End Notes
This section contains responses from people kind enough to correct my thoughts. I have posted their replies for your reference.
Lov Grover
It might help to point out that even quantum computers have their limitations.
When quantum computing was first being invented, it was hoped that it would be able to solve all conceivable problems just through the parallelism of quantum mechanics. Such a scheme would do a brute force search and would not need to use any of the structure of these problems. These hopes were dashed in 1995 by the [BBBV] paper, this proved that the best improvement that such an unstructured search scheme could provide was a square-root speedup. Later on I achieved this speedup in the quantum search algorithm.
Helen Quinn
All angular momentum of rotating objects in quantum theory comes in integer units of Planck's constant h/(2pi), which has dimensions of angular momentum. The angular momentum of each of your skater analogies is actually somewhere of order 10^33 in these units. Each type of particle has a characteristic intrinsic amount of angular momentum. 1/2 unit for any electron or quark and one unit for any photon or W or Z boson. So when we say a photon has spin 1 we are telling you the absolute magnitude of its internal angular momentum. But that is not the full story, as you can see from your skaters, a skater spinning to the left has a different angular momentum than a skater spinning to the right, and furthermore, if we go beyond skaters on a level surface we have to define the direction in space of the axis about which the skater spins left or right.
So the language spin up and spin down tells us about the direction of the spin about some previously defined axis. Now the peculiarities of quantum mechanics come into play. In the case of the skater, or say a spinning tennis ball you would think there is a continuous infinity of choices of the orientation of the spin in space --you could pick any direction for the axis and then have any amount of spin about that axis. Since you have 10**33 or so units of spin to play with this is pretty nearly true, but when we have only a few units, say J units of spin (where J can be 1/2 or 1 or 3/2 etc, depending on the particle), then in general once we have chosen an axis there are only 2J+1 ways put some fraction of the spin of the particle along the axis, so for spin 1/2 it can be +1/2 or -1/2, which we also call up and down, and for a massless spin 1 particle it can be +1,0 or -1.
These rules are not something you can ''understand'' they come out of the mathematics of the quantum theory, in which not just the total spin but also its orientation choices are quantized. For a massless spin 1 particle, such as a photon, there is one more oddity. First is a particle is massless it is never "at rest" --it is always travelling at the speed of light, so the only axis that it is reasonable to talk about for orienting the spin of this particle is the direction in which it is moving. It turns out that a photon can have spin +1 or -1 along this axis, but the state with spin 0 along this axis simply does not exist --a massless spin 1 particle has fewer possible spin orientaions than a massive one. Again I cannot explain to you by any classical physics analogy why this is true, but this is what the quantum theory says and what we observe to be true. So once again it has only two possible spin states, and you may sometimes see these called spin up and spin down, although that language is usually reserved for the spin 1/2 case.
When we chose the direction of travel of the particle as the axis along which we label the spin states we also have another term --helicity -- for those states, and we talk of the photons as having helicity plus and helicity -1. The plus orientation is right handed (ie a rotation in the direction you can make by putting right thumb along the direction of travel and curling your fingers, and the helicity minus states have the opposite sense to their rotation --they are left handed.
Another thing which may puzzle you in this story (it is indeed quite remarkable) is that we say the electron only has two spin state, but what if I choose a different axis --can't I make spin=+1/2 and -1/2 states along any axis? The answer is yes, but they are not independent (distinct) states from the spin =1/2 and -1/2 along the first axis you chose. Once I pick any axis the state spin =1/2 along any other axis can be written in terms of some admixture of the two states =1/2 and -1/2 along my first axis. Surprising, puzzling and utterly unexplainable with classical physics analogies!
This may answer your question, but I'll bet it still leaves you a little puzzled. Welcome to the world of quantum physics!
John Townsend
Intrinsic spin of a particle, which is angular momentum, has no classical analog. The electron, even though it appears to be a point particle, has intrinsic spin, for example. And, as you note, photons do also. If you want to stop by, I am happy to chat about this. Two of our seniors, Jeremy Liu and Seth Foreman, are also intrigued by photon spin. They are trying to do an experiment in which photon spin is transferred to a disk, which then rotates, thus converting angular momentum that is indeed tough to get your brain around into angular momentum with which you are familiar. You may want to talk with them as well.
Bill Tanksley
The hardest part of this request is that "spin" isn't really like anything in the classical-mechanics world. It's an intrinsic property of the particle; changing the property is actually changing the particle, not merely changing the particle's motion.
The usual explanation is a metaphor. Picture a photon as a tiny globe, rotating about its axis as it speeds along its way (metaphor note: the photon doesn't actually rotate! It does have an axis and a spin, but those are properties of the electron, not movements). Unlike a normal globe, a photon's axis MUST be at 90 degrees to its motion, and like a real globe, once the photon has a axis and a direction the axis can't be changed without also changing the direction (it's like a gyroscope).
The direction of this axis is the direction of the photon's "polarization". There are an infinite number of possible polarizations (just point the electron's axis in a different direction); polarized light is light in which all the photons have axises pointing in the same two directions (up and down).
Piers Coleman
Think of it classically. A photon is a particle of light. Light can be divided into two polarizations- you can do this classically in two ways, equivalent- either two different plane polarizations (separated out by a polarizing crystal), or in two different circular polarizations (right or left handed)- filtered out using a chiral crystal. Well- photons are similar, they can be regarded as either left handed or right-handed, Those which are RH, have a spin in the direction of motion, those LH, spin in the opposite direction of motion. It turns out that when you take a spinning particle to the speed of light, its spin either points forwards or backwards....
Professor Anthony E. Siegman
A photon is, rigorously speaking, a quantum state of a single, isolated (or nearly isolated) physical system (e.g., the electromagnetic field distribution in a single resonant cavity mode) whose quantum mechanical properties are described (fully) by a single simple harmonic oscillator hamiltonian.
Entangled states can occur only, however, in physical systems that have at least twice as many degrees of freedom as a SHO. For example, two coupled SHOs, or an atom with multiple degrees of freedom.
Quantum systems can be analyzed rigorously only by treating the entire system as a whole. The term "photon" is neither clear nor rigorously defined with respect to an entangled system.
When people talk about "entangled photons", there really aren't two clearly defined separate or independent photons, there's only the (more complex) quantum state of a (more complex) combined quantum system. Applying ideas or concepts derived from simple rigorously defined photon systems to this more complex system can get you in trouble.
