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Excited states of noninteracting Fermion and Bose systems

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The bosons and the distinguishable particles can be in the same single particle states, the fermions must all be in different states. If you had only one particle then it could occupy a state with energy 0, 1, 2, ...etc. For the many particle system we can denote any state by specifying how many particles are in the single particle states. So, an example of a five particle state could be the state in which there are 4 particles in the single particle state with energy 0 and one in the single particle state with energy 1. Now, for Fermions this state is not allowed as there are more than one particles in the same state. For distinguishable particles the state is not uniquely defined by just specifying how many particles are occupying which single particle states. If there are 4 particles in the single particle state with energy 0 and 1 particle in the single particle state with energy 1, then there are 5 different possibilities for the particle with energy 1.

Let's use the following notation. n0, n1, n2, n3, n4, ...etc. denote the number of particles in the single particle state with ...

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