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Free electron conservation of probability

  1. Feb 24, 2012 #1
    1. The problem statement, all variables and given/known data
    Say you have the equation:
    [itex] i \dfrac{\partial u}{\partial t} = -\beta \frac{\partial^2 u}{\partial x^2}[/itex]
    How does one show that the probability change pr. unit time equals zero for ANY wave function which obeys the equation.

    2. Relevant equations
    dP/dt = 0 and the above equation.


    3. The attempt at a solution
    I Fourier transformed the equation from x-> k and got a solution:
    [itex]u(k) = c(k)\exp(-\beta k^2 i t)[/itex]
    And here its easy to show that dp/dt = 0, but is it sufficient. Say i wanted to transform like t-> k, then i would have 2 linear independent solutions and that can also be shown true.

    My question is: Is my argument strong enough to make sure that all wave functions obeying the equation satisfies dp/dt = 0
     
    Last edited: Feb 24, 2012
  2. jcsd
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