# How does a boson interact with a fermion?

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?

• I think that the title may be misleading: as I understand, the question is "which are the properties of a many-body wave function for a system comprised of both fermionic and bosonic particles?". E.g. a gas of Helium (bosonic) and Lithium (fermion). Is my interpretation correct? Oct 22, 2022 at 21:03

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 $$\frac{e}{m}\vec{p}\cdot \vec{A},\qquad ({\rm non-relativistic\;QM})\;.$$
• Am I correct that there must be an even number of fermions but can be any number of bosons in the interaction? For example in $\bar \psi \gamma^\mu \psi A_\mu$ there are two fermions ($\bar\psi$ and $\psi$) and one boson $A_\mu$. Aug 31, 2020 at 10:33