# Is it physically meaningful to combine a photon's position state space with its polarization state space?

Mathematically, we can take the tensor product of a photon's position state space $$\mathscr{H}_{pos}$$ with that of its polarization state space $$\mathscr{H}_{pol}$$. Then, in the resulting Hilbert space we have vectors like $$|\psi_1\rangle=|pos_1\rangle\otimes|pol_1\rangle$$ and $$|\psi_2\rangle=|pos_2\rangle\otimes|pol_2\rangle$$.

But is it physically possible to have a superposition state of $$|\psi_1\rangle$$ and $$|\psi_2\rangle$$, e.g. $$|\phi\rangle=0.8|\psi_1\rangle+0.6|\psi_2\rangle$$?

## 1 Answer

Yes, this is perfectly reasonable. Why wouldn't it be?

As a general rule, there is no rule of electromagnetism that dictates that the electromagnetic field must have a uniform polarization throughout all space. Such beams are certainly possible (say, a loosely focused gaussian beam with uniform linear or circular polarization), but they are special solutions. If you combine two of those solutions (with distinct polarizations and distinct spatial dependence) then you'll get a nontrivial polarization field (i.e. the polarization will have a nontrivial spatial dependence), and there's nothing wrong with that.

• Thank you. Sorry if I'm asking the obvious, but I don't have background in physics. Along the same lines, I guess it is then also meaningful to combine the electron's position state with its spin, and have superpositions of such combined states, right? – Tamás V Jan 28 at 12:26
• Yes. With very few exceptions, for any two accessible states of any system, their superposition is a physically allowed state. – Emilio Pisanty Jan 28 at 12:35