Fermions vs Bosons
In particle physics, there are many ways to classify fundamental particles. One classification is into fermions and bosons. Electrons are an example of fermions while photons are examples of bosons. The difference between fermions and bosons is that two bosons can be in the same quantum state, but two fermions can never occupy the same state (think about electrons in an atom - all of them have at least one quantum number different).
Imagine that you have two bosons and two fermions, and three energy levels available to them: , and ( is not a multiple of ). How many more possible energies can a system of two bosons have than a system of two fermions?
The answer is 3.
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1 solution
Fermions cannot occupy the same energy state, so each state can be populated at most with one fermion. There are three ways to do this: <1,1,0>, <1,0,1> and <0,1,1> (first number corresponds to the number of particles in zero energy state, second to the number of particles in E 1 energy state etc).
Bosons can share the same state, so the following configurations are possible in addition to the ones available to fermions: <2,0,0>,<0,2,0>,<0,0,2>. That is six configurations in total. Therefore, a pair of bosons can have three more energy states than a pair of fermions.