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Don't feel disturbed by the cryptic form of the question. Take some time to analyse it. The real important part deals with a particle in a 3D box and degeneracy. The rest is a cryptic way to hide the energy level of interest. Degeneracy means that more than one wave function corresponds to the same energy. The cause for degeneracy is symmetry. There is a lot of symmetry present in a cubic box. For a particle being in a cube, this means that the general form of the wave function is analogous to a one-dimensional box.
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The corresponding energy is given by
| En1n2n3 = (n12+n22+n32) |
h2
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Because of the fact that the movements in x-, y- and z-direction are independent, the wave function is just the product of the wave functions describing the movements in those directions. This means
Now we move to the cryptic part of the exercise. The energy depends on the quantum numbers. From the restriction on the quantum numbers, being positive integers, follows that the lowest energy will be found for n1 = 1, n2 = 1,n3 = 1
| E111 = (12+12+12) |
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The energy of interest is three times higher
| En1n2n3 = 9 |
h2
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= (n12+n22+n32) |
h2
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The unknown quantum numbers can be found by solving following equation.
None of the quantum numbers can be larger than 2, because then the sum would always be larger than 9. So each must be 1 or 2. Some trial and error then leads to three possibilities n1 = 1, n2 = 2, n3 = 2, n1 = 2,n2 = 1, n3 = 2, and n1 = 2, n2 = 2, n3 = 1. Hence the degeneracy of the level that has an energy three times that of the lowest level is three.
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