## Probability current

**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.

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[…] seen the concept of the probability current in one dimension. A generalization to 3-d gives the […]