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How to Normalize the Wave Function in a Box Potential

In your quantum physics course, you may be asked to normalize the wave function in a box potential. Here's an example: consider the wave function

image0.png

In the x dimension, you have this for the wave equation:

image1.png

So the wave function is a sine wave, going to zero at x = 0 and x = Lz. You can also insist that the wave function be normalized, like this:

image2.png

By normalizing the wave function, you can solve for the unknown constant A. Substituting for X(x) in the equation gives you the following:

image3.png

Therefore,

image4.png

which means you can solve for A:

image5.png

Great, now you have the constant A, so you can get X(x):

image6.png

Now get

image7.png

You can divide the wave function into three parts:

image8.png

By analogy with X(x), you can find Y(y) and Z(z):

image9.png

So

image10.png

equals the following:

image11.png

That's a pretty long wave function. In fact, when you're dealing with a box potential, the energy looks like this:

image12.png
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