# How is a spherical electromagnetic wave emitted from an antenna described in terms of photons?

When an atenna transmits radiowaves isn't it true that the electromagnetic pulse is radiated away from the accelerating electron as a spherical wave in all directions simultaneously, and if so how can the associated photon be "everywhere" on this rapidly expanding sphere?

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Suggest renaming the title of the question to be more "informative". – Kostya Jan 17 '11 at 16:14

## 4 Answers

First of all, note that any realistic antenna emits a gigantic amount of photons at the radio frequencies. The energy of a single photon is $E=hf$ where $h=6.626\times 10^{-34}$ Js which is tiny, so if you have frequencies of order "just a few Hertz", the energy of one photon will be a tiny fraction of one Joule. Antennas consume much more energy than that.

So in reality, you emit trillions of photons that fly in all the directions more or less uniformly - well, vertical dipole antennas emit mostly in the horizontal directions etc. The number of photons is so huge that it makes no sense to talk about individual photons: classical electromagnetism described by Maxwell's equations is a totally satisfactory approximation for all practical purposes (and even many impractical ones).

But if you designed a similar experiment where you would only emit one photon, there would be one photon going away and its position i.e. direction would be undetermined. A photon is a particle that always respects the laws of quantum mechanics, including the uncertainty principle. If the frequency and angular momentum of the photon is known, its position - direction in which it propagates - is completely unknown.

The photon is described by a probabilistic wave whose dependence on the space is pretty much the same as the dependence of $E+iB$ of a classical electromagnetic wave that you obtain if you emit lots of photons in the same state. The probability that the photon will be found at a point is proportional to the energy density $(E^2+B^2)/2$ of the classical electromagnetic wave that you emit by the same antenna if the number of photons is very large.

But for a single photon, you can't predict in what direction it will go. That's a basic feature of quantum mechanics that the evolution is not deterministic and outcomes of experiments can only be predicted probabilistically. If you know that you have only emitted one photon, the distant detectors will only detect one photon in one particular direction - but you can't be sure in which direction it will be. Again, for a dipole antenna, the nearly-horizontal directions will be preferred - the ratios of probabilities in different directions will follow the energy density of the corresponding classical electromagnetic wave.

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Ok Lubos, so I think if I understand you correctly the classical laws of electromagnetism are actually describing the "collective" behavior of what in practice can be considered an infinite number of photons being "spit out" in all directions "simultaneously" or in a more restricted set of directions depending on the configuration of the antenna. Whereas quantum mechanics is used to describe the case of a single photon. So obviously I was confusing the case of "infinite" number of photons with the case of a single photon. Many thanks for your help! – BuckyBadger Jan 17 '11 at 22:43
For some reason, my instinct is that a spherical electromagnetic wave cannot be emitted by an antenna. Instead, they can only be emitted by a charge. I guess that's cause I always think of an antenna as an object that has no net charge. – Carl Brannen Jan 19 '11 at 2:42
@BuckyBadger , I hope it is clear to you that the classical wave distributions are a limiting case of the quantum mechanical description. Quantum mechanical solutions are always  there, except it is not practical or reasonable to use its formalism when the  limiting classical formalism is more than adequate. – anna v Feb 4 '11 at 15:28
"If the frequency and angular momentum of the photon is known, its position - direction in which it propagates - is completely unknown". Surely Lubos wrote this in haste. The relevant uncertainties for this problem are the initial position of the "photon", presumed to be completely known, and its final momentum, presumed to be completely unknown. In fact it's a little more suttle: even for a fixed initial position, we can specify one axis along which the momentum is known to be zero; but that's as much as I can say in 600 characters or less. – Marty Green Aug 6 '11 at 23:57

What follows is not a proper answer to your question, but the statement of a fact worth knowing. There is no way to create a (perfectly) spherical wave: "A spherically symmetric vacuum solution to Maxwell's equations is always static." (Pappas, Am. J. Phys., 52 (255), 1984.) Also: H.F. Mathis: "A short proof that an isotropic antenna is impossible", Proc. IRE, 59 (979), 1951. It's an amusing application of Brouwer's "hairy ball" theorem.

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One of the lessons quantum mechanics supposedly teaches us is that we should be cautious about asking questions that cannot be answered experimentally. Not that we should never do it, just that we should be careful. I think this question and the subsequent answers are a good example of this principle being disregarded.

How can a "photon" be everywhere at once on an expanding spherical surface? First of all, I would dismiss those people who make a major point of the issue of spherical symmetry. Everyone knows that an e-m wave is not spherically symmetrical. It is so obvious that those who deal with the subject will use the term "spherical" to describe the next best thing, the familiar donut shape of a dipole radiator. S-wave or p-wave, the question stands: how can the photon be everywhere at once?

Second, I disagree with those who say the question is wrong because an antenna emits billions of photons. There are in fact antennas which regularly emit light in quantities similar to one photon's worth of energy; these antennas are called "atoms" and they are everywhere. The question stands: how can a photon emitted by an atom be everywhere at once on a spherical surface? In fact, this is very close to the original form of the EPR paradox.

When Einstein posed the question in 1935, no one at first seriously considered that it might be tested experimentally. The EPR paradox went through a number of transformations before it dawned on people that it might be put testable. Among these transformations we can list Bohm, who recast it in the form of two electrons in the spin singlet state; and Feynmann, who analyzed the two-photon decay of positronium. Neither of these models were, then or now, amenable to experimental testing. After Bell's analysis in 1964, people were motivated afresh to look for experimental manifestations, and found something workable in parametric down-conversion. But that's another story.

The basic problem with the question as posed here is: how would you measure it? The theory tells us that the photon spreads out as a "spherical" wave. But Copenhagen, in one form or another, tells us that the photon is detected at a single point. How do we know? Many, notably Feynmann, would say that the click in a photomultiplier tube tells us when a photon has been detected. But the detailed physics of a detector event can be interpreted in different ways; all we can say with relative certainty is that the probability of a detector going off is proportional to the square of the incident field. And this is entirely conisistent with the photon's energy being spread over a spherical surface. It is very difficult to establish that a click in the photodetector is necessarily associated with the abosorption of one full photon's worth of energy.

Some will undoubtedly say it is obvious that when a photomultiplier tube clicks, it must have absorbed a photon. To those people, I would ask: what experiment can you propose to demonstrate that a photomultiplier tube will never click when exposed to less than one photon's worth of energy? Others will object that once a detector clicks, a second detector will never go off at the same time; this shows that the whole photon "collapsed" into the first detector. But experimentally, this hypothesis is notoriously difficult to demonstrate. The reason is simply that we still do not have a working pea-shooter for photons that reliably produces one photon at a time.

To answer the original question, I would say that the wave from an antenna, even at "atomic" antenna, spreads out "spherically"; and that there is no experiment which can conclusively show that the emitted "photon" ever appears concentrated at a single point.

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Whether or not photons should be taken into account when calculating antenna parameters is irrelevant to the natural wish to get a picture of what is going on with the photons when an antennais tranmitting or receiving. Unfortunately, experiments involving photons and antennas are presumably difficult. Nevertheless, if we can look for gravity waves, we should be able to figure out how to observe low-energy photons coming out of a wire.

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You obviously lack basics on partice-wave dualism. – Georg Aug 3 '11 at 16:10
I most certainly do not lack basics. I am talking about the physical photon picture behind transmission and reception by an antenna. That is an area where a quantum description is seldom discussed, mainly because classical electrodynamics is the appropriate calculational framework to use for getting answers in such low-frequency radiation. But I am looking for a quantum picture, not a specific tool for making calculations. – Ralph Dratman Aug 4 '11 at 5:41
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