Do free particles have quantum numbers? What are they?
Do they have quantum numbers? Of course, why wouldn't they? (The set of) Quantum numbers are the minimum set of conserved quantities (i.e. quantities that commute with the hamiltonian) you need to describe a system. Though what I think you're probably asking is do free particles have QUANTIZED (or discrete) quantum numbers. Well, it depends how much you specify. The position and momentum of a free particle are continuous quantities, but if you want to be nitpicky things like its charge aren't. I'm going to take a leap and assume that that's the question you're really asking (i apologize if it is not). The position and momentum of a free particle are continuous not discrete. They still have quantum numbers though. Remember that things like n,l and m_s are the quantum numbers OF A HYDROGEN-LIKE atom, they are not THE quantum numbers. What the quantum numbers of a system are depend on the system and how specific you want to be.
Perhaps, it might be easiest to consider a particle in a box, in which case you get a quantum number like n where the energy E is proportional to n^2 and the spacing between level is dependent on the radius of the box. Now take the radius of the box out to infinity. You get a continuum. You could still say n is a quantum number, but it's no longer quantized.
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