# What is a Photon

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Mentor
Hi Guys and Gal's

In answering a question in general physics I came across the following which explains at a reasonably basic level what a photon is, spontaneous emission etc at the level of basic QM with a bit of math:
http://www.physics.usu.edu/torre/3700_Spring_2015/What_is_a_photon.pdf

IMHO its much better than the usual misleading hand-wavey stuff and even if you don't follow the math would allow a general gist to be had.

Thanks
Bill

OmneBonum, fluidistic, exponent137 and 4 others

For those who want to really know, including all the fine points that the question includes, a collection of relevant articles on the topic can be found here:

The Nature of Light: What Is a Photon?
Optics and Photonics News, October 2003
http://www.osa-opn.org/Content/ViewFile.aspx?Id=3185

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kith and vanhees71
Gold Member
Isn't true that a photon doesn't exist until we make a measurement?

Homework Helper
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Isn't true that a photon doesn't exist until we make a measurement?

I think it exists but it exists in a superposition of states, that is a state that is beyond our classical understanding.

Gold Member
I think it exists but it exists in a superposition of states, that is a state that is beyond our classical understanding.

Yes, that is my understanding too. But can we distinguish between a photon field and an electron field before a measurement is done? IOW, if all of these fields are in a superposition, how can we tell that an electron field and a photon field are not the same field manifesting as a particular object according to what we wish to measure?

how can we tell that an electron field and a photon field are not the same field
The strength of the electromagnetic photon field and the current density of the electron field are state dependent. But the fields exist independent of the particular state, and are know to be distinct because of the way they appear in QED. For example they differ in spin, and hence in the form the basic observable field values take (field strength resp. current density).

The e/m field excitations manifest themselves as observable photons only in the moments they are detected by a counter or screen.

ComplexVar89 and tionis
Gold Member
The strength of the electromagnetic photon field and the current density of the electron field are state dependent. But the fields exist independent of the particular state, and are know to be distinct because of the way they appear in QED. For example they differ in spin, and hence in the form the basic observable field values take (field strength resp. current density).

The e/m field excitations manifest themselves as observable photons only in the moments they are detected by a counter or screen.

Thanks, A. Neumaier. A few questions, if I may: what is the difference between a wavefunction and a field? Could I say that in the universe there is only one wavefunction that manifests as different fields, or do different fields have their own wavefunction, and if so, what is/are the difference between them other than exhibiting different values at the moment of measurement? Also, you said that the fields are ''known to be distinct because of the way they appear in QED,'' but is this a prediction of the theory before measurement?

Thanks, A. Neumaier. A few questions, if I may: what is the difference between a wavefunction and a field?
It is precisely the same difference in quantum field theory as between a wave function and position in case of single particle quantum mechanics. The wave function defines in both cases a pure state, whereas the components of position resp. the fields averaged over a region of observation define the primary observables.

do different fields have their own wavefunction
No. Different fields are like different particles in a molecule. They define different observables but there is only one wave function for the molecule, not one for every particle.

Gold Member
It is precisely the same difference in quantum field theory as between a wave function and position in case of single particle quantum mechanics. The wave function defines in both cases a pure state, whereas the components of position resp. the fields averaged over a region of observation define the primary observables.

Huh? I'm sorry I don't understand what you said. Are you saying the wavefunction is a pure state i.e, an undefined probabilistic state of the fields that doesn't have any properties until it is measured?

No. Different fields are like different particles in a molecule. They define different observables but there is only one wave function for the molecule, not one for every particle.

OK, so there is only one wavefunction for all the fields out there?

Huh? I'm sorry I don't understand what you said. Are you saying the wavefunction is a pure state i.e, an undefined probabilistic state of the fields that doesn't have any properties until it is measured?
A wavefunction describes a pure state, which is well-defined once the wave function is given. It determines the measurable field expectations. If the state is known, these field expectations are known, too, and can be checked by measurement.
OK, so there is only one wavefunction for all the fields out there?
Yes, if the state is pure. Otherwise there is only a density operator. We can never know enough about the state of a macroscopic system to decide which is the case, and effectively use always density operators. Pure states are useful only for very small systems.

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tionis
Gold Member
A wavefunction describes a pure state, which is well-defined once the wave function is given. it determines the measurabler field expectations.

Yes, if the state is pure. Otherwise there is only a density operator. We can never know enough about the state of a macroscopic system to decide which is the case, and effectively use always density operators. Pure states are useful only for very small systems.

Thanks, Neumaier. I don't want to drag you into a never-ending series of questions and it's ok if you don't wish to answer anymore, but can you please break it down for me? In the hierarchy of all things quantum, what entity rules supreme? Is it the wavefunction, then the fields, then the particles/observables? I'm not still clear on that. Like for example, when physicists such as yourself search for a TOE, what exactly are you looking for? Is it an explanation of the physicality of the wavefunction, or whatever the wavefunction manifests into, meaning the observables?

Mentor
In the hierarchy of all things quantum, what entity rules supreme?

The field is a field of operators. It operates on something more general than the usual space in QM, called a Fock space:
https://en.wikipedia.org/wiki/Fock_space

Thanks
Bill

tionis
In the hierarchy of all things quantum, what entity rules supreme?
The observables (i..e, the field operators) and the relations between them (commutation rules, field equations, equations of motion, operator product expansions) are the primary thing - without these one has no theoretical framework to speak about anything. They are independent of any state and therefore have a universal form. Having a TOE means knowing just this. From it one can determine (in principle) the particle content and the possible decays and reactions between particles, and scattering cross sections involving probabilities, which are the next important thing.

The expectations and correlation functions are the next important thing. They depend on the state, which is in general something very complex to which only coarse approximations can be determined experimentally. These determine what we actually find where in the world. The general laws for them (derivable without knowing more than the existence of a state) tell how macroscopic objects flow and deform.

vortextor, tionis and bhobba
Gold Member
Thank you both. Until today, I thought the wavefunction was the most important and fundamental object in quantum mechanics, but how could it possibly be if all it is is an infinite number of probabilities that do not manifest until it's is acted upon by the field operators? At least that's what I gathered from your replies. If that is somewhat correct, then I have no further questions.

Mentor
Thank you both. Until today, I thought the wavefunction was the most important and fundamental object in quantum mechanics,

Have a look at Gleason's Theroem:
http://www.kiko.fysik.su.se/en/thesis/helena-master.pdf

That would tend to suggest operators are the fundamental thing.

Thanks
Bill

tionis
Gold Member
Have a look at Gleason's Theroem:
http://www.kiko.fysik.su.se/en/thesis/helena-master.pdf

That would tend to suggest operators are the fundamental thing.

Thanks
Bill

Too advanced for me, but the consensus seems to be that operators are the fundamental objects, so yeah. Thanks.

Gold Member
Great article. No better way to introduce photons then talking about harmonic oscillators. A simple harmonic oscillator is anything with a linear restoring potential. Simple things like a spring, or a string with tension, or a wave. Erwin Schrödinger described mathematically how a harmonic oscillator stores energy and how to calculate how much energy it stores.

For the photon, the value of this restoring potential is the Planck constant (h). Planck’s constant, relates the amount of energy stored in a photon to its wavelength (λ). Planck’s constant tells you the amount of time it takes the photon to undergo one cycle of whatever its doing given that the photon has a specific amount of energy. The equation E for energy = h / λ, tells us that a photon with low energy will take much longer to complete one cycle of the wave then a photon with high energy.

View attachment 194839

My favorite model of a photon as a harmonic oscillator is as an expanding and contracting ball of energy flying though the air. It immediately lends understanding to the particle and wave nature of the photon. A photon of a specific wavelength has a specific energy. The properties of a photon change periodically over time and distance depending on the wavelength. High energy photons oscillate very fast and store a lot of energy, low energy photons oscillate very slowly. Photons can be in the same place at the same time, sometimes reinforcing each other, sometimes cancelling each other out and it looks like there are no photons at all. The uncertainty principle: ΔxΔp ≥ h/4π falls from this. The photon is either big affecting a wide area, or it’s tiny and only affecting one small area, it cannot be both at the same time. The importance of visualizing the photon as a harmonic oscillator cannot be overstated.

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Fooality
Jilang
What determines if it big or tiny? The energy?

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This is also not the entirely correct way to view photons, I'm sorry to say. First of all as massless quanta with spin ##1## there's no position operator. So the naive uncertainty relation, valid for massive particles, doesn't make sense to begin with. For a review, see

http://www.mat.univie.ac.at/~neum/physfaq/topics/position.html

The most easy way to introduce photons in a correct way is to look at the free classical electromagnetic field and fix the gauge completely, i.e., you choose the radiation gauge for the four potential ##A^{\mu}##, which consists in the two constraints
$$A^0=0, \quad \vec{\nabla} \cdot \vec{A}=0.$$
and then quantize the remaining two physical transverse components canonically. This leads to the mode decomposition
$$\hat{\vec{A}}(x)=\sum_{\lambda= \pm 1} \int_{\mathbb{R}^3} \frac{\mathrm{d}^3 \vec{p}}{\sqrt{(2 \pi)^3 2 |\vec{p}|}} [\vec{\epsilon}(\vec{k},\lambda) \hat{a}(\vec{p},\lambda) \exp(-\mathrm{i} x \cdot p) + \text{h.c.}]_{p^0=|\vec{p}|}.$$
The total energy and momentum, i.e., the Hamiltonian and the momentum of the em. field are defined as the normal ordered expressions
$$\hat{P}^{\mu} = \sum_{\lambda= \pm 1} \int_{\mathbb{R}^3} \mathrm{d}^3 \vec{p} p^{\mu} \hat{a}^{\dagger}(\vec{p},\lambda) \hat{a}(\vec{p},\lambda), \quad p^0=|\vec{p}|.$$
the annihilation and creation operators fulfill the commutator relations
$$[\hat{a}(\vec{p},\lambda),\hat{a}^{\dagger}(\vec{p}',\lambda')]= \delta^{(3)}(\vec{p}-\vec{p}') \delta_{\lambda \lambda'}.$$
Each single mode indeed fulfills the commutator relations of the harmonic oscillator, i.e., the electromagnetic field is equivalent to an infinite number of uncoupled harmonic oscillators. Thus the Fock states, i.e., the common occupation-number eigenstates of the number operators ##\hat{N}(\vec{p},\lambda)=\hat{a}^{\dagger}(\vec{p},\lambda) \hat{a}(\vec{p},\lambda)## (where only a finite number of occupation numbers is different from 0, ##N(\vec{p},\lambda)\in \{0,1,2,\ldots \}##.

Now we can unanimously define what a photon is: It's a single-photon state, i.e.,
$$|\psi \rangle=\sum_{\lambda=\pm 1} \int_{\mathbb{R}^3} \mathrm{d}^3 \vec{p} \hat{a}^{\dagger}(\vec{p},\lambda) \phi(\vec{p},\lambda) |\Omega \rangle,$$
where ##|\Omega \rangle## is the vacuum state, for which all ##N(\vec{p},\lambda)=0## (ground state of the system).

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fluidistic, tionis, Mentz114 and 2 others
Gold Member
What determines if it big or tiny? The energy?

Yes, for sure. High energy photons have a tiny wavelength and store a lot of energy, low energy photons have a very large wavelenght.

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Again, these statements are very misleading, not only but particularly for photons. One should emphasize that one cannot think about quanta, particularly massless quanta like the photon, in terms of classical fields ("waves") or particles. There is no wave-particle dualism, there is no position operator for photons and thus you cannot define in a reasonable way what a photon's position is. All these ideas are gone from modern physics for more than 90 years now!

The only way to describe the observable facts about photons is relativistic quantum field theory, i.e., in this case QED. It's the most accurate theory concerning the comparison between theory and experiment we have today. QED tells you that all you can measure are detection probabilities for photons, and this is expressed in terms of (gauge invariant) correlation functions. For a very good introduction to the quantum-optics aspects, see

M. O. Scully, M. S. Zubairy, Quantum Optics, Cambridge University Press

For the high-energy particle physics aspects I recommend

Schwartz, M. D.: Quantum field theory and the Standard Model, Cambridge University Press, 2014

fluidistic, tionis and bhobba
maline
Now we can unanimously define what a photon is: It's a single-photon state, i.e.,
|ψ⟩=∑λ=±1∫R3d3⃗p^a†(⃗p,λ)ϕ(⃗p,λ)|Ω⟩,|ψ⟩=∑λ=±1∫R3d3p→a^†(p→,λ)ϕ(p→,λ)|Ω⟩,​
|\psi \rangle=\sum_{\lambda=\pm 1} \int_{\mathbb{R}^3} \mathrm{d}^3 \vec{p} \hat{a}^{\dagger}(\vec{p},\lambda) \phi(\vec{p},\lambda) |\Omega \rangle,
where |Ω⟩|Ω⟩|\Omega \rangle is the vacuum state, for which all N(⃗p,λ)=0N(p→,λ)=0N(\vec{p},\lambda)=0 (ground state of the system).
Would you mind describing this a bit in words? I see you are integrating over momentum, so clearly this photon will not have a definite frequency. What is it that defines this a a single photon? In a state with multiple photons, how would you define the number of photons? Is it in fact always a whole number?

Gold Member
Again, these statements are very misleading, not only but particularly for photons. One should emphasize that one cannot think about quanta, particularly massless quanta like the photon, in terms of classical fields ("waves") or particles. There is no wave-particle dualism, there is no position operator for photons and thus you cannot define in a reasonable way what a photon's position is. All these ideas are gone from modern physics for more than 90 years now!

This is PhD stuff. So everyone from a master's degree down is pretty much wrong or unaware (unless they visit PF or read the books) of what the correct explanation of a photon is?

weirdoguy
This is PhD stuff.

No it's not :P In Poland you can learn it on your first year of masters.

Gold Member
What about in the US? When do you learn what's really going on? I mean, they teach you all the classical stuff then they tell you is not accurate enough or it's wrong?

Mentor
What about in the US? When do you learn what's really going on? I mean, they teach you all the classical stuff then they tell you is not accurate enough or it's wrong?

Hold on.

This is a well known issue.

You are given half truths to start with that gradually gets corrected as you learn more. It happens in physics, and expecially QM, all the time eg the wave particle duality. Feynman commented on it - he didn't like it - but couldn't see any way around it.

Its just the way things are.

The reason I posted the link is it should be accessible to people who have done a proper first course in QM. Its not the last word - its just better than the usual hand wavy stuff about what photons are.

Thanks
Bill

tionis
Gold Member

Since this is a thread about the photon, let me ask a question: are Maxwell's equations correct, or are they too discarded when you get further in your studies? Are they replaced by that equation Vanhees posted?
Now we can unanimously define what a photon is: It's a single-photon state, i.e.,
$$|\psi \rangle=\sum_{\lambda=\pm 1} \int_{\mathbb{R}^3} \mathrm{d}^3 \vec{p} \hat{a}^{\dagger}(\vec{p},\lambda) \phi(\vec{p},\lambda) |\Omega \rangle,$$
where ##|\Omega \rangle## is the vacuum state, for which all ##N(\vec{p},\lambda)=0## (ground state of the system).

Mentor
Since this is a thread about the photon, let me ask a question: are Maxwell's equations correct, or are they too discarded when you get further in your studies? Are they replaced by that equation Vanhees posted?

Of course they aren't correct - its replaced by QED. Nor is QED correct - its replaced by the elecroweak theory. And due to the Landau pole its quite possible the electroweak theory isn't correct either - but research into that seems ongoing and various opinions have been expressed about it on this forum over the years. My view is it likely isn't.

All I can see Vanhees has done, like the link I posted, is rewrite Maxwells equations in a different form that make quantization a snap.

Classically Maxwell's equations pretty much MUST be correct or our understanding of fundamental physics is way off eg SR would be wrong:

And some quite general symmetry considerations make it very unlikely SR is wrong
http://www2.physics.umd.edu/~yakovenk/teaching/Lorentz.pdf:

QFT is even more constraining - gauge symmetry more or less implies QED.

Thanks
Bill

tionis
Gold Member

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This is PhD stuff. So everyone from a master's degree down is pretty much wrong or unaware (unless they visit PF or read the books) of what the correct explanation of a photon is?
Well, I never understood why I had to learn "old-fashioned quantum theory" and then had to unlearn it (and the QM1 lecture is usually taught in the 4th-5th semester and not only in graduate school).

tionis and bhobba
Gold Member
Well, I never understood why I had to learn "old-fashioned quantum theory" and then had to unlearn it (and the QM1 lecture is usually taught in the 4th-5th semester and not only in graduate school).

I suppose the ''old-fashioned'' way eases you into the new like bhobba says, but just the thought of having to wait all those years to learn the most accurate picture is daunting, but I look forward to it. Must be nice to be you guys and have all that knowledge.

Mentor
Well, I never understood why I had to learn "old-fashioned quantum theory" and then had to unlearn it (and the QM1 lecture is usually taught in the 4th-5th semester and not only in graduate school).

I have said it before, and will say it again, teaching QM along the lines of the following is much more rational:
http://www.scottaaronson.com/democritus/lec9.html

Thanks
Bill

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Mentor
I suppose the ''old-fashioned'' way eases you into the new like bhobba says, but just the thought of having to wait all those years to learn the most accurate picture is daunting, but I look forward to it. Must be nice to be you guys and have all that knowledge.

Sometimes you need to do it the learn a half truth then unlearn it way, and sometimes not. In QM a lot of it has to do with the semi historical way its usually presented and could be replaced with something much more rational.

My view of physics, based heavily on symmetry, was very hard won from reading bits here and there and fitting it together.

In that journey the following book was crucial:
https://www.amazon.com/dp/0750628960/?tag=pfamazon01-20

As one reviewer said:
'If physicists could weep, they would weep over this book. The book is devastingly brief whilst deriving, in its few pages, all the great results of classical mechanics. Results that in other books take take up many more pages. I first came across Landau's mechanics many years ago as a brash undergrad. My prof at the time had given me this book but warned me that it's the kind of book that ages like wine. I've read this book several times since and I have found that indeed, each time is more rewarding than the last.

The reason for the brevity is that, as pointed out by previous reviewers, Landau derives mechanics from symmetry. Historically, it was long after the main bulk of mechanics was developed that Emmy Noether proved that symmetries underly every important quantity in physics. So instead of starting from concrete mechanical case-studies and generalising to the formal machinery of the Hamilton equations, Landau starts out from the most generic symmetry and dervies the mechanics. The 2nd laws of mechanics, for example, is derived as a consequence of the uniqueness of trajectories in the Lagragian. For some, this may seem too "mathematical" but in reality, it is a sign of sophisitication in physics if one can identify the underlying symmetries in a mechanical system. Thus this book represents the height of theoretical sophistication in that symmetries are used to derive so many physical results.

The difficulty with this approach, and the reason why this book is not a beginner's book, is that to the follow symmetric arguments, one really has to have already mastered vector calculus.'

And that is the whole issue. In order to get to the deep and powerful beauty of physics one must have a certain amount of mathematical maturity - you must be eased into it.

Thanks
Bill

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dsatkas and tionis
In answering a question in general physics I came across the following which explains at a reasonably basic level what a photon is, spontaneous emission etc at the level of basic QM with a bit of math:
http://www.physics.usu.edu/torre/3700_Spring_2015/What_is_a_photon.pdf

This is also not the entirely correct way to view photons, I'm sorry to say. First of all as massless quanta with spin ##1## there's no position operator. So the naive uncertainty relation, valid for massive particles, doesn't make sense to begin with. For a review, see
http://www.mat.univie.ac.at/~neum/physfaq/topics/position.html

I came to quantum optics via 'second quantization' as per bhobba's reference, but have recently become aware of the deficiencies in that approach:

http://www.worldscientific.com/worldscibooks/10.1142/9251

Not being a mathematician, please forgive any mis-statements- I am interested in what the above authors have to say, but don't fully understand what they are saying and so comments are appreciated. As best I can tell, they claim that:

1) The Fock representation is only defined on simply-connected spaces. This is sufficient for many optical cavities, but there are more complex topologies: ring-shaped and bowtie-shaped cavities, branching networks, etc., and the Fock space may not be applicable to those.

2) The field operators live in an infinite dimensional space, and many of the operators are unbounded.

The authors seem to have developed a completely different approach, based on algebraic quantization.

Unfortunately for me, I am trying to parse statements like "The one-photon Hilbert space is given by HT=PTL2(Λ,ℂ3)" and "As the C*-algebra of observables we have chosen the Weyl algebra W(ET, ħ Im(.|.)). The free, diagonalized transversal Maxwell dynamics is given by the one-parameter group of Bogoliubov *-automorphisms αtfree, t∈ℝ, αtfree(Wħ(f))=Wħ(eitS/ħf), ∀f∈ET".

It's slow going...

vanhees71