Required math: calculus
Required physics: Schrödinger equation
Reference: Griffiths, David J. (2005), Introduction to Quantum Mechanics, 2nd Edition; Pearson Education – Problem 1.14.
Using the Schrödinger equation we can derive an interesting quantity called the probability current. Using the probabilistic interpretation of the wave function, the probability of a particle being between and is
The rate of change of this probability can then be expressed in terms of spatial derivatives using the Schrödinger equation:
We can now apply integration by parts to each term.
Adding these terms together, we get
If we define the probability current as
we can write the rate of change of probability as
As an example, if the wave function is given by
(we’ve taken the constant in front so that it normalizes the wave function), then
So for the probability current we get
A bit of an anti-climax after all that mathematics.