**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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changGreat solution. For (b), you can also convince yourself that Psi*del(Psi*)/del(x) = psi \times d(psi)/dx (sorry for the terrible typesetting, but Psi here is the capital psi and psi is the small psi, del for partial derivative and d for definite derivative) and you get psi \times d(psi)/dx – psi\times d(psi)/dx = 0 inside the parenthesis for J.

gwrowePost authorYou can use Latex to make the math easier to read – see here.