kith said:
This is a question of interpretation. As I wrote, in the Copenhagen interpretation, each photon is in polarization eigenstate, because the density operator of an unpolarized beam is interpreted as a statistical mixture of particles with definite states.
I've been looking into this over the past few days, and I will admit that the situation is murkier than I thought. A few things:
(1) I'm pretty sure the Copenhagen interpretation is inconsistent with "hidden variables" theory -- which incorporates the idea that quantum systems have definite values of all physical parameters before and after measurement. In other words, in the Copenhagen interpretation, a system does not acquire a definite value of momentum, polarization, etc. until you measure it.
(2) I'm pretty sure that there is no requirement in the postulates of quantum mechanics that a quantum system be in a "pure" state.
If the system is in a pure state, then you can make certain calculations, but there is no law of nature that says that the system
must be in a pure state.
(3) To that end, in 1927 John von Neumann invented the concept of quantum density matrices and the density operator. This generalizes the computations of quantum mechanics to states which are mixed, i.e. can be represented by the density matrix
ρ=Ʃp
i|ψ
i><ψ
i|
(4) Many texts discuss the density matrix in terms of the classical analogue, although you have to read carefully not to be deceived. Quantum systems described by a density matrix are quantum, not classical. In particular there is no "law of large numbers" for the quantum density matrix formalism. A single photon can be in a mixed state.
(5) For unpolarized light in particular, the idea that the density matrix implies that each photon has a definite polarization is incoherent. For example, in the {x,y} eigenbasis (={x-polarization, y-polarization}), one way to represent unpolarized photons is with the density matrix
\rho=\frac{1}{2}\left( \begin{array}{cc}<br />
1 & 0 \\<br />
0 & 1 \\<br />
\end{array} \right)
Classically, this describes a system with a statistically large number of particles, half of which are in state x, and half in state y. Quantum mechanically, we might be tempted to use the same words, but it is more correct to say that the state is such that there is a 50% probability that a measurement will find the polarization is x, and 50% for y.
Here's the rub: if we change the basis to {R, L} (={right circularly polarized, left circularly polarized}), then the density matrix doesn't change. The state is such that there is a 50% probability that a measurement will find the polarization is R, and 50% for L. But quantum mechanically,
this is the identical mixed state.
So you see the problem with the "semi-classical" interpretation? Classically, we think that the collection of photons is half x, half y, and that if we examine each one, we can find which is which. In the classical view, the photon actually has an x-polarization, and is not circularly polarized. Quantum mechanically, though, we are agnostic about the state prior to measurement. We only claim knowledge of the state after the polarization has been measured.
(6) In the case of photons, it's difficult to say why the Copenhagen interpretation (my version, not yours) is forced on us. Classically, because the beam has no net polarization, one is tempted to conclude what fisico30 said: that the beam has a definite polarization at any time, but it switches from one state to another very rapidly. Quantum mechanically, the beam is just a stream of photons in the unpolarized mixed state. But it takes a subtle experiment to see why the classical view is untenable.
(7) At the risk of playing bait-and-switch, this is easier to discuss in terms of systems with polarized atoms, like the paradigmatic Stern-Gerlach (SG) experiment. In the atomic polarization case, we can have x, y, and z polarizations, each with +,- states. The SGx device is designed to sort the beam into x+ or x- atoms, ditto SGy, SGz. Now, classically, an unpolarized beam of atoms is composed of atoms of definite but random polarization. Suppose we send an unpolarized beam of atoms into an SGz detector. Half come out z+, half z-. Classically, half of the z+ atoms are in the x+ state as well, half x-, and if we send the z+ beam into an SGx device, this what we observe. But the crucial difference between classical and quantum measurements is that if we send the doubly-filtered beam into an SGz device again, we see z- atoms coming out -- even though we selected only z+ atoms in the first device. It is this peculiar behavior of quantum mechanical systems that leads us down the path of defining eigenstates, operators, amplitudes, expectation values, and the probabilistic Copenhagen interpretation, which is agnostic about states prior to measurement.
Along these lines, imagine a Stern-Gerlach device for photons. An SGxy device sorts the photons into x and y polarizations. An SGRL device sorts the stream into R and L polarizations. If we send an unpolarized stream of photons into an SGxy device, half come out x, half y. Now send the x stream into an SGRL device. Half of those come out R, half L. Now send the x,R stream into an SGxy device -- and we will find y photons coming out. Why does this not surprise us? I would argue that this result does not surprise us because we already had a workable wave theory of light long before we were confronted with the quantum mechanical properties of photons. And in the case of EM wave propagation, we already have the idea that polarizing filters project the wave into a different state. In the case of atoms, the idea of discreteness was around from the inception. It took awhile before we got the point about the wave nature of atoms.
But here's the point: an unpolarized stream of atoms is represented by a density matrix, and it is not just "an unpolarized stream of atoms" but literally "a stream of unpolarized atoms". It's not that half the atoms are in state |z+> and half in |z->. The state of each atom is not determined until we measure it. In the same way, "an unpolarized stream of photons" is really "a stream of unpolarized photons". The polarization is not set until we measure it.
At least, that's what makes sense to me. I'm open to other interpretations or insights.
BBB
