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At the end of this nice video (https://youtu.be/XiHVe8U5PhU?t=10m27s), she says that electromagnetic wave is a chain reaction of electric and magnetic fields creating each other so the chain of wave moves forward.

I wonder where the photon is in this explanation. What is the relation between electromagnetic wave and photon?

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    $\begingroup$ Please see my answer here. You can understand Willis Lamb's frustration and the waves and normal modes describe the electromagnetic field. Photons are then the changes of number state of each normal mode - they are like the discrete "communications" the whole EM field has with the other quantum fields of the World that make up "empty space". One can reinterpret this statement as Maxwell's equations being the propagation equation for a lone "photon", but only in terms of propagation equations for the mean of electric and magnetic field .... $\endgroup$ – WetSavannaAnimal Dec 19 '13 at 0:48
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    $\begingroup$ ...observables when the EM field is in a superposition of $n=1$ Fock states (so it is "one photon propagating"). $\endgroup$ – WetSavannaAnimal Dec 19 '13 at 0:49
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Both the wave theory of light and the particle theory of light are approximations to a deeper theory called Quantum Electrodynamics (QED for short). Light is not a wave nor a particle but instead it is an excitation in a quantum field.

QED is a complicated theory, so while it is possible to do calculations directly in QED we often find it simpler to use an approximation. The wave theory of light is often a good approximation when we are looking at how light propagates, and the particle theory of light is often a good approximation when we are looking at how light interacts i.e. exchanges energy with something else.

So it isn't really possible to answer the question where the photon is in this explanation. In general if you're looking at a system, like the one in the video, where the wave theory is a good description of light you'll find the photon theory to be a poor description of light, and vice versa. The two ways of looking at light are complementary.

For example if you look at the experiment described in Anna's answer (which is one of the seminal experiments in understanding diffraction!) the wave theory gives us a good description of how the light travels through the Young's slits and creates the interference pattern, but it cannot describe how the light interacts with the photomultiplier used to record the image. By contrast the photon theory gives us a good explanation of how the light interacts with the photomultiplier but cannot describe how it travelled through the slits and formed the diffraction pattern.

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    $\begingroup$ This is news because all QM teachers told me that photons abstractions, proposed by QED, which is more exact than wave discription. However, this should not stop us from figuring out how two are related. Actually quanta = particles. $\endgroup$ – Val Dec 20 '13 at 18:09
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    $\begingroup$ @Val The way we actually calculate things in QED is with a perturbative expansion that involves photons. The underlying exact theory is one of several completely quantum fields. $\endgroup$ – Kevin Driscoll Dec 20 '13 at 19:26
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    $\begingroup$ There is a sense in which the classical description of light is retrieved as the classical limit of a coherent state of photons. I would say that this would be an appropriate answer to "where is the photon in the classical wave theory of light?" $\endgroup$ – Prahar May 4 '16 at 18:01
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    $\begingroup$ @Prahar Yes, but you just said it yourself - that's not the reality. That's just "how it fits in the models"- it doesn't help you outside of the constraints of the models, and that's exactly what the OP is asking here. In the classical wave theory of light... there's no photons. Not one per wave, not "infinite amounts" per wave, just no photons, period. $\endgroup$ – Luaan May 5 '16 at 11:32
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    $\begingroup$ I think that "excitation of a field instead of waves and particles" is one interpretation, and probably not the most popular one. Many people view fields only as a handy mathematical tool. $\endgroup$ – Helen - down with PCorrectness Sep 30 '18 at 7:56
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In this link there exists a mathematical explanation of how an ensemble of photons of frequency $\nu$ and energy $E=h\nu$ end up building coherently the classical electromagnetic wave of frequency $\nu$.

It is not simple to follow if one does not have the mathematical background. Conceptually watching the build up of interference fringes from single photons in a two slit experiment might give you an intuition of how even though light is composed of individual elementary particles, photons, the classical wave pattern emerges when the ensemble becomes large.

single photon

Figure 1. Single-photon camera recording of photons from a double slit illuminated by very weak laser light. Left to right: single frame, superposition of 200, 1’000, and 500’000 frames.

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In 1995 Willis Lamb published a provocative article with the title "Anti-photon", Appl. Phys. B 60, 77-84 (1995). As Lamb was one of the great pioneers of 20th century physics it is not easy to dismiss him as an old crank.

He writes in the introductory paragraph:

The photon concepts as used by a high percentage of the laser community have no scientific justification. It is now about thirty-five years after the making of the first laser. The sooner an appropriate reformulation of our educational processes can be made, the better.

He finishes with these comments:

There is a lot to talk about the wave-particle duality in discussion of quantum mechanics. This may be necessary for those who are unwilling or unable to acquire an understanding of the theory. However, this concept is even more pointlessly introduced in discussions of problems in the quantum theory or radiation. Here the normal mode waves of a purely classical electrodynamics appear, and for each normal mode there is an equivalent pseudosimple harmonic-oscillator particle which may then have a wave function whose argument is the corresponding normal-mode amplitude. Note that the particle is not a photon. One might rather think of a multiplicity of two distinct wave concepts and a particle concept for each normal mode of the radiation field. However, such concepts are really not useful or appropriate. The "Complementarity Principle" and the notion of wave-particle duality were introduced by N. Bohr in 1927. They reflect the fact that he mostly dealt with theoretical and philosophical concepts, and left the detailed work to postdoctoral assistants. It is very likely that Bohr never, by himself, made a significant quantum-mechanical calculation after the formulation of quantum mechanics in 1925-1926. It is high time to give up the use of the word "photon", and of a bad concept which will shortly be a century old. Radiation does not consist of particles, and the classical, i.e., non-quantum, limit of QTR is described by Maxwell's equations for the electromagnetic fields, which do not involve particles. Talking about radiation in terms of particles is like using such ubiquitous phrases as "You know" or "I mean" which are very much to be heard in some cultures. For a friend of Charlie Brown, it might serve as a kind of security blanket.

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    $\begingroup$ Wow, Lamb is actually making my rethink me admittedly amateur perspective on the matter. This quote blew my mind: " It is very likely that Bohr never, by himself, made a significant quantum-mechanical calculation after the formulation of quantum mechanics in 1925-1926." $\endgroup$ – electronpusher Mar 23 '17 at 7:54
  • $\begingroup$ This is not within the mainstream physics models at present, but a peculiar proposal not validated or supported by model calculations and predictions. $\endgroup$ – anna v Aug 3 '18 at 3:54
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    $\begingroup$ @anna_v to the limited extent I understand it, I believe that if you read the whole paper and not just the snippet I quoted here you would agree that Lamb's is mainstream physics with mainstream interpretation. $\endgroup$ – hyportnex Aug 3 '18 at 19:54
  • $\begingroup$ @annav, then again, the chosen answer interpreting everything as fields is not necessarily mainstream physics for many physicists (or, more importantly, not necessarily correct). I think this reference deserves a reading. $\endgroup$ – Helen - down with PCorrectness Sep 30 '18 at 8:01
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    $\begingroup$ @Helen Inmy opinion quantum field theory has very many calculational successes in describing particle physics, where it is mainstream . One could argue about its region of validity, as with many mathematical models. For example QCD has more success with lattice QCD as the expansions of perturbative field theory do not work. I do not think that there is a problem with photons in the standard model, and photons are their own antiparticle. So I will not go to the trouble of reading the paper ( no link provided so it means a library or a paywall) where a prominent physicist discusses new theory $\endgroup$ – anna v Sep 30 '18 at 8:49
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The photon dilemma

It is postulated by Planck that energy is quantized. Owing to classical electromagnetic theory light is an electromagnetic field. This field satisfies a wave equation traveling at the speed of light. Hence, light is an electromagnetic wave. Light consists of photons; and thus each photon carries a unit of energy. This behavior is demonstrated by the photoelectric and Compton effects. Since light is an electromagnetic energy, photons must also carry electromagnetic field and a unit of it. While photons are quantum objects, light still governed by Maxwell's classical theory. The photon model is not critically consistent with Maxwell equations, since it has a dual nature. In fact light as a wave is well described by Maxwell. Recall that Maxwell's equations don't involve the Planck's constant, and thus can not describe the particle nature of the photon. A complete Maxwell's equations should involve this missing element. In quantum electrodynamic paradigm, the photon is brought to interact with the electrons by invoking the idea of minimal coupling where electrons and photons exchange momentum. The photon appears as a mediator between charged particles.

At the same time while a moving charged particle has its self electric field, and magnetic field that depend on the particle velocity, the photon, the carrier of the electromagnetic energy is void of these self-fields because it has no charge and mass. Thus, a charge-less photon can't have electric and magnetic fields accompanying its motion.

The appropriate Maxwell's equations should then incorporate the photon linear momentum as well as its angular momentum. In such a case the new Maxwell's equations can then describe the dual nature of the photon. Like electric charge, the angular momentum is generally a conserved quantity. The question is how one can correct for these photon proprieties? One way to achieve that is to employ quaternions that generically allow many physical properties to be joined in a single equation. This is so because the quaternion algebra is so rich, unlike the ordinary real numbers. To this end we employ the position-momentum commutator bracket, and invoked a photon wavefunction. This wavefunction is constructed from the linear complex combination of the electric and magnetic fields.

The outcome of the bracket yields three equations defining the photon electric and magnetic fields in terms of its angular momentum. These equations turn out to be very similar to those fields created by a moving charge. Thus, the electric and magnetic fields of the photon doesn't' require a charge for the photon. It is intriguing that the photon has no charge and mass but has electric and magnetic fields as well as energy. These fields should also satisfy Maxwell's equations. Doing so, yields additional electric and magnetic charge and current densities for the photon. The emergent Maxwell's equations are now appropriate to describe the photon as a quantum particle. These additional terms in Maxwell's equations are the source in describing the photon quantum electrodynamics behavior. Some emergent phenomena associated with topological insulator, Faraday's rotation effect, Hall effect and Kerr's effect could be examples of this contribution terms to Maxwell's equations.

Here are the quantized Maxwell's equations incorporating the photon linear and angular momentum. These are the electric and magnetic fields due to the photon as a particle: \begin{equation} \vec{L}\cdot\vec{E}=-\frac{3\hbar c}{2}\,\Lambda\,, \qquad\qquad \vec{L}\cdot\vec{B}=0\,, \end{equation} and \begin{equation} \vec{B}=-\frac{2}{3\hbar c}\,(\vec{L}\times\vec{E})\,,\qquad\qquad\vec{E}=\frac{2 c}{3\hbar}(-\Lambda\,\vec{L}+\vec{L}\times\vec{B})\,. \end{equation} And these are the new Maxwell's equations: \begin{equation} \vec{\nabla}\cdot\vec{E}=-\frac{4c}{3\hbar}\,\,(\vec{B}-\frac{1}{2}\,\mu_0\vec{r}\times\vec{J})\cdot\vec{p}+\frac{2}{3\hbar c}\,\vec{E}\cdot\vec{\tau}+\frac{\partial \Lambda}{\partial t}\,,\qquad \vec{\nabla}\cdot\vec{B}=\frac{4}{3\hbar c}\,\,\vec{E}\cdot\vec{p}+\frac{2}{3\hbar c}\,\vec{B}\cdot\vec{\tau}\,, \end{equation} and \begin{equation} \vec{\nabla}\times\vec{B}=\frac{1}{c^2}\,\frac{\partial\vec{E}}{\partial t}+\frac{2}{3\hbar c}\left(\Lambda\vec{\tau}+\vec{B}\times\vec{\tau }-\frac{\vec{P}}{\varepsilon_0}\times\vec{p}\right)-\vec{\nabla}\Lambda\,, \end{equation}

\begin{equation} \vec{\nabla}\times\vec{E}=-\frac{\partial\vec{B}}{\partial t}-\frac{2c}{3\hbar}\left(\mu_0\vec{J}\times\vec{L}+\frac{\vec{\tau}}{c^2}\times\vec{E}+2\Lambda\,\vec{p}\right)\,, \end{equation} where \begin{equation} -\Lambda=\frac{1}{c^2}\,\frac{\partial\varphi}{\partial t}+\vec{\nabla}\cdot\vec{A}=\partial_\mu A^\mu\,. \end{equation} In the standard electrodynamics $\Lambda=0$ represents the Lorenz gauge condition.

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  • $\begingroup$ This answer is very confused. The photon does not "carry" the electric or magnetic fields, it is the Standard Model mediator of the EM interaction. It looks like you've confused classical and quantum concepts. Maxwell's equations do not need to incorporate anything quantum mechanical -- they are purely classical equations. I also don't know where those equations have come from. $\endgroup$ – Zorawar Nov 20 at 17:29
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In order to understand the wave particle dualism you have simply to understand what time is:

In 1905, the Newtonian unique time concept was replaced by a twofold time concept of observed coordinate time and proper time - the observed time is relative and observer-dependent, and it is derived from the intrinsic proper time of the observed particle ("The time measured by a clock following a given object"). Proper time is the more fundamental time concept.

You can understand the wave particle dualism if you consider the simplest case of a photon, that is a photon moving at light speed c. The spacetime interval of such photons (which corresponds to their proper time) is zero. That means that the event of emission and the event of absorption are adjacent in spacetime, the emitting mass particle is transmitting the momentum which is called photon directly to the absorbing mass particle, without any spacetime between them. That means that the particle characteristics are transmitted directly without need for any intermediate massless particle.

However, for observers the zero spacetime interval is not observable, e.g. between Sun and Earth are observed to be eight light minutes, even if the spacetime interval of the path of the photon is zero. In spite of the direct transmission of a momentum between two mass particles, observers observe an electromagnetic wave which is filling the gap of eight light minutes.

In summary, particle characteristics are transmitted directly according to the principles of spacetime intervals and proper time, whereas the wave is transmitted according the principles of the observed spacetime manifold.

Now you will ask: What about photons which are moving slower than c (through gravity fields and through transparent media)? The answer is that here quantum effects such as nonlocality are implied. But it is important to notice that the limit case of photons in vacuum moving at c may be explained and understood classically, without need for any quantum theory.

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What are photons?

Photons get emitted every time when a body has a temperature higher 0 Kelvin (the absolute zero temperature). All bodies, surrounding us (except black holes) at any time radiate. They emit radiation into the surrounding as well as they receive radiation from the surrounding. Max Planck was the physicist who found out that this radiation has to be emitted in small portions, later called quanta and even later called photons. Making some changes in the imagination of how electrons are distributed around the nucleus, it was concluded that electrons get disturbed by incoming photons, by this way gain energy and give back this energy by the emission of photons. And photons not only get emitted from electrons. The nucleus, if well disturbed, emits photons too. Such radiations are called X-rays and gamma rays.

What is electromagnetic radiation?

EM radiation is the sum of all emitted photons from the involved electrons, protons and neutrons of a body. All bodies emit infrared radiation; beginning with approx. 500°C they emit visible light, first glowing in red and then shining brighter and brighter. There are some methods to stimulate the emission of EM radiation. It was found out that beside the re-emission of photons there is a second possibility to generate EM radiation. Every time, an electron is accelerated, it emits photons. This explanation helps to understand what happens in the glow filament of an electric bulb. The electrons at the filament are not moving straight forwards, they bump together and running zig-zag. By this accelerations they lose energy and this energy is emitted as photons. Most of this photons are infrared photons, and some of this photons are in the range of the visible light. In a fluorescent tube the electrons get accelerated with higher energy and they emit ultraviolet photons (which get converted into visible light by the fluorescent coating of the glass). Higher energy (with higher velocity) electrons reach the nucleus and the nucleus emits X-rays. As long as the introduced energy is a continuous flow, not one is able to measure an oscillation of EM radiation.

What are EM waves?

Using a wave generator it is possible to create oscillating EM radiation. Such radiations are called radio waves. It was found out that a modified LC circuit in unit with a wave generator is able to radiate and that it’s possible to filter out such a modulated radiation (of a certain frequency) from the surrounding noisy EM radiation.

from Wikipedia

So the wave generator has a double function. The generator has to accelerate forward and backward the electrons inside the antenna rod and by this the photons of the radio wave get emitted, and the generator makes it possible to modulate this EM radiation with a carrier frequency. It has to be underlined that the frequency of the emitted photons are in the IR range and sometime in the X-ray range. There is an optimal ratio between the length of the antenna rod and the frequency of the wave generator. But of course one can change the length of the rod or one can change the frequency of generator. This changes the efficiency of the radiation to the needed energy input only. To conclude from the length of the antenna rod to the wavelength of the emitted photons is nonsense.

What is the wave characteristic of the photon?

Since the electrons in an antenna rod are accelerated more or less at the same time, they emit photons simultaneous. The EM radiation of an antenna is measurable and it was found out that the nearfield of an antenna has two components, an electric field component and a magnetic field component. This two components get converted in each other, the induce each other. At some moment the transmitting energy is in the electric field component and otherwise the energy is in the magnetic field component. So why not conclude from the overall picture to the nature of the involved photons? They are the constituents which make the radio wave.

from Wikipedia

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    $\begingroup$ The two components do not induce each other, though it's a common misconception (that's what I've been taught in school as well :-). Because of how wide that misconception is, animations now usually show both the electric and magnetic field in phase, to prevent confusion. $\endgroup$ – Luaan Jul 21 '16 at 9:20
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    $\begingroup$ The final figure here shows the $E$ and $B$ fields oscillating a quarter-turn out of phase. For waves in vacuum that's incorrect; $E$ and $B$ should be in phase. $\endgroup$ – rob Dec 15 '16 at 19:50
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    $\begingroup$ @HolgerFiedler If the fields are a quarter-turn out of phase, the average value for the Poynting vector is zero and the wave is not transmitting any energy. $\endgroup$ – rob Dec 16 '16 at 6:51
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    $\begingroup$ @rob Than how the energy transfer in the near field of an antenna work? And how a standing EM wave inside a box work? $\endgroup$ – HolgerFiedler Dec 16 '16 at 8:07
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    $\begingroup$ Those would make good follow-up questions; I don't know if I can answer completely in a comment. $\endgroup$ – rob Dec 18 '16 at 5:42
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You report that in the video it is stated that an electromagnetic wave is "a chain reaction of electric and magnetic fields creating each other so the chain of wave moves forward." I disagree with this view. There is just one wave, that of the vector potential or more generally of the four potential. The electric and magnetic fields are just derivatives of the vector potential and do not "create each other".

Rejecting this explanation we then arrive at your deeper question: "What is the relation between electromagnetic wave and photon?"

Until a few years ago I shared the opinion of Willis Lamb, that the photon is a fictive particle. I finally changed my mind because such an explanation cannot account for low intensity diffraction experiments. Indeed, how can a single atom or molecule absorb a wave that is much larger that it? Note that I don't intend to fork off a discussion on this here but want to give my interpretation. This is that the vector potential describes the probability of a photon being absorbed, just like the Schrödinger and Dirac wave functions do for an electron. Indeed the Maxwell equations in vacuum can be written as a wave equation that closely resembles the Klein-Gordon equation. This interpretation implies that the photon indeed exists as a particle, much smaller than an atom and at least as small as a nucleon.

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  • $\begingroup$ "how can a single atom or molecule absorb a wave that is much larger that it? ", the same question can be asked how can an electrically small antenna ($"dimension"<< \lambda$), say, a Hertz dipole absorb an essentially infinite plane wave. It can, I have seen it; all waves all the way down, no photons needed... $\endgroup$ – hyportnex Nov 8 '18 at 15:55
  • $\begingroup$ @hyportnex your argument can easily be used to support the photon concept. $\endgroup$ – my2cts Nov 8 '18 at 19:49
  • $\begingroup$ I have not seen any attempt neither do I believe that, say, a 5cm long ferrite loaded loop antenna's operation at around 550kHz can be usefully explained via photons and quantum physics but, please, go ahead. $\endgroup$ – hyportnex Nov 8 '18 at 20:53
  • $\begingroup$ @hyportnex your example pertains to the limit of many photons. That is why no QM is needed. $\endgroup$ – my2cts Nov 8 '18 at 23:30

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