Required math: calculus
Required physics: Schrödinger equation
Reference: Griffiths, David J. (2005), Introduction to Quantum Mechanics, 2nd Edition; Pearson Education – Problem 2.19.
The rate of change of probability of a particle in a given range of can be written as the difference in probability current at the two ends. The current is defined as
For the free particle, a stationary state is given by
The probability current for this state is found by working out the derivative:
So we get
(The complex exponentials cancel out in .) Since the current is positive, it ‘flows’ in the positive direction. Note that the current is independent of , so the probability of a particle being found in any given range of is constant. (Actually, as we’ve seen, a free particle can’t exist in a single stationary state since such a state cannot be normalized.)