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Conservation of probability

  1. Oct 17, 2011 #1
    1. The problem statement, all variables and given/known data
    I have a wavefunction [itex]\psi = \pi^{-1\over 4}(1+it)^{-1\over 2} \exp{({-x^2\over 2(1+it)})}[/itex]

    I want to show that it satisfies the conservation of probability.

    2. Relevant equations
    [itex]\partial_t P +\partial_x j =0[/itex] --(*)

    3. The attempt at a solution
    I calculated the probability distribution to be [itex]P=\pi^{-1\over 2}(1+t^2)^{-1\over 2} \exp{({-x^2\over (1+t^2)})}[/itex] and the probability current [itex]j=ix\pi^{-1\over 2}(1+t^2)^{-3\over 2} \exp{({-x^2\over (1+t^2)})}[/itex]

    This gives [itex]\partial_t P = -t\pi^{-1\over 2}(1+t^2)^{-5\over 2} (t^2-2x^2+1)\exp{({-x^2\over (1+t^2)})}[/itex] and [itex]\partial_x j = i\pi^{-1\over 2}(1+t^2)^{-5\over 2} (t^2-2x^2+1)\exp{({-x^2\over (1+t^2)})}[/itex]

    But how are they do they satisfy (*)?

  2. jcsd
  3. Oct 17, 2011 #2
    If equation (*) is satisfied then probability IS conserved. And your derivatives of P and of j show just that.
  4. Oct 17, 2011 #3

    Thanks, I know that! The problem is notice how the 2 partial derivatives are not exactly equal! My aim is to fit them into (*).
  5. Oct 17, 2011 #4
    Sorry!! I missed the extra t in the derivative of P.
  6. Oct 17, 2011 #5

    Also the extra [itex]i[/itex] in the 2nd expression :(
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