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Old Jan 20th 2010, 08:58 AM   #1
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particle degeneracy

I'm trying to calculate the degeneracy of the first few levels of a free particle confined to a cube of length a.

To start, I think I need to consider the energy states of a particle in a 3-D infinite well (the picture attachment).

But I'm not sure where I'm supposed to go from there...
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Old Jan 20th 2010, 09:50 AM   #2
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The eigenfunctions are sine waves of the right wavelength. The energy operator is essentially a second derivative. The directions x,y,z act independently, so you will have a complete eigenfunction that's a product of an X part, a Y part and a Z part. To see the degeneracy, just find out how many ways there are to write a particular energy. For example:

Energy
1 = 1^2+0+0 (x contribution =1, y=0,z=0)
1 = 0 + 1 + 0
1 = 0 + 0 + 1, so degeneracy = 3 here.

For higher values, there may be other ways to write the energy, for instance:

2 = 1 + 1 + 0 ( & permutations)
3 = 1 + 1 + 1 (only 1 way )
etc.
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