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Homework Help: Simple exponential multiplication (electron interfernce)

  1. Jan 8, 2010 #1
    1. The problem statement, all variables and given/known data

    Briefly, the question asks to prove how the interference of 2 electrons (travelling in opposite directions as 1-D waves) would affect the probability of finding each electron in free space. My issue has to do with the first step in the solution.


    2. Relevant equations

    [tex]\Psi_{1}[/tex] = [tex]\Psi_{0}[/tex] [tex]e^{jkx}[/tex]
    [tex]\Psi_{2}[/tex] = [tex]\Psi_{0}[/tex] [tex]e^{-jkx}[/tex] (Note change in direction)

    Hence the interference of these two functions will be given by their sum:
    [tex]\Psi_{total}[/tex] = [tex]\Psi_{0}[/tex] [tex]e^{jkx}[/tex] [tex]\Psi_{0}[/tex] ยท [tex]e^{-jkx}[/tex] (i)
    [tex]\Psi_{total}[/tex] = 2[tex]\Psi_{0}[/tex] cos (kx) (ii)


    3. The attempt at a solution

    I just don't understand how to get (ii) from (i)... thanks for looking!
     
  2. jcsd
  3. Jan 8, 2010 #2
    A correction (You put a multiplication sign instead of an addition one. I think that was your mistake):

    [tex]\Psi_{total}=\Psi_0(e^{jkx}+e^{-jkx})[/tex]
    Use Euler's formula, and just see that the imaginary sine terms cancel, while the real cosine terms add up.

    [tex]e^{i\theta}=\cos \theta + i\sin \theta[/tex]
     
  4. Jan 8, 2010 #3
    Ah I see! I tend to forget simple steps like that which involve such simple identities. Thanks!
    (PS: That was a typo on my part)
     
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