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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:


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 nth quantum state is


Therefore, the result is


where exp (x) = ex.

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