# 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(L

) = 0_{x}

So the fact that

tells you right away that B must be 0, because cos(0) = 1. And the fact that X(L* _{x}*) = 0 tells you that X(L

*) = A sin(*

_{x}*k*

*L*

_{x}*) = 0. Because the sine is 0 when its argument is a multiple of*

_{x}this means that

And because

it means that

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 = E* _{x}* + E

*+ E*

_{y}*. Following*

_{z}you have this for E* _{y}* and E

*:*

_{z}So the total energy of the particle is E = E* _{x}* + E

*+ E*

_{y}*, which equals this:*

_{z}And there you have the total energy of a particle in the box potential.