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What is the physical dimension/unit of Probability current?

  1. May 10, 2016 #1
    1. The problem statement, all variables and given/known data
    Question:
    What is the physical dimension of Probability Current for a particle in 1 dimension? (Quantum Mechanics)

    2. Relevant equations
    Quantum mechanical Probability Current:
    07e268a6edab33f979b674b6b05b6d08.png

    3. The attempt at a solution
    I know the physical dimension of mass, that is kg. If I know every dimension, I can try some things and I can find the dimension of the observable. But now, I'm stuck. I guess that the wave function has no dimension, because it is very related to probability. But what's the case with $$ \frac{\partial \Psi}{\partial x} $$?
     
  2. jcsd
  3. May 10, 2016 #2

    BvU

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    Hello Feest, :welcome:

    m-1 I would dare to venture .... (on your last question)
     
  4. May 10, 2016 #3

    haruspex

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    But h has dimension of action. Taking Ψ to be dimensionless, j would appear to have dimension of velocity.
     
  5. May 10, 2016 #4

    BvU

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    Actually ##j## is a probability current density.
     
  6. May 10, 2016 #5

    haruspex

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    Just realised you meant m as metres, not mass. The question is for dimension, so I would answer L-1 there.
    But what do you think the dimension of j is, if not LT-1?
     
  7. May 10, 2016 #6

    BvU

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    With $${\partial \rho\over \partial t} + \nabla\cdot {\bf j} = 0
    $$ where ##\rho = \psi^*\psi ## and ##\int \rho \, d\tau = 1##, I would end up with L-2T-1 :smile:

    [edit] And now I have to backtrack to 1 dimension as cleary stated in post #1 (o:)) ending up with T-1 :smile:
     
  8. May 10, 2016 #7

    BvU

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    So here we must react also: It is related to probability in the sense that $$\int \psi^*\psi\,d\tau = 1 $$where the integral is over all space. The ##1## is dimensionless: a genuine probability. So guess again !
     
  9. May 11, 2016 #8
    This is a nice question to answer. It took me some time but I think:
    ∫ψ∗ψdτ=1 has no dimensions.
    But as you integrate over space, and in this case this is one-dimensional, the dimension of ψ∗ψ is multiplied with [length]. And then ψ has a dimension of [(1/length)^½]?
     
  10. May 11, 2016 #9

    BvU

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    An unavoidable conclusion, isn't it ? I never worried about the wave function having a dimension (always considered it as dimensionless) and would have liked to keep it that way. But -- unless we are being corrected -- this is what comes out !
     
  11. May 11, 2016 #10
    So thank you everyone!

    Conclusion: the dimension of the probability current is [1/time]?

    These physical dimensions used to help me give interpretations to different physical quantities. But in QM, that's not (yet) the case. :)
     
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