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Real and Complex Parts of a Wave Function

  1. Sep 21, 2007 #1
    1. The problem statement, all variables and given/known data

    Just a snipit of one of my homework problems. I'm trying to find out what [tex]\Psi \frac{\partial \Psi^{*}}{\partial x}[/tex] equals to help me find out what the probability current for a given free particle is.

    2. Relevant equations
    [tex]\Psi = Ae^{i(kx-\frac{\hbar k^{2}t}{2m})}[/tex]


    3. The attempt at a solution

    I view [tex]\Psi^{*}[/tex] as the complex part of the given wave function; but in this case there is no real part, it's all complex. Does that mean the real part is zero? If so then [tex]\Psi \frac{\partial \Psi^{*}}{\partial x} = 0[/tex]. If [tex]\Psi = \Psi^{*}[/tex], then the larger equation I'm trying to calculate comes out to be zero because it's:

    [tex]\Psi \frac{\partial \Psi^{*}}{\partial x} - \Psi^{*} \frac{\partial \Psi}{\partial x}[/tex]

    So what am I missing here? Does it actually have a zero probability current because it's a "free particle" (whatever that really means)?
     
  2. jcsd
  3. Sep 21, 2007 #2

    Kurdt

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    I think you've been misinformed somewhere. The starred notation means the complex conjugate (certainly if this is quantum mechanics HW) of the function. Basically to find the complex conjugate of a complex function you reverse the sign in front of any i. For example:

    If we have a general complex number z = a + ib then it has a complex conjugate of z* = a - ib.

    There is more information about complex conjugates here:

    http://mathworld.wolfram.com/ComplexConjugate.html
     
    Last edited: Sep 21, 2007
  4. Sep 21, 2007 #3
    Kurdt - Ah, brilliant. Ok, yes, I do remember it being the conjugate. I think in my mind I had the idea that it dealt with something complex (I don't mean complicated; but [tex]i[/tex] ), and so I must have mentally given it the value of the complex portion of the wave function. Thanks for that clarification.

    So, if I understand correctly, then in the case listed I would have

    [tex]\Psi = Ae^{i(kx - \frac{\hbar k^{2}t}{2m})}[/tex]
    and
    [tex]\Psi^{*} = Ae^{-i(kx - \frac{\hbar k^{2}t}{2m})}[/tex]​

    If that's correct, then I think I can figure it out. Thanks for the help. I'll jump back on later if I need more help; but for now it's off to class.
     
  5. Sep 21, 2007 #4

    Kurdt

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    Yeah thats it basically.
     
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