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Determining the Energy Levels of a Particle in a Box Potential

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.

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