Tuesday, March 8, 2022

Where is the angular momentum "stored" in an electromagnetic wave?

Let us consider a rotating electric dipole.


                       ^
                       |      motor
                         + === O === -
                                                 |
                                                 v


A motor makes the dipole to rotate. The electromagnetic wave carries away angular momentum from the motor.

If we put an absorbing dipole close to the emitting one, the absorbing dipole starts to rotate and we can harvest energy from it. The absorbing dipole harvests angular momentum, too.

What kind of a model might describe the  angular momentum and its absorption? In a wave, parts of the wave are coupled together through the wave equation. We probably cannot describe the system with particles flying independently of each other.


                        __________
                  _ /                      \_


A jumping rope of children does carry angular momentum from one end to the other. Could this describe the coupling of the two rotating dipoles? The rope carries energy, angular momentum, and also linear momentum which tries to push the receiving end away. Qualitatively, the rope looks like a good model.

We believe that the Poynting vector of the electromagnetic wave describes the angular momentum. However, does it tell us in an intuitive way how the angular momentum is transmitted?


The angular momentum of a photon


The absorbing dipole receives angular momentum in units of the reduced Planck constant ħ. Do we have an intuitive model for this?

Classically, angular momentum is a continuous quantity. The receiving dipole would accelerate gradually. There would not be a discrete unit ħ. At the same time, the emitting dipole would lose angular momentum gradually.

What if we do not have receiving dipole? Then we interpret that the angular momentum lost by the emitting dipole went to the electromagnetic wave.

In quantum mechanics, a measurement makes the wave function of the receiving dipole to collapse. The probability of measuring an additional unit ħ in its angular momentum grows with time.

We interpret that the receiving dipole "absorbed a photon" if we see that its angular momentum grew.

The classical limit states that the if very many photons are emitted and absorbed, and if the dipoles are macroscopic, then the classical model must correctly describe the process. Consequently, we can calculate in classical physics, and only at the end map the result to the language of quantum mechanics.


Conclusions


Where is the angular momentum of a classical electromagnetic wave stored? We believe that the Poynting vector describes the phenomenon correctly. The angular momentum is distributed smoothly in the wave.

The jumping rope of children might be a qualitative classical model.

Where is the angular momentum of a single photon stored? It is distributed in the wave. In a wave function collapse, that angular momentum is found in a receiving dipole. There is no need to describe the storage of angular momentum in a more detailed way. We do not need to talk about the photon as a particle, or claim that the photon somehow "rotates" or "orbits in a circular path". The "photon" is essentially the wave function collapse.

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