Is the Wave Function of a Photon the Same as a Classical Wave?

[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) ?

 

[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.

 

[bhobba]

hello, is the wave function of a photon is as same as the classical wave exhibited by a group of photons ?

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

 

[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.

 

[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 ?

 

[Demystifier]

One possible definition of the photon wave function can be found in
http://lanl.arxiv.org/abs/1205.1992
Sec. 8.3.3.5.

 

[bhobba]

So how do we explain the diffraction pattern formed by emitting the photons one by one in the two-slits experiment ?

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

 

[athosanian]

But be aware, without going into the details it's wrong.

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

 

[vanhees71]

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

 

[athosanian]

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

Thanks a lot ! I will search it.

 

[Demystifier]

Thanks a lot ! I will search it.

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

 

[vanhees71]

A very nice article has been written by Zeilinger in Nature:

http://www.nature.com/nature/journal/v433/n7023/abs/nature03280.html

 

[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 !

 

[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.

 

[athosanian]

What do you mean by "the experiments proves the random behavior not just holds for ensembles".

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.

 

[FunkyDwarf]

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

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.

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?

 

[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).

 

[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)

 

[bhobba]

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)

We have ways and means of stopping people asking certain questions
http://arnold-neumaier.at/physfaq/topics/position.html



Thanks
Bill

 

[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 Schrödinger 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...

 

[bhobba]

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

The summery is the math which I don't entirely understand either - my knowledge of QFT isn't quite as good as I would like it. Bottom line - in QFT the position operator, hence defining a position, is problematical for massless particles. Without a position operator defining a wave-function, which is the expansion in terms of position eigenstates is impossible. Hence you can't speak of the wave-function of a photon.

Also, to clarify, when most people say photon, do they mean a wave packet or a plane wave?

Most people don't know QFT, so when most people speak about a photon what they speak about is wrong. Its neither a wave or wave-packet. Its an excitation of a quantum field.

Thanks
Bill

 

[FunkyDwarf]

Well you summary seems a simple restatement of the result, which I already grok =P

Additionally, my understanding is it is not systemic to QFT specifically. For instance, one can use the field operator to 'create' a particle 'at position x', why does this not make sense for a photon? Sure, you'd need an infinite number of wavemodes to do it, but the same is true for an electron, is it not? Also, if i have photons in a cavity, they are said to be localized surely? I'm being facetious, clearly, but you see my point.

 

[bhobba]

What do you already crock? The link I gave that showed for massless particles position is not an operator? Because if it isn't you can't have a wave-function which, by definition, is the expansion of the state in the eigenstate of position.

What I am getting the feeling here is this is going around in circles. I point to an advanced article a bit beyond my current understanding of QFT that explains position is not an operator for massless particles like a photon, you ask for a précis of it, I point out its technically a bit above where I am currently at with QFT, and you still want to argue about it.

If position is not an operator you can't have a wave-function in the usual sense.

Thanks
Bill

 

[yoron]

Bhobba, isn't it also a question of scales? A wave is one type of scale description, a 'photon' is its opposite, or limit. Maybe this could be a question? Scale wise, how 'small' can one define something to being a wave? Excitations in a field is to me a 'whole description' from what I naively call a 'container definition' of a 'wave universe', whereas 'a photon' is a discrete entity of momentum and energy. Although, if one can prove a photon to be a 'wave packet' experimentally then it becomes muddy.

 

[bhobba]

Bhobba, isn't it also a question of scales?.

There is no wave or particle - there is only a wave-function, or even more fundamentally, a state.

Thanks
Bill