represented by a pure momentum state, which implies that the photon is uniformly spread over space
There are representations and then there are other representations. If the photon is spread over space, what is spread? Momentum? Or what we must say is the spread is composed of spacetime points. It begs the question. If that which is spread over space has angular momentum which is not spin,intrinsic angular momentum, that which is spread must be undergoing angular momentum. It is E x B that rotates. The spreading or smearing is a relativistic effect, but the photonic entity's Lagrangian is quantized to Planck's constant. To claim the photon is not particulate means the action integral is infinite. QED is accurate because the point-like idealization of the photon is effective. This is an old controversy. We either try to preserve the semi-classical picture or the quantum picture. In transmission photonics you stay with semi-classical, because the quantum theory DeBroglie Wave momentum 1-form picture is not as helpful.
Then there is also a position space where the position is initially localized to a point and the momentum is spread out over momentum space.
As above what is initially localized to a point? It looks like position space and momentum space are disjoint. Nature isn't. We use this way of representation since it is consistent with experiments and relations of reality done in the macro world.
A pure position state evolves immediately into a non-local position wave function.
Position, spacetime point, is not a state. States are pure numbers and are the parameters that govern transition from one spacetime point to another. You are mixing classical and quantum representations.
In either representation one can construct a wave packet.
The wave packet is the attempt to represent both representations in a unified picture which humorously attempts to retain the macro classical picture in an ad hoc way. You don't need a Fourier Transform; it is merely a mathematical convenience. The uncertainty relation falls out of the wave packet picture and is governed by the unit of quantum action, Planck's constant.
Arguing that spin requires spatial extent seems an incorrect understanding of spin.
There is the quantum number, spin, n*h/2*Pi, which you have mentioned, and then there is angular momentum of the tangible energy field. When we talk about the nucleus its spin is the total angular momentum. Please give me your definition of spin.
A particle's spin determines its angular momentum and is not associated with physical extension of the quantum particle. An electron with a spin 1/2...
You mean intrinsic angular momentum. There is historical confusion caused by the evolution of these concepts and also due to the translation of classical picture into quantum picture. Uhlenbeck and Goudsmidt innocently started the confusion trying to get away from 2 eigenfunction models.
The quantum number or operator, intrinsic angular momentum, is caused by an unknown mechanism. It may yet drop out of string theory or M + N theory as a geometrodynamical entity. What I'm talking about is k*E x B per unit volume. It is areal and it rotates. |
