At first, the answer seems obvious, as E/h = f, where E is the energy of the photon, h is the Planck constant, and f is the frequency of the photon. But then one realizes that the photon would need an infinite duration in order to have a single frequency (e.g., Fourier transform relation, and Heisenberg uncertainty principle).
The probability doctrine of quantum mechanics (QM) asserts that the indetermination, of which we have just given an example, is a property inherent in Nature, and not merely a profession of our temporary ignorance from which we expect to be relieved by a future better and more complete theory.
Such more complete theory appears to be Quantum Field Theory (QFT). The Heisenberg uncertainty principle in QM may then have to be reexamined.
Obviously, then, E/h = f is not the correct answer.
Nothing is infinite in Nature. We can't wait forever to measure a photon, and nothing can. The Universe would not exist.
The answer is to realize that something is wrong with the QM picture of a photon. The frequency of a photon is defined by its physical conditions in QFT, not by itself.
And it is not described by a Fourier transform either, which is a mathematically "continuous" procedure -- with the hypothesis of infinitely close frequencies -- and should never be used to represent a discrete phenomena, or artifacts of the interpolation will appear.
As Juan Weisz asks -- why is QFT better than QM? The answer may be relevant here, as QM is subjective but QFT is intersubjective. Like math, it is not enough to be subjective, as follows.
QM is based on two deep untruths, as revealed by Nature, in addition to the rather formalist easy-to-solve fact that QM is not combining the principles of Lorentz invariance (SR-MINKOWSKI-EINSTEIN). They are:
1. One needs to abandon the single-particle approach of QM (subjectivity). In any relativistic quantum theory, particle number need not be conserved, since the relativistic dispersion relation in SR, that E^2 = c^2p^2 + m^2c^4, implies that energy can be converted into particles and vice versa. This requires a multi-particle framework (intersubjectivity), a many-body interaction with SR included and uses QM. It is a many-body-relativistic-QM, not just QM.
2. Unitarity (basically, preserving the inner product) and causality cannot be combined in a single-particle approach, requires intersubjectivity.
QFT solves these two problems by using a different approach:
A. The fundamental entities are not the particles, but the field, an abstract object that penetrates spacetime.
B. Particles appear as the vibrations of the field.
The physical model of the photon, for example, is given as a vibration of the EM field, and follows QFT. Then, in QFT the frequency of the photon does NOT depend on the photon itself and only (that would be subjective), but on its physical conditions in a many-body-relativistic-QM (intersubjective). Then, that intersubjectivity can obtain objectivity to different observers, in different experiments, at differing spacetimes.
See https://www.researchgate.net/post/What_is_wrong_with_Quantum_Mechanics
and
Sean Carroll recently at https://www.youtube.com/watch?v=rBpR0LBsUfM and after the 45 minute mark specially.
SEG and all: You wrote, "the frequency and the duration (= total length) of the composite electromagnetic wave determine the energy." That is not true, and we see that in the photoelectric effect of a photon, for example.
In the photoelectric effect, an electron within a material absorbs the energy of one photon and if it acquires more energy than a certain frequency threshold (e.g., giving the ionization limit in terms of frequency, not duration), it is ejected from the material. This was explained by Einstein who, in 1922, was awarded the Nobel Prize in Physics 1921, for "his discovery of the law of the photoelectric effect".
According to Einstein, only the frequency of the photon determines the photon's energy, and this is the QM model.
From general properties on an infinite wave
v= lambda/T
v speed of the wave, lambda wave length, T the time period of one oscilation.
This translates automatically omega = vk
k is wavector., omega angular frequency. For enm waves omega =ck
Now multiply both sides by hbar (h divided by 2 pi)
hbar omega =hf =cp (em waves)
(You get cp also from special relativity.)
Because de Broglie is valid also for photons.
On the other hand people talk about a packet. Inconsistent with an infinite wave.
Then what one says I guess in is something of a wave packet, long enough for
wave properties, but which zooms in fast into a material medium, because the speed of light is fast. There is an approximation.
JW and all: Yes, there is an approximation. What the question asks, and you may wish to answer, is different. How is this frequency determined? It cannot be as Einstein and QM say, because the photon would have to be infinite in duration. And it may not follow Heisenberg's uncertainty principle either, as revealed in QFT (see link). It seems to depend on the environment, the physical conditions, such as the boundary conditions -- not just on the photon by itself, intrinsically.
All: This question shows that following the (e.g.) book "Quantum Mechanics" by Eugen Merzbacher, once influential and authoritative on QM, is today a slippery slope in many parts.
The simultaneous appearance of wave and particle aspects no longer exists in QM. The only thing that exists in Nature, as we know from QFT, is quantum waves, NOT particles and NOT waves as in Fourier analysis. The Heisenberg uncertainty principle principle in QM will, thus, have to be modified.
There is only one (not two or even three) case of photon interference in the two-slit experiment, and that is the case that is often neglected -- the quantum wave case. See the two-slit experiment at low-intensity, for example at: https://www.youtube.com/watch?v=GzbKb59my3U
There would not be a single frequency, but a narrow window of frequencies, with f located somewhere in or near.
delta (f) delta (t) roughly equal to one.
delta (t) lifetime of photon.
JW and all: QFT is THE fundamental theory of this world (QM is non-relativistic). QFT is also THE most precisely tested theory we have. All "particles" of the SM are described by QFT. So, according to our best theories of physics, the fundamental building blocks of matter are not particles, but "quantum fields".
More at:
https://www.thphys.uni-heidelberg.de/~hebecker/QFTI/n1.pdf
https://www.youtube.com/watch?v=zNVQfWC_evg
I think you are confused and should refer to what items in QFT you specifically refer to. QFT is very vast and mixes up all sorts of theories.
This sort of generality does not contribute anything.
You are selling wind.
JW and all: JW wrote, "... a narrow window of frequencies, with f located somewhere in or near."
In QFT we do not talk like that. This is specific , as JW asks, not a criticism of JW's text. See also the header text.
QFT is the modern language of old "Condensed Matter Theory" and "Cold Atom Physics", even though it is just an effective description of the underlying many-body-relativistic-QM.
QM is based on two deep untruths, as revealed by Nature and further laid out in the header, in addition to the rather formalist easy-to-solve fact that QM is not combining the principles of Lorentz invariance.
In QFT therefore, the frequency of the photon does NOT depend on the photon, but on its physical conditions in a many-body-relativistic-QM.
Ed
I dont have any particular pick about general arguments for the need of
a many body relativistic quantum theory etc, These are questions QFT attemps to solve, but with questionable success, hard and questionable mathematics.
But I do object to the vagueness of the need for a single photon frequency, in light about your own comments that a single frequency wont do the job with a finite wave train.
....a photon frequency does not depend on the photon....
is self contradictory.
If everything is a wave , why some waves have mass and others not.?
P. D. The mass of different elementary particles is readily found listed, so do not give me a story of massive particles not existing.
If not tell me how a wave packet does the trick.
You are no more concrete than previously.
I strongly disagree about QFT being the most tested theory. To start with it is not just one theory but a whole pack of ideas. Right now it is more a need than a success.
I have used canonical QFT, do know something about changing the wave function into an operator. Is that your underlying field?
Perhaps you are thinking in terms of specific topics in QED, with predictions with an unheard amount of accuracy? The QED sector is rather more cemented, and you can follow how a photon follows from that math. You have to respect Maxwell in vacuum to start you off though.
JW: You wrote, "I strongly disagree about QFT being the most tested theory. " See, for example, the reference I gave above: https://www.thphys.uni-heidelberg.de/~hebecker/QFTI/n1.pdf and go learn a couple things. It might take you a year, so talk to you then. Take your time, it is innocuous to physics.
I did take a glance, at this short study.
The author talks in futuristic terms about QFT.(ie. for electron field)
Mostly it is known stuff, but there are opinions that you copy,
that I would not agree with
In any case I would not rely on any single source.
There is quite a lot of variation in content, style and taste in QFT
It is not finished or totally agreed on stuff. You cannot form a one source good understanding.
All: We can flip a light switch and point out that, despite a person's supposed comprehensive understanding of classical and quantum physics, they still cannot explain what is happening.
Where does the light comes from? The emission and absorption of photons is a quantum process for which particle number is not conserved; it is an everyday phenomenon which cannot be explained without QFT.
So says, for example, Matt Schwartz of Harvard, at the seminal book "Quantum Field Theory and the Standard Model".
This is exemplified by my question here, that asks "What is the frequency of a single photon?" Unless one considers the photon physical conditions in a many-body-relativistic-QM, which is a QFT, the frequency of that photon is, literally, "blowing in wind" and cannot be given by QM considerations where (actually, not true) particle number would be conserved. One is forced into considering a many-body-relativistic-QM, a QFT -- QM would not do it.
The frequency of the photon, then, does NOT depend on the photon itself, locally. No considerations of Fourier transform of the isolated photon will give its frequency without taking into account its global physical environment -- for example, its boundary conditions.
All: To clarify a question by PM on unitarity (mentioned in the question header), hermitian matrices have real eigenvalues whose eigenvectors form a unitary basis. For real matrices, hermitian is the same as symmetric.
Examples of hermitian matrices include Pauli matrices. Any matrix which is not hermitian can be expressed as the sum of a hermitian matrix and a antihermitian matrix. The general rule is that if the classical variable is real, the quantum operator is hermitian.
All: Another question by PM reaches us. Besides understanding the photon as a many-body-relativistic-QM, and its frequency as non-local, what else is a QFT useful for?
Today, the Standard Model of particle physics, which accounts for all observed phenomena on length scales larger than ∼ 10−18 meters, is a QFT. There are no "particles" in the Universe, or classical waves, just quantum waves -- and QFT models this as explained in this question header. The "particles" are vibrations of the fields, where the fundamental entities are not the "particles", but the field of each type (e.g., photon, electron, whatever) as an abstract object that penetrates spacetime in QFT, for the whole universe.
General Relativity can also be viewed as a QFT, or an ‘effective quantum field theory’ describing the long-distance behavior of a massless spin 2 "particle", the graviton, as it interacts with the "particles" from the rest of the SM.
All: In some cases, there are the kinetic energy of atoms, and plasma temperature, that are considered continuous -- whereas a sequential quantum, step-wise, process is proposed in QFT.
And, a multiphoton (seemingly continuous levels of energy being accessible from a single photon that is NOT absorbable per se) band is considered (erroneously, without QFT) to somehow “be able” to generate laser action to a lower level, whereas a sequential multi-photon pumping process is proposed in QFT.
There are other examples, where one “sees” broad ground state energy bands (in infrared excimers, such as XeI), or calculates broad bands in H atoms near ionization (Rydberg states), or calculates a shift of quantum-well levels when dressed by a lower frequency photon, all of which show a “continuous spectrum” of energies, and, yet, they are not classical (i.e., actually continuous) when viewed in QFT terms.
We see that the appearance of “analog” or “continuous” is not real, but an illusion created by smeared sub-bands, each one quantum. The underlying reality is quantum, or digital, as in QFT. It is not described by a Fourier transform, which is continuous.
In philosophy, this can be seen as setting the stage of two limiting cases, 'what is there to see' representing the Kantian concept of Natur, quantum, versus 'what one seems to see' representing the Kantian Wirklichkeit, analog and quantum, critically separating diverse interpretations in philosophy, physics, and mathematics.
We can consider the QFT processes that "look" analog (seemingly “continuous) and quantum, in the experience (Wirklichkeit) -- but the underlying reality (the Natur) and mathematics, is only quantum.
The confusion about the photon should end soon. It is a matter of definition.
Alternatives are:
a)
A photon is a bundle of frequencies of infinite wave trains used to describe a finite wave train or wave packet.
b) A photon has exactly the energy hf needed to describe an infinite wave train,
and multiple photons intervene when this wave packet is absorbed in mater.
How do people vote?
Let us not be axiomatic, you should show what it is in QFT, and why this is better.
Black body radiation already shows some features you are interested in, how the number of them is not constant but varies with temperature.(The bose factor)
Ed Gerck
Dear Ed,
I am very surprised why you are not talking about the physical model of a photon? What do your equations, matrices and formulas describe? Which thing? I thought a lot about the structure of the photon and agree with Louis de Broglie. A photon is composed of two neutrinos (neutrinos are toroidal vortices of an electromagnetic field). The frequency of ring rotations of tori is f in the Planck formula.
Read more in the article NEUTRIN LIGHT THEORY OF LOUIS DE BROIL AND MODELING PHOTON AND ELECTRON in my profile.
Yours
Valeriy Pakulin
PS
You can see the animation of photon on my site http://gravity.spb.ru
VP and all: Thank you, VP, but I did address the physical model of a photon. The physical model is given as a vibration of the field, and follows QFT.
The photon is not made of neutrinos, or whatever, that also have mass. A single photon, as all experiments show, has no mass. Two or more photons, in a many-body-relativistic-QM a.k.a. a QFT, can have mass -- and that is straightforward to calculate. QFT is a good approach for describing both old "particle physics" (nothing is a particle in Nature) and statistical physics systems, because in both cases we are interested in (relatively) long-distance or macroscopic properties.
The rest you can see at (recommended, with a good list of books also):
https://sites.krieger.jhu.edu/jared-kaplan/files/2016/05/QFTNotes.pdf
I think maybe one of you is thinking of the electron-antielectron anihalation in the presence of matter, giving rise to two gamma rays in opposite directions.?
I accept that the photon is constructed by superposition of vibrations in a definite manner. Every fundamental particle have there own mode of vibration.
Now let us focus on some collection of photons with different energy. Here intrinsic property, like spin, is same for all particles. Hence the reason which gives intrinsic angular momentum is also same. So the vibration which make this particle "photon" is also same.
What I am trying to say is that the frequency which gives energy is not same as frequency that makes the particle photon.
The energy is decided by the situation by which it is credited and it is stored in the form of rate of change of phase.
The sub topic in QFT dealing with photons is QED, in which it is true that they have sharply defined frequency f, so you deal with many of them.(ie. Bose statistics)
However the starting point in QED is Maxwells equations in vacuum.
I would not call this the only possible point of view.
JW: You may be mistaken, think of it... it is your own doubt. But physics no longer shares the doubts you have here, and it has much more doubts -- but no one believes in particles anymore, or that QED is a false view.
Read the question head, and please stop insisting on this across different RG threads, which I manage. Denying my subjects will not make them false, just experiments could... and they have already spoken affirmatively. Change, evolve. Participate positively.
All: The physical model behind this approach to vibrations and quantum mechanics, as not tied only to the microscopic view, to an atomic view of matter that we find has further subdivisions, is the Euler-Lagrange equation, harking back to the Sturm-Liouville differential equation, but In Galois fields.
For a limited time, my home-page has two free full-text papers on it -- as offered through Amazon and Kindle books -- for any RG members, just for the asking, in PDF. Sent me please a PM, and I will send back. At least until June 30.
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