# Find the Wave Function of a Particle in an Infinite Square Well over Time

In quantum physics, you can use the Schrödinger equation to see how the wave function for a particle in an infinite square well evolves with time*.* The Schrödinger equation looks like this:

You can also write the Schrödinger equation this way, where H is the Hermitian Hamiltonian operator:

That’s actually the *time-independent* Schrödinger equation. The time-dependent Schrödinger equation looks like this:

Combining the preceding three equations gives you the following, which is another form of the time-dependent Schrödinger equation:

And because you’re dealing with only one dimension, *x*, this equation becomes

This is simpler than it looks, however, because the potential doesn’t change with time. In fact, because E is constant, you can rewrite the equation as

That equation makes life a lot simpler — it’s easy to solve the time-dependent Schrödinger equation if you’re dealing with a constant potential. In this case, the solution is

Neat. When the potential doesn’t vary with time, the solution to the time-dependent Schrödinger equation simply becomes

the spatial part, multiplied by

the time-dependent part.

So when you add in the time-dependent part to the time-independent wave function, you get the time-dependent wave function, which looks like this:

The energy of the *n*th quantum state is

Therefore, the result is

where exp (*x*) = *e** ^{x}*.