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In water waves, the wave is transmitted through vertically moving water particles which face no displacement nonetheless the wave is moving. So does the same happen with photons in electromagnetic waves or does photons move along the wave?

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Really, photons are the wave. What makes a wave in the classical sense is a large number of photons all averaging together.

Your question is an obvious guess to make -- other wave phenomenon is a result of local interactions within some medium, so electromagnetic waves must be the same. For a while, people guessed that there was a medium that carried electromagnetic waves, and they called it the aether. Turns out, it doesn't exist.

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  • $\begingroup$ Like your answer. Little expansion: Photons are oscillating electric and magnetic dipoles propagating with c. A big number of photons produced synchronously like in a radio antenna are detectable as an EM wave. $\endgroup$ – HolgerFiedler Feb 19 '17 at 7:02
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For waves like string or water, there are a bunch of particles that are stuck together, and because they pull on each other, we see wavelike effects.

Electromagnetic waves are typically said to be "made up of photons," but this is not exactly true.

Quantum mechanically we can represent a classical electromagnetic wave as: $E= cos(\omega t)$ as this thing called a "coherent state." Such a state is actually a quantum mechanical superposition of being one photon, two photons, three photons...all the way up to infinite photons.

probability of a coherent state being in a particular amount of photons as a function of $\alpha$: $$e^{-\alpha^2/2}\frac{\alpha^n}{\sqrt{n}}$$ where $\alpha$ is the average expected amount of photons for that state.

Often what you see as light (particularly coherent light like laser light) is a quantum-mechanical superposition of photons. So its the sum of a bunch of different possibilities that are creating what you see. Additionally, these photons aren't "talking" to each other in the same way that molecules on a rope are tugging on each other. While it's a collection of them that form a classical electric field, each photon possibility isn't tugging on other possibilities.

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