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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:

image0.png

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

image1.png

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

image2.png

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

image3.png

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

image4.png

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

image5.png

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

image6.png

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

image7.png

the spatial part, multiplied by

image8.png

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:

image9.png

The energy of the nth quantum state is

image10.png

Therefore, the result is

image11.png

where exp (x) = ex.

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