Question

Consider a composite object such as the hydrogen atom. Will it behave as a boson or a
fermion? Argue in general that objects containing an even/odd number of fermions will behave
as bosons/fermions

Answer 1

Suppose if we look at the hydrogen atom in particular. It is composed of a proton and an electron, both of which are fermions. The proton and electron are not, of course, identical particles, but now suppose we have two hydrogen atoms. The two protons are identical fermions, just as are the two electrons. However, when analyzing a system of two hydrogen atoms, the relevant question is what happens to the state vector if we exchange the two atoms. In doing so, we exchange both the two protons and the two electrons. Each exchange multiplies the state vector by −1, so the net effect of exchanging both protons and both electrons is to multiply the state vector
by (−1)² = 1. In other words, a hydrogen atom acts as a boson, even though it is composed of two fermions.
In general, if we have a compound object containing n fermions, then the state vector for a system of two such objects is multiplied by (−1)ⁿ when these two objects are exchanged. That is, a compound object containing ane ven number of fermions behaves as a boson, while if it contains an oddn umber of fermions, it behaves as a fermion.

A compound object consisting entirely of bosons will always behave as a boson, no matter how many such bosonic particles it contains, since interchanging all n bosons just multiplies the state vector by (+1)ⁿ = 1.