Physics Asked on November 20, 2020
We have antisymmetric wavefunctions for a fermionic systems and symmetric wavefunctions for bosonic systems that give us a hint that quantum states can be occupied by a single fermion(or none) while multiple bosons can occupy single quantum state(including zero bosons in a state). This shows how fermions behave in themselves i.e. in a pure fermionic gas and how bosons behave in themselves i.e. in pure bosonic gas. Let us take a system of one fermion and one boson. I want to know how these two behave in presence of each other. What can we say about this system?
They will interact through their Hamiltonian. For example, if the fermions are electrons and the bosons are electromagnetic radiation, the interaction Hamiltonian could be $$ bar psi gamma^mu psi A_mu,qquad ({rm relativistic; case})$$ or $$ frac{e}{m}vec{p}cdot vec{A},qquad ({rm non-relativistic;QM});.$$
If the interaction Hamiltonian is zero, they will both evolve independently, each in their own world, without interacting.
But I think you are asking for specifically if their interactions are affected by the boson/fermion statistics. I guess this would affect the form of the interaction Hamiltonian. The spin-statistics theorem says that fermions have to be half-integer spin and bosons integer. This severely limits the choices of Lorentz-invariant/rotation,translation-invariant interaction Hamiltonians. Of course, zero is always allowed.
Answered by Eric David Kramer on November 20, 2020
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