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How to Find a Wave-Function Equation in an Infinite Square Well

Infinite square well, in which the walls go to infinity, is a favorite problem in quantum physics. To solve for the wave function of a particle trapped in an infinite square well, you can simply solve the Schrödinger equation.

Take a look at the infinite square well in the figure.

A square well.
A square well.

Here’s what that square well looks like:

image1.png

The Schrödinger equation looks like this in three dimensions:

image2.png

Writing out the Schrödinger equation gives you the following:

image3.png

You’re interested in only one dimension — x (distance) — in this instance, so the Schrödinger equation looks like

image4.png

Because V(x) = 0 inside the well, the equation becomes

image5.png

And in problems of this sort, the equation is usually written as

image6.png

So now you have a second-order differential equation to solve for the wave function of a particle trapped in an infinite square well.

You get two independent solutions because this equation is a second-order differential equation:

image7.png

A and B are constants that are yet to be determined.

The general solution of

image8.png

is the sum of

image9.png
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