In quantum mechanics, the probability current describes the movement
of the probability of finding a particle at given point in space.
In other words, it treats the particle as a heterogeneous fluid with density .
Now, the probability of finding the particle within a volume is:
As the system evolves in time, this probability may change, so we take
its derivative with respect to time , and when necessary substitute
in the other side of the Schrödinger equation to get:
Where we have defined the probability current as follows in
Let us rewrite this using the momentum operator
as follows, noting that is
simply the velocity operator :
Returning to the derivation of , we now have the following
By removing the integrals, we thus arrive at the continuity equation
This states that the total probability is conserved, and is reminiscent of charge
conservation in electromagnetism. In other words, the probability at a
point can only change by letting it “flow” towards or away from it. Thus
represents the flow of probability, which is analogous to the
motion of a particle.
As a bonus, this still holds for a particle in an electromagnetic vector
potential , thanks to the gauge invariance of the Schrödinger
equation. We can thus extend the definition to a particle with charge
in an SI-unit field, neglecting spin:
- L.E. Ballentine,
Quantum mechanics: a modern development, 2nd edition,