1. Jan 12, 2015

athosanian

hello, is the wave function of a photon is as same as the classical wave exhibited by a group of photons ?
If no, what is the relationship between the wave function associated with a single photon and the classical wave which describes the behavior (such as diffraction) of light (a group of photons) ?

2. Jan 12, 2015

Jazzdude

No, the classical electromagnetic wave is not related like that to photons. First of all, the photon does not have a wavefunction in the sense you are suggesting. There is no way to expand the state of a photon in the position basis to get a meaningful state description. The relationship between the quantum field of which photons are the excitations and the classical EM field is given by field operators and their expectation values. In order to understand what this means you will have to study quantum field theory or at least the quantisation of the electromagnetic field.

It's also not very clear what a single photon is even supposed to be. There are different definitions of what a photon is, but you will mostly get two answers depending on whether you ask an experimentalist or a theorist. 1) A single photon is what makes the detector click exactly once and is absorbed at the same time. 2) A single photon is the state of the EM quantum field with particle number 1. Multiple photon states are defined accordingly.

These two definitions are not identical and it's not obvious if they're even compatible. If you are pedantic about the field state in 2) being an eigenstate of the particle number operator they're surely incompatible. More relaxed interpretations can bring the concepts closer together, but it's still a very difficult concept.

So asking about the field of a single photon is both difficult to answer and not even that well defined. It's however certainly not the classical EM field.

3. Jan 12, 2015

bhobba

It's a difficult issue.

It has been discussed on this forum before and generates a lot of robust discussion.

At the risk of simply regurgitating that you may find the following helpful:
http://arxiv.org/ftp/quant-ph/papers/0604/0604169.pdf

Thanks
Bill

Last edited: Jan 13, 2015
4. Jan 12, 2015

blue_leaf77

There are a couple of representations for describing the state of a photon. Two of them are number state (or Fock state) which Jazzdude has given a brief intro about, and coherent state. Number state is the eigenstate of photon Hamiltonian (i.e. photon energy) which happens to be the same as harmonic oscillator hamiltonian.
While coherent state is defined to be linear superposition of many number states. And if you look for the expectation value electric field in coherent states, you will find that the expectation value forms a sinusoidal function w.r.t phase. If i'm not mistaken this is why some people regards coherent state to be the closest quantum representation of light the its classical counterpart. Simply because in classical EM wave, the E field is sinusoidal in time.

5. Jan 13, 2015

athosanian

Thanks a lot ! It seems there is no ordinary wave function associated with a single photon to describe its position. So how do we explain the diffraction pattern formed by emitting the photons one by one in the two-slits experiment ?

6. Jan 13, 2015

Demystifier

7. Jan 13, 2015

bhobba

Well here is one way:
http://arxiv.org/ftp/quant-ph/papers/0703/0703126.pdf

The idea is the slits change the momentum of the photon.

But be aware, without going into the details it's wrong. Unfortunately physics is sometimes like that - what you learn at a less advanced level needs to be corrected later.

Thanks
Bill

8. Jan 13, 2015

athosanian

Is there any universal accepted explainations for the diffraction pattern formed by the one-by-one photon emission in two slits experiment ?

9. Jan 17, 2015

vanhees71

Yes, it's called quantum electrodynamics and one of the most accurate mathematical models about nature every discoved :-).

10. Jan 19, 2015

athosanian

Thanks a lot ! I will search it.

11. Jan 19, 2015

Demystifier

Or even better, search about quantum optics, which in this case is the relevant branch of quantum electrodynamics.

12. Jan 19, 2015

vanhees71

13. Jan 19, 2015

athosanian

Demystifier and vanhees71, Thank you very much, I have download the paper by Zeilinger in Nature. The author mentioned the single photon two-slits experiemnts in his article. But he just emphasiezed that the experiments proves the random behavior not just holds for ensembles, but also for individual particle, following Feynman. However, the author did not further explain how the interference pattern in the two-slits experiments is formed (the photon does not have the ordinary wave function as electrons and can not be explained by interference of the wave function). I will search more information in the quantum field theory, quantum electrodynamics and quantum optics. Thanks a lot !

14. Jan 20, 2015

vanhees71

What do you mean by "the experiments proves the random behavior not just holds for ensembles". Of course, each single event is random, i.e., you cannot predict where a individual photon hits the photodetector. The interference pattern is built up by many photons, all equally prepared and send through the double slit.

The physics is precisely what Feynman describes for massive particles in the beginning of Vol. III of the Feynman Lectures. The advantage to start quantum mechanics with non-relativistic massive particles is that you can describe single particles with a wave function, which is not strictly possible for relativistic massive particles and impossible for massless particles. For photons you need quantum field theory.

15. Jan 23, 2015

athosanian

I just cited the words from the authoer Zeilinger. In the history it seems some scientists do not accepted the statistical behavior of a single particle and just thought it was the behavior exhibited by masive particles.

16. May 9, 2015

FunkyDwarf

Sorry for thread necro, but one quick question regarding the comment:

I'm curious as to the distinction between equations for fields and for wavefunctions (or equivalently, solving for a field or a wavefunction). For instance, the Weyl equation is for massless spin 1/2 particles, so if you solve that in free space or in some potential, do you not get a wavefunction? Or is the distinction that it is simply artificial to consider the wavefunction in this way, as you are really considering only one mode of an oscillator in isolation?

17. May 9, 2015

vanhees71

You can solve all kinds of equations, which is a nice mathematical exercise. The only problem is, if you want to make sense of the equations and their solutions as physics. In relativistic quantum theory of interacting particles you cannot make sense of the "first-quantization approach", i.e., a single-particle description doesn't make sense, because you can always create and destroy particles (as long as the conservation laws allow it). Only in the non-relativistic limit, such a single-particle description makes sense, and there is no sensible non-relativistic limit for massless particles (or quanta for that matter).

18. May 9, 2015

FunkyDwarf

Right, but in principle the Dirac equation can be used to calculate properties of electrons in various fields, despite that it is a full relativistic treatment and formally particle number may not be conserved (Dirac sea etc). My question is more why specifically the fact that a particle is massless is important (obviously, no rest frame exists, but I do not see a direct line from that to this issue, but i'm probably missing something stupid)

19. May 9, 2015

bhobba

We have ways and means of stopping people asking certain questions
http://www.mat.univie.ac.at/~neum/physfaq/topics/position.html

Thanks
Bill

20. May 9, 2015

FunkyDwarf

I don't suppose you can summarize for me? The typesetting on that page makes my eyes bleed and baby jeebus cry =P

Also, to clarify, when most people say photon, do they mean a wave packet or a plane wave? Clearly for the latter, as for any particle even in the Schrodinger case, it is non integrable and so it makes no sense to talk about position. Or is this perhaps the issue: photons are only ever plane waves and only interact by being absorbed (a photon in a 'potential' will not get trapped, except maybe in GR and then it's only some weird effective potential), thus this wave packet idea doesn't apply? I realize I am being sloppy with my terminology, but...

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