Probability Current - Connection With Classical Mechanics

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:

$\begin{align} \bold{j} & = \frac{\hbar}{2mi}\left(\Psi^* \bold{\nabla} \Psi - \Psi \bold{\nabla}\Psi^*\right) \\ & = \frac{\hbar}{2mi}\left(R e^{-i S / \hbar } \bold{\nabla}R e^{i S / \hbar } - R e^{i S / \hbar } \bold{\nabla}R e^{-i S / \hbar }\right) \\ & = \frac{\hbar}{2mi}\left \\ \end{align}$

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.

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