**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.)

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GeorgeIf I understood the theory well, a free particle cannot exist in a stationary state so we should probably have the integral over k form of the wave equation? Are we allowed to use stationary states as a mathematically useful trick which does not have any physical meaning, and getting right results? The integral form shouldn’t have given the same result?