The attempt at a solution
    I tried a bunch of stuff already but I can't seem to get to the answer.

    Attempt 1. I evaluated ##\psi## at ##0## and at ##a## for both the final equation and the general equation and tried to see if I could come to some conclusion based on equating these, but no luck there.

    Attempt 2. I tried working backwards and seeing if I could use the sine identity ##\sin(a-b) = \sin(a)\cos(b)-\cos(a)\sin(b)## but it only seems to make things more complicated.

    Could someone just give me a hint?
     
  2. jcsd
  3. Jan 12, 2017 #2

    vela

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    Try multiplying ##\psi(x)## by ##\frac{\sin ka}{\sin ka}##.
     
  4. Jan 12, 2017 #3
    I see 3 ways to do something with what you've suggested. Here is one attempt:

    ##\psi(x) = \frac{Asin(kx)sin(ka)+Bcos(kx)sin(ka)}{sin(ka)}##
    ##=Asin(kx) + \frac{B[(1/2)(sin(ka+kx)+sin(ka-kx))]}{sin(ka)}##
    ##=Asin(kx)+\frac{A}{2(e^{iKa}-cos(ka))}[sin(ka+kx)+sin(ka-kx)]##
    ##=Asin(kx)+\frac{e^{-iKa}A}{2(1-e^{-iKa}cos(ka))}[sin(ka+kx)+sin(ka-kx)]##

    This would almost be great if it weren't for that ##sin(ka+kx)## term. I don't know what to do with it.
     
  5. Jan 12, 2017 #4

    vela

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    There's a reason I said to multiply ##\psi## by ##\frac{\sin ka}{\sin ka}## rather than just the last term. See what you can do with the first term.
     
  6. Jan 13, 2017 #5
    I expanded it out to this, but nothing cancels nicely.

    ##\psi (x) = \frac{A(sin(kx)sin(ka)-cos(kx)cos(ka))}{sin(ka)}+\frac{B((1/2)(sin(kx)cos(ka)+cos(kx)sin(ka)-sin(kx-ka)))}{sin(ka)}##

    Should I have gone a different route?
     
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