References: Shankar, R. (1994), Principles of Quantum Mechanics, Plenum Press. Section 5.3, Exercises 5.3.2 – 5.3.4.
Here are a few examples of probability current.
Example 1 Suppose the wave function has the form
where is a complex constant and is a real function of position and time. Then the probability current is
In particular, if itself is real, the probability current is always zero, so all the stationary states of systems like the harmonic oscillator and hydrogen atom that we’ve studied show no flow of probability, which is what we’d expect since they are, after all, stationary states.
Example 2 Now the wave function is
where the momentum is constant. In this case we have
This gives a probability current of
The probability density is
Thus the current can be written as
Classically, the momentum is , so the current has the same form as . This is similar to the electromagnetic case where the electric current density where is the charge density and is the velocity of that charge. The probability density can be viewed as “probability” moving with velocity .
Example 3 Now consider a one-dimensional problem where the wave function consists of two oppositely-moving plane waves:
In this case, we have
The probability current separates into two terms, one for each direction of momentum.