Quantum Physics For Dummies, Revised Edition
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In quantum physics, to be able to determine the energy levels of a particle in a box potential, you need an exact value for X(x) — not just one of the terms of the constants A and B. You have to use the boundary conditions to find A and B. What are the boundary conditions? The wave function must disappear at the boundaries of the box, so

  • X(0) = 0

  • X(Lx) = 0

So the fact that

image0.png

tells you right away that B must be 0, because cos(0) = 1. And the fact that X(Lx) = 0 tells you that X(Lx) = A sin(kxLx) = 0. Because the sine is 0 when its argument is a multiple of

image1.png

this means that

image2.png

And because

image3.png

it means that

image4.png

That's the energy in the x component of the wave function, corresponding to the quantum numbers 1, 2, 3, and so on. The total energy of a particle of mass m inside the box potential is E = Ex + Ey + Ez. Following

image5.png

you have this for Ey and Ez:

image6.png

So the total energy of the particle is E = Ex + Ey + Ez, which equals this:

image7.png

And there you have the total energy of a particle in the box potential.

About This Article

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About the book author:

Steven Holzner is an award-winning author of technical and science books (like Physics For Dummies and Differential Equations For Dummies). He graduated from MIT and did his PhD in physics at Cornell University, where he was on the teaching faculty for 10 years. He’s also been on the faculty of MIT. Steve also teaches corporate groups around the country.

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