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I am currently trying to learn some basic quantum mechanics and I am a bit confused. Wikipedia defines a photon as a quantum of light, which it further explains as some kind of a wave-packet.

What exactly is a quantum of light?

More precisely, is a quantum of light meant to be just a certain number of wavelengths of light (something like "1 quantum = a single period of a sine wave" perhaps?), or is the concept completely unrelated to wavelengths? In other words, how much is a single quantum?

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No. It's intimately tied in with the concept of uncertainty, and how that's represented in waves. Check out this question: – Mike Dunlavey Dec 21 '11 at 1:53
Also, an excellent place to start wrapping your head around this is to study the double-slit experiment. – Mike Dunlavey Dec 21 '11 at 2:16
Thanks for the suggestions. The double-slit video is quite interesting. – Dejan Govc Dec 21 '11 at 2:40
Never feel bad about feeling confused about photons. Willis Lamb (of Lamb shift fame) said that licenses should be awarded to qualify people to use the word "photon". The paper was called anti-photon. Although I should add that, in contrast to his abstract, I do, in fact, believe in photons ! – twistor59 Dec 21 '11 at 7:43
@Dejan: ok, but the accepted answer gives a description of a photon as a wave-packet of E and B fields, which is grossly incorrect, and will mislead others. A single photon is purely quantum mechanical, and is described by a quantum superposition of E and B fields which has no definite classical analog in any circumstance. This is different from, say, a single pion, where the wavefunction can be formally described by a solution to the classical pion-field equation of motion, because the pion can be nonrelativistic. – Ron Maimon Dec 29 '11 at 16:31
up vote 10 down vote accepted

There are two meanings usually attached to the word "quantum" in quantum theory, one colloquial and one technical.

As you know, electromagnetic radiation behaves in ways characteristic of both waves and particles. For non-specialists, it's easy to think of a particle as being a "unit" of the wave, and since "quantum" means a unit of something, the word has gotten associated with "particle." But in reality, the idea of a particle isn't precisely defined. When people talk about a particle of light, the EM field associated with what they probably mean could be described as a wave packet, which you can think of as an electromagnetic wave that is localized to some small region in space. For example, something like this:

wave packet

This is just an example, of course; wave packets can have all sorts of shapes.

The more precise, technical meaning of "quantum" has to do with Fourier decomposition. As you may know, any function can be decomposed into a sum of sine waves (or complex exponentials),

$$f(x) \propto \int e^{ikx}\tilde f(k)\mathrm{d}k$$

For any given momentum $k$, the amplitude $\tilde f(k)$ represents the contribution of the sine wave with that frequency to the overall wave. Now, classically the value of $[\tilde f(k)]^2$ at each $k$ represents a bona fide contribution to the energy of the light. But the assumption that makes quantum theory quantum is that $[\tilde f(k)]^2$ instead represents the probability that there is a contribution to the energy of the light coming from that frequency. The actual contribution that can come from any given frequency can only be one of a set of specific values, which are integer multiples of some unit $\hbar c/k$. "Quantum" is the word for that unit of energy.

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A quantum of light is a particle of light which can disappear, giving its energy to an atomic or particle system, or appear, taking energy away from a particle or atomic system. A quantum of light of wavelength $\lambda$ is the minimum amount of energy which can be stored in an electromagnetic wave at that wavelength, which is Planck's constant h times the frequency. The photon is not related to the wave in any concrete way, the classical wave is a superposition of a large number of photons which are coherent.

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....not necesarily a large number of photons, but definitely an indeterminate number of photons, since field amplitude does not conmute with energy and/or mode occupation number – lurscher Dec 21 '11 at 3:54
@lurscher: No, a large number is the more precise statement. An indeterminate small number doesn't work to produce a definite field quantity, while a large definite number of photons can still produce a field whose local phase fluctuations are tiny, meaning that if you measure the phase at one point, the phase at a distant point collapses to a consistent wave. – Ron Maimon Dec 21 '11 at 9:39
@lurscher: What does $[\hat{n},\hat{a}]$ have to do with Ron Miamon's reply? I'm not sure I understand your statement. – Antillar Maximus Dec 22 '11 at 17:48
@Antillar: the point is, when does a photon have a field description? He is saying that the limit needs not just a large number, but an indeterminate number of photons, just like the limit of "definite position" in a Harmonic Oscillator needs a "large indeterminate energy level". This is technically true, but I think it is better to just say "large number", because the relative phase can still be ok after a measurement, like after a position measurement of a large N HO, the particle oscillates. It's a minor issue, and the main point is unaltered. – Ron Maimon Dec 22 '11 at 18:58

Just a remark that might be helpful to understand what photon is: the "wavelengths of light" seems to be just a theoretical value calculated with the help of Planck model. What can really be measured in the experiment is the momentum/energy of photon, not the wavelength. For instance, the "colour" of the photon is fully determined by its momentum.

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