# Are the photons in a standing wave moving at $c$?

So, in a standing wave, two superimposed waves produce a wave that remains stationary, with its particles resonating back and forth, right? For instance, in a water wave, two waves moving in opposite directions meet in the middle, and the water molecules only move up and down; the waves are no longer moving longitudinally anymore.

However, photons always move at $$c$$, right? If you have two light waves that meet each other to produce a standing wave, what happens to the photons? If the wave's fluctuations stop moving along the path of the beam and start just moving "up and down", wouldn't that imply that the photons carrying the light wave have stopped and are just moving "up and down" in place, at a speed determined by the amplitude and frequency of the standing wave? Are the amplitude and frequency of light waves always related to each other so that the speed of photons of a standing wave remains $$c$$?

The photons always move at $$c$$.

To relate photons to a classical picture of light, one good way is to proceed as follows.

1. Express the light field you are interested in as a sum of plane waves: $$E({\bf r},t) = \sum_i E_i \cos({\bf r}\cdot{\bf k}_i - \omega_i t + \phi_i)$$ where $$E_i$$ are electric field amplitude, $${\bf k}_i$$ are wave vectors (they point in the direction of travel of the wavefronts of the given plane wave), $$\omega_i$$ are angular frequencies and $$\phi_i$$ are phase offsets. These plane waves are still classical.

2. Treat each classical plane wave as a flow of photons. The photons in each given classical plane wave have energy $$\hbar \omega_i$$ and momentum $$\hbar {\bf k}_i$$, and their density is sufficient to give the energy density of the wave you are treating. Their arrival times are random in the sense that the number crossing a plane in some given time interval is described by a Poissonian probability distribution.

In the case of a standing wave, the conclusion is that there are photons travelling in both directions.

(I guess for completeness one should add that you can also, if you like, use the term 'one photon' for a case where there is a total occupation number 1 for a superposition of various plane waves all with the same frequency. In this case that photon is itself in a superposition of travelling in various directions. Its motion in each direction is still at the speed of light.)

• This applies to mechanical standing waves as well (aside from the "photon" part). The waves are always moving; just the sum at a node is always zero. Aug 17, 2021 at 15:03