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How to Normalize the Wave Function in a Box Potential

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2016-03-26 14:06:28
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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

About This Article

This article is from the book: 

About the book author:

Dr. Steven Holzner has written more than 40 books about physics and programming. He was a contributing editor at PC Magazine and was on the faculty at both MIT and Cornell. He has authored Dummies titles including Physics For Dummies and Physics Essentials For Dummies. Dr. Holzner received his PhD at Cornell.