Understanding Recoil in Photon Emission and the Role of Quantum Electrodynamics

[Danyon]

This is the second time I've asked this question, I thought I'd add some extra details. Consider a single accelerating electron, this electron emits a single photon wave which radiates out spherically in a superposition, What direction and what time does the electron recoil if there is no defined direction for the photon before it's wavefunction collapses? also, what would happen if say a laser is pointed at an empty region of space, such that the photons wave function never collapses, does the laser recoil?

 

[jtbell]

Consider a single accelerating electron, this electron emits a single photon wave which radiates out spherically in a superposition, What direction and what time does the electron recoil if there is no defined direction for the photon before it's wavefunction collapses?

So long as the photon does not have a well-defined momentum (including direction), neither does the recoiling electron have a well-defined momentum/direction. They are entangled. When/if you measure the photon momentum/direction sufficiently precisely, then you also know the recoiling electon's momentum/direction, and vice versa.

This assumes that the electron's initial momentum/direction is known sufficiently precisely.

 

[Nugatory]

What direction and what time does the electron recoil if there is no defined direction for the photon before its wavefunction collapses?

When you say "its wavefunction" you're thinking of the situation as if the photon has its own wave function separate from that of the electron. That's not how quantum mechanics works; instead we have a single wavefunction for the entire system consisting of an electron and a photon (warning - this is a huge oversimplification that you'll have to unlearn when it comes time for QFT, but it works for now). Until this wavefunction collapses neither the electron recoil nor the photon momentum are defined; the collapse assigns definite values to both.

also, what would happen if say a laser is pointed at an empty region of space, such that the photon's wave function never collapses, does the laser recoil?

Again, we have one wave function that covers the laser and the photon. Measuring the recoil of the laser collapses the whole thing.

 

[vanhees71]

It's in fact most easily (and only correctly!) answered in terms of QED. The most simple setup is the scattering of a charged particle (electron) on an external electromagnetic field (e.g., a heavy atomic nucleus which you can approximately treat a classical static field). Then there is certain probability for the electron just being scattered elastically, emitting one photon, emitting two photons, and so on. The leading-order bremsstrahlung diagram is the one where one photon is emitted on the electron line, and energy and momentum are always conserved (when the recoil to the heavy nucleus is considered of course) conserved. Note however, that you have to take into account the infrared problems of the bremsstrahlung. You have to add also the one-loop correction to the vertex for elastic scattering, which is at the same order as the tree-level single-photon bremsstrahlung diagram (an example for the Block-Nordsieck theorem).

The other questions cannot be answered from first principles. Concerning the time at which the electron recoils, this is a meaningless idea since you cannot even unambigously define what an electron might be in the transient state of interaction. Only a delay time due to scattering can be defined by the energy derivative of the corresponding scattering phase shifts.

Also, I don't understand what you mean by "collapse of the wave function" for a photon. It's even questionable, whether there is a collapse of a wave function in a physical sense at all, where the very concept of wave function is applicable. For photons there is no such concept of a wave function.

Last but not least a Laser doesn't emit photons but (a very good approximation to) coherent states of the em. field.