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Probability Current for Free Particle Wave Function

  1. Feb 19, 2008 #1
    [SOLVED] Probability Current for Free Particle Wave Function

    1. The problem statement, all variables and given/known data
    Find the probability current, J for the free particle wave function. Which direction does the probability current flow?

    2. Relevant equations
    [tex]J(x,t) = \frac{ih}{4\pi m}\left(\Psi \frac{\partial \Psi^{*}}{\partial x} - \Psi^{*} \frac{\partial \Psi}{\partial x}\right)[/tex]

    [tex]\Psi_{k}\left(x, t\right) = Ae^{i\left(kx - \frac{hk^{2}}{4\pi m}t}\right)[/tex]

    3. The attempt at a solution
    I won't take the time to put my math into Latex, but I come up with

    [tex]J(x,t) = \frac{A^{2}hk}{2\pi m}[/tex]

    Is this correct or did I do the complex conjugate wrong?
    How would I find the probability current flow direction?
    Last edited: Feb 19, 2008
  2. jcsd
  3. Feb 19, 2008 #2
    I just read that the direction is simply the sign of J(x,t) ( - corresponds to left and + corresponds to right). If this is so, that would be great. Can anyone confirm? (it wasnt exactly a textbook source)
  4. Feb 19, 2008 #3


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    Yes, that's correct (and you can tell that the wavefunction you have is a plane wave travelling to the right since the sign of the x and t terms in the exponential have opposite signs). Your current looks good if A is assumed real (you should really have |A|^2 there, not A^2 since a gets complex conjugated).
  5. Feb 19, 2008 #4
    Great, thank you very much.
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