What is the physical dimension/unit of Probability current?

  • #1

Homework Statement


Question:
What is the physical dimension of Probability Current for a particle in 1 dimension? (Quantum Mechanics)

Homework Equations


Quantum mechanical Probability Current:
07e268a6edab33f979b674b6b05b6d08.png


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 $$ \frac{\partial \Psi}{\partial x} $$?
 

Answers and Replies

  • #2
BvU
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Hello Feest, :welcome:

m-1 I would dare to venture .... (on your last question)
 
  • #3
haruspex
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Hello Feest, :welcome:

m-1 I would dare to venture .... (on your last question)
But h has dimension of action. Taking Ψ to be dimensionless, j would appear to have dimension of velocity.
 
  • #4
BvU
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Actually ##j## is a probability current density.
 
  • #5
haruspex
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Actually ##j## is a probability current density.
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?
 
  • #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:
 
  • #7
BvU
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I guess that the wave function has no dimension, because it is very related to probability.
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 !
 
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  • #8
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 !
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]?
 
  • #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 !
 
  • #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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