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PeterDonis
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Go read my post #28. I already addressed this question there.QuantumCuriosity42 said:In the equation E=h f, could you please tell me what is E and what is f really?
Go read my post #28. I already addressed this question there.QuantumCuriosity42 said:In the equation E=h f, could you please tell me what is E and what is f really?
There is no EM wave associated with an individual photon. You're still mixing up two different theories of light. Consider diffraction:QuantumCuriosity42 said:At a simpler level, without delving too deep into advanced theory, I'm trying to understand why, in the equation E=hf (for an individual photon's energy), the energy is dependent on the harmonic frequency of the wave (I don't think my question is ambigous?)
As above, E is the energy associated with an individual photon, and ##f## is the emergent frequency when a sufficient number of photons are involved for classical wavelike behaviour to be observed.QuantumCuriosity42 said:Maybe I should start more basic. In the equation E=h f, could you please tell me what is E and what is f really?
Sometimes different people ask similar questions, but seeks an answer of different types.QuantumCuriosity42 said:My burning question is: why is this the case? Is there something fundamentally ingrained in nature that dictates the energy to be quantized in this manner, specifically in terms of a sine wave frequency? Why not in terms of a square wave, or any other waveform for that matter?
I am well aware that sine waves possess unique properties such as orthogonality and smoothness, and they are prevalent in numerous physical phenomena. However, my intellectual curiosity yearns for a deeper understanding — is there a more profound or fundamental reason behind the photon’s energy being quantized in this specific way?
I have spent years searching for answers, sifting through articles online, and reaching out to professors, but the answers I found were either too surface-level or they just skirted around the question. I’m at my wits' end here, and I am earnestly hoping that this community might offer a new perspective or point me towards resources that can finally put this longstanding query to rest.
Any thoughts, references, or guidance would be immensely appreciated.
See conjugate varibels, it relates to how E is "defined". But the question is of course, why do we "choose" to define it that way?QuantumCuriosity42 said:Maybe I should start more basic. In the equation E=h f, could you please tell me what is E and what is f really?
Any free electromagnetic field can be decomposed into plane-wave modes, and so you can for the field operators of the quantized theory.QuantumCuriosity42 said:At a simpler level, without delving too deep into advanced theory, I'm trying to understand why, in the equation E=hf (for an individual photon's energy), the energy is dependent on the harmonic frequency of the wave (I don't think my question is ambigous?) That is precisely Planck's relation, and I've struggled to find a satisfactory explanation online.
Of course there is an em wave associated with an "individual photon" (where "individual" has to be taken as grain of salt since photons are of course indistinguishable in the usual sense of QT). A single-photon Fock state represents a corresponding mode of the electromagnetic field. In the usual momentum-helicity basis it's a plane em. wave with sharp wave vector ##\vec{k}## and helicity ##\lambda##. One can interpret single-photon states only in this way. A naive particle picture doesn't work for photons, which are massless spin-1 quanta, which don't admit a full-fledged position observable, i.e., a photon cannot be prepared in a "localized state".PeroK said:There is no EM wave associated with an individual photon. You're still mixing up two different theories of light. Consider diffraction:
please don'tFra said:I will just note that used to ask myself the same question, and its a good one, I don't think it's confused at all, on the contrary, I think it's related to the paradigm we use in physics. But this might fit better in the interpretational forum IMO, but perhaps someone can move it there.
QuantumCuriosity42 said:Maybe I should start more basic. In the equation E=h f, could you please tell me what is E and what is f really?
This was a simple question. I don't get your answer. I guess an answer appropriate for the level of QuantumCuriosity42 could start likeFra said:See conjugate varibels, it relates to how E is "defined". But the question is of course, why do we "choose" to define it that way?
(i.e. giving a context in which E=h f has meaning). And then explain the meaning of E and f in that context, i.e. f is the frequency of the classical electromagnatic field. The energy exchange with the walls happens only in discrete energy packets, and E is the amount of energy in such a packet.hutchphd said:In the original Planck formulation, I believe this was a statement about energy exchange with the walls in a cavity in a solid which was the model for a black body radiator at temperature T.
Was it?gentzen said:This was a simple question.
But, not in the sense of a classical EM wave, satisfying Maxwell's equations. Otherwise, a single photon would exhibit classical EM behaviour.vanhees71 said:Of course there is an em wave associated with an "individual photon"
Yes of course there is.QuantumCuriosity42 said:TL;DR Summary: I've been on a multi-year quest, diving into internet resources and consulting professors, trying to grasp why photon energy is quantized in terms of sine wave frequency (E=h⋅ν), and not any other waveform. Despite understanding the unique properties of sine waves, I’m still in search of a deeper, more fundamental explanation. Any insights or resources to finally put this question to rest would be immensely appreciated!
Is there something fundamentally ingrained in nature that dictates the energy to be quantized in this manner
They are not eigensolutions for the free fields.QuantumCuriosity42 said:Why not in terms of a square wave, or any other waveform for that matter
Study enough to do field theory. This does not mean watch more videosQuantumCuriosity42 said:I am earnestly hoping that this community might offer a new perspective or point me towards resources that can finally put this longstanding query to rest.
They are the eigenstates of the quantized EM field in free space. One starts with the appropriate Lagrangian for the EM field (usually using the vector potentials). This produces equations that look much like many simple harmonic oscilllators and therefore proceeds using that usual formalism. The "number of photons n" corresponds to energy level of the appropriate oscillation mode $$E_{k,n}=\hbar \omega_k(n+\frac 1 2)$$QuantumCuriosity42 said:Could you expand on why these trigonometric functions are particularly suited for describing systems along a line? Is it tied to their properties, or is there a deeper physical reasoning?
Books about advanced quantum mechanics, preceeded by books about elementary quantum mechanics. See topQuantumCuriosity42 said:Would you be able to provide some resources where I can delve deeper into these topics?
no these are exact eigensolutions for the quantized free fieldQuantumCuriosity42 said:So, is the photon's energy as described by E=h*f just an approximation
Study tbooks. These are eigenmodes of harmonic oscillatorsQuantumCuriosity42 said:You are correct, but my original doubt remains. Why energy increases in relation to harmonic frequency.
Of course, it's a generic quantum state of the em. field, which cannot be in any way approximated by a classical theory (neither by a classical point-particle theory nor by classical electrodynamics).PeroK said:But, not in the sense of a classical EM wave, satisfying Maxwell's equations. Otherwise, a single photon would exhibit classical EM behaviour.
Careful. The "photons" you describe here are a coherent state of the quantum EM field, which is not an eigenstate of photon number or energy. It has a definite frequency ##f##, but there is no definite energy ##E## corresponding to ##f##. So the Planck relation ##E = hf## has no meaning for this case since ##E## is not well-defined.PeroK said:However, when the pattern has built up we see that the photons collectively can be associated with a classical frequency ##f##.
Quibble: Because photons do not interact directly, this interaction ( "detection"?) is central to the counting problem associated with light in a cavity "black body" at a defined temperature. So I disagree or we are drowning in the semantic sea?PeterDonis said:But that sense of the term "photon" has nothing to do with the Planck relation.
The case of light in a cavity is a different physical scenario from the case of light incident on a detector and being detected as discrete dots ("photons"). In a cavity we do not detect any photons, we just measure the black body temperature and the intensity of radiation at various frequencies. And of course the Planck relation was originally introduced by Planck to explain the curve of intensity vs. frequency, since the classical prediction was known to be egregiously wrong (the "ultraviolet catastrophe"). But the Planck relation in that case has no relationship to any detections of individual photons.hutchphd said:Quibble: Because photons do not interact directly, this interaction ( "detection"?) is central to the counting problem associated with light in a cavity "black body" at a defined temperature.
I understand the distinction you're making between the classical EM wave and the quantum view of light. I also grasp that when we talk about frequency, we are referring to the classical EM wave.PeroK said:There is no EM wave associated with an individual photon. You're still mixing up two different theories of light. Consider diffraction:
There is a classical theory, describing light as an EM wave, where the diffraction pattern is explained by Huygens principle and an analysis using the classical wavelength of the light.
There is a quantum theory, where light is described probabilistically, which results in the same diffraction pattern. Moreover, in this theory, light interacts with matter in discrete quanta - called photons. And if we do diffraction with very low intensity light, we can see the diffraction pattern building up photon by photon. Note that each photon appears on the detection screen probabilistically. So, although each photon has an associated frequency they do not all diffract by the same angle.
However, when the pattern has built up we see that the photons collectively can be associated with a classical frequency ##f##.
And, if we also measure the energy of each photon, we find that ##E = hf##.
This is one example of how we see that the quantum theory is the fundamental theory, with the classical theory emerging as an approximation.
The classical EM wave is a similar case. The wave only appears as a result of the probabilistic behaviour of a sufficiently large number of photons. The individual photons are not themselves waves - and don't inherently have a wavelength and frequency. However, when the resulting phenomenon of light is studied, the energy of the photons corresponds to classical wavelengths and frequencies related to the energy.
Understanding this fully requires a study of the mathematics that underpins both theories. As, ultimately, the equivalence of the two theories where they overlap is a mathematical one.
I suggest you study Feynmans book fully, as this describes how classical wavelike phenomena emerge from a probabilistic quantum theory where light has no inherent wavelength or frequency at the fundamental level.As above, E is the energy associated with an individual photon, and ##f## is the emergent frequency when a sufficient number of photons are involved for classical wavelike behaviour to be observed.
That equation is itself, therefore, something of a mixture of two theories of light.
Could you point me to a demonstration or a source where it is shown that the diffraction pattern can be specifically associated with a classical sinusoidal wave of a certain frequency?PeroK said:PPS a related question is how can an electron have a wavelength and frequency? It's the same answer: when you apply the probabilistic quantum theory to a particle with mass, such as an electron, you get behaviour such as diffraction. Again, however, only when you do an experiment with a large number of electrons. And, the resulting diffraction pattern can be associated with that of a classical wave of a certain wavelength and frequency.
The only difference is that classically we associate light as a wave and an electron as a particle. When light exhibits particle-like behaviour or an electron exhibits wavelike behaviour we are surprised. But, ultimately, both behaviours are just two sides of the quantum coin.
Firstly, I apologize for any misunderstandings or lack of clarity in my previous communications. I understand the foundational role of sines and cosines in solving the wave equations. My inquiry stems from a philosophical angle more than a purely mathematical one: Even if sines and cosines are solutions, I think we could decompose an electromagnetic wave using a basis other than sines and cosines. Why is the frequency in Planck's relation specifically tied to the frequency of a sine wave and not, for instance, a square or triangular wave? Is the sine/cosine (of infinite frequencies) basis the only orthogonal and periodic one? Perhaps my question borders on the philosophical, and the answer might not be rooted strictly within the current bounds of physics.Vanadium 50 said:Wish you hadn't said "quibble", because now I can't complain about quibbling without it sounding like I am complaining about you specifically. I'm not, but I think we (as a group) are,
Trying to summarize, our answer is;
The OP seems not to accept this. However, his objection is far from clear. He writes a lot...A LOT..and closes with restarting his original question. That leaves us only two options: a) repeat what we wrote, or b) quibble about each others answers.
- Sines and cosines are the solutions to the relevant differential equations.
- Sums of sins and cosines are also solutions to the relevant differential equations.
- However, sums of sines and cosines do not have a single well-defined energy. The only states that have a single energy are sines of one frequency (I drop the cosines here - a cosine is just an offset sine). For both sides of E = hf to be well-defined, E and f need to be single-valued.
If the OP can succinctly - succinctly - state his objection to the answers, it may be worth another circle around. Otherwise, this is probably hopeless.
To the extent this question is well-defined (I have already posted multiple times about the issues with it), the answer has already been given by @Vanadium 50: sinusoidal waves are the eigenstates of energy, i.e., they are the states that have a definite energy. Any state for which the Planck relation can be well-defined must have a definite energy, otherwise the ##E## in the relation has no meaning. So any state for which the Planck relation can be well-defined must be a sinusoidal wave.QuantumCuriosity42 said:Why is the frequency in Planck's relation specifically tied to the frequency of a sine wave
Of course you can. But you won't be decomposing it in a basis of states with definite energy.QuantumCuriosity42 said:I think we could decompose an electromagnetic wave using a basis other than sines and cosines.
We don't discuss philosophy here.QuantumCuriosity42 said:philosophical,
That question has been answered already and is covered in any undergraduate textbook on EM. The wavelike solutions to Maxwell's equations are (sinusoidal) oscillating electric and magnetic fields. This led to Maxwell concluding that light was EM radiation in the first place.QuantumCuriosity42 said:However, my core question remains: Why is this emergent frequency specifically related to sinusoidal (harmonic) waves? Why not square waves, triangular waves, or any other waveform for that matter? I'm curious about the inherent nature of light that leads us to describe its frequency using sinusoidal waves as opposed to any other shape.
The Strange Theory of Light and Matter, which you said you'd already glanced at.QuantumCuriosity42 said:Also, in what Feynman book can I see what you say about how classical wavelike phenomena emerge from a probabilistic quantum theory?
The diffraction of light depends only on the wavelength. This is covered in numerous online physics material, such as the Khan Academy.QuantumCuriosity42 said:Could you point me to a demonstration or a source where it is shown that the diffraction pattern can be specifically associated with a classical sinusoidal wave of a certain frequency?
Yes, so provided OP understands the answers so far (?) it seems the residual question is:Vanadium 50 said:They are the logical consequences of the equations, and even philosophers don't doubt these consequences.
@PeterDonis is right with giving the caveat that a "very dim laser light" is not a single-photon state but still a coherent state. It's with good approximation a single-mode coherent state with a frequency ##\omega##, but it's not an eigenstate of the electromagnetic field energy nor a photon-number eigenstate. The photon number is Poisson distributed. If the intensity, i.e., the mean photon number is (much) smaller than one, it's "mostly a vacuum state", and you see randomly single points at a time on the screen is just due to the fluctuations of this field, but it's not a single-photon source.hutchphd said:Quibble: Because photons do not interact directly, this interaction ( "detection"?) is central to the counting problem associated with light in a cavity "black body" at a defined temperature. So I disagree or we are drowning in the semantic sea?
I fear we are in fact adrift on the semantic sea.PeterDonis said:That said, even in the cavity case the radiation is a coherent state, not a Fock state, and so it is not in an eigenstate of energy and the E in the Planck relation still does not describe "the energy of a photon" in any sense that corresponds to an actual observation. It is an abstract "energy" that appears in the distribution function.
But if you do that with a state that is not an energy eigenstate, such as a coherent state, you get either a superposition of multiple energy eigenstates with different amplitudes, or an expectation value (depending on what you mean by "projection"). Neither of these are "the energy of a photon".hutchphd said:When the thermodynamic counting is done, the associated parameter E involves projection onto the energy eigenbasis.
I disagree. See above.hutchphd said:But that is the "energy of a photon" in my mind.
For a Fock state, yes, as you say. But that does not mean the term "the energy of a photon" can be used in a meaningful sense for states that are not Fock states, which is what other posters in this thread are trying to do.vanhees71 said:The energy of a single photon is of course well defined