Connection With Classical Mechanics
The wave function can also be written in the complex exponential (polar) form:
where R and S are real functions of r and t.
Written this way, the probability density is
and the probability current is:
The exponentials and R∇R terms cancel:
Finally, combining and cancelling the constants, and replacing R2 with ρ,
If we take the familiar formula for the current:
- ,
where v is the velocity of the particle (also the group velocity of the wave), we can associate the velocity with ∇S/m, which is the same as equating ∇S with the classical momentum p = mv. This interpretation fits with Hamilton-Jacobi theory, in which
in Cartesian coordinates is given by ∇S, where S is Hamilton's principal function.
Read more about this topic: Probability Current
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