I Why is Photon Energy Quantized in Terms of Sine Wave Frequency?

[QuantumCuriosity42]

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!

Hello everyone,

I've been grappling with a concept for years, diving into internet resources and pestering professors, yet I still find myself tangled in confusion. I'm reaching out in hopes that someone here can shed light on a question that has been haunting my thoughts regarding the nature of light and the quantization of photon energy.

As per my understanding, the energy of a photon is expressed through the equation E = h f), where E represents energy, h is Planck’s constant, and f is the frequency of the associated wave. This relation appears to imply that energy is quantized in terms of the frequency of a sine wave.

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.

Thank you so much in advance!

 

[Vanadium 50]

Is your question why is EM radiation described by sines and cosines? Or is it why is EM radiation quantized?

 

[Drakkith]

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?

Not sure. My question would be, if it isn't quantized in terms of a sine wave, how else could you quantize it? All other waveforms can be composed of sums of sine waves, so I'm not sure you could get away from them.

 

[QuantumCuriosity42]

Is your question why is EM radiation described by sines and cosines? Or is it why is EM radiation quantized?

Thanks for your response, and for helping to clarify the focus of my question. I understand that electromagnetic (EM) waves can be described using a variety of function bases due to the principles of Fourier analysis, which is why we commonly use sines and cosines for this purpose. However, my curiosity is rooted in the relationship between photon energy and the frequency of EM waves.

The formula E=hν intriguingly ties the energy of a photon directly to the frequency of the EM wave it is associated with. What captivates me is the peculiar coincidence that the energy of photons, the quantum particles of light, is related to the frequency of the harmonic components of the EM wave they constitute.

I am aware that there are numerous other bases of orthogonal functions, such as wavelets, Hermite functions, and Legendre polynomials, that can also be used to decompose signals. Despite this, the photon energy relation specifically hinges on frequencies derived from a harmonic basis. This raises the question: why does nature exhibit a preference for the harmonic basis when it comes to defining the quantum properties of light? Is there a deeper reason for this, possibly rooted in the fundamental structure of space-time or the intrinsic properties of photons themselves?

Any insights, resources, or directions for further reading on this peculiar aspect of quantum mechanics would be immensely appreciated, as I am eager to deepen my understanding of this phenomenon.

 

[QuantumCuriosity42]

Not sure. My question would be, if it isn't quantized in terms of a sine wave, how else could you quantize it? All other waveforms can be composed of sums of sine waves, so I'm not sure you could get away from them.

Thanks for your input. I agree that any waveform can indeed be decomposed into a series of sine waves, making them a natural choice for analyzing oscillatory behavior. However, as I've mentioned in another response, there are various other bases of orthogonal functions that can be employed to decompose signals, not limited to Bessel functions, Legendre polynomials, Hermite polynomials, and Chebyshev polynomials.

Each of these function sets has its own unique characteristics and is better suited for specific types of problems. For instance, Bessel functions are particularly useful in solving problems with cylindrical symmetry, while Legendre polynomials are often employed in problems with spherical symmetry. Hermite and Chebyshev polynomials also find applications in various branches of physics and engineering.

If we focus on periodic functions, we have alternatives like the Walsh functions, which can be used to decompose signals in terms of square waves instead of sine waves. The associated Walsh-Hadamard transform provides a different perspective on signal decomposition compared to the Fourier transform. (Or in a general case, a wavelet transform).

This brings us back to the main crux of my question: Given that there are numerous orthogonal bases available to decompose signals, why does the quantization of photon energy specifically relate to the frequencies of harmonic functions? It seems like an extraordinary coincidence, and I’m trying to understand if there is a more profound reason behind this specific relationship.

I am interested in exploring whether this unique characteristic of light has deeper implications about the nature of space-time, quantum mechanics, or the properties of photons themselves.

 

[anuttarasammyak]

In quantum mechanics chosen bases depend on corresponding observables.
Sinusoidal functions are eigenfunction of observable Energy. When you apply different bases for expansion of states, these bases are superpositon of various energy states which is often inconveniet in physical insights.

 

[Drakkith]

You might find some mathematical relationship that makes sense, but I suspect that in the end it's a matter of "that's just the way it is".

 

[QuantumCuriosity42]

In quantum mechanics chosen bases depend on corresponding observables.
Sinusoidal functions are eigenfunction of observable Energy. When you apply different bases for expansion of states, these bases are superpositon of various energy states which is often inconveniet in physical insights.

I understand and agree that sinusoidal functions can serve as eigenfunctions for the observable energy in certain quantum systems. However, as I've delved deeper into the topic and consulted various resources, it's clear that this isn't universally true for all quantum systems. The eigenfunctions of the Hamiltonian (or energy observable) depend heavily on the specifics of the potential and the boundary conditions in the system.

For instance, while sinusoidal functions might be the appropriate eigenfunctions for a particle in an infinite potential well or a free particle, there are many quantum systems where other functions (like Hermite polynomials for the quantum harmonic oscillator) serve as the energy eigenfunctions.

 

[QuantumCuriosity42]

You might find some mathematical relationship that makes sense, but I suspect that in the end it's a matter of "that's just the way it is".

I do find it quite unsettling that nature, has chosen to associate energy with the frequency of harmonic sine and cosine waves. These functions are indeed very special and unique in many ways. It's both fascinating and somewhat mysterious that these specific waveforms have such a profound connection to the fundamental properties of our universe. Additionally, it's intriguing that the convention we use to decompose waves (sine and cosine basis), aligns with this natural choice.
And not just that, but our ears too, they are sensible to the frequency of the armonics (sine and cos decomposition). It just does not make any sense to me.

 

[Vanadium 50]

Each of these function sets has its own unique characteristics and is better suited for specific types of problems. For instance, Bessel functions are particularly useful in solving problems with cylindrical symmetry, while Legendre polynomials are often employed in problems with spherical symmetry. Hermite and Chebyshev polynomials also find applications in various branches of physics and engineering.

And sines and cosines are appropriate for systems along a line. And light travels along a line.

 

[PeterDonis]

This relation appears to imply that energy is quantized in terms of the frequency of a sine wave.

It implies no such thing. If the frequency spectrum is continuous, as it is for light traveling in free space, so is the energy spectrum.

 

[anuttarasammyak]

it's clear that this isn't universally true for all quantum systems. The eigenfunctions of the Hamiltonian (or energy observable) depend heavily on the specifics of the potential and the boundary conditions in the system.

Sinusoidal waves serve as bases of energy in free space, V=0, which is familiar in many optical systems. If we are lucky enough, we can find energy eigenfunction for specific V, e.g. harmonic oscillator, square well, hydrogen atom,etc. In most practical cases we cannot find explicit eigenfunction so instead use method of perturbation. Sinusoidal waves, i.e. free motion in free space, are often used as base of perturbation in elementary particle phyisics as you see in Feynman diagrams.

 

[PeroK]

I've been grappling with a concept for years, diving into internet resources and pestering professors, yet I still find myself tangled in confusion.

Perhaps that shows that your chosen method to learn physics is unsound.

The main issue is that photons are the quanta of the quantized EM field. That is part of QED, the quantum theory of light. Whereas, electromagnetic waves are part of the classical theory of light, as described by Maxwell's equation.

Your question is really this:

If we have a quantized EM field with large number of photons of a given energy:

A) how do we show that this is approximated by a classical EM field of monochromatic light waves?

B) how is the frequency of those light waves related to the energy of the photons?

 

[vanhees71]

A) from a QED point of view classical electromagnetic waves are entirely different states of the electromagnetic field, so-called coherent states. They are states with an undetermined number of photons, and in no way you can understand them as some "stream" of classical particles.

The coherent states for a single mode $(\omega,\vec{k},\lambda)$ are defined as eigenstates of the corresponding annihilation operator for this mode. The eigenvalues, $\alpha$, are complex. For $|\alpha| \gg 1$ this state can be very well approximated as a classical em. plane wave. The probability to find a given number of photons is described by a Poisson distribution.

B) The energy density is described by the operator $\hat{\mathcal{u}}=:\hat{\vec{E}}^2/2 + \hat{\vec{B}}^2/2:$, analogous to classical electrodynamics.

If expressed in terms of plane-wave modes this implies that each corresponding photon has an energy $\hbar \omega$ and momentum $\hbar \vec{k}$, as in "old quantum theory" a la Planck, Einstein, and de Broglie.

 

[QuantumCuriosity42]

And sines and cosines are appropriate for systems along a line. And light travels along a line.

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?

As you mentioned, we choose sines and cosines because they are fitting for systems along a line. However, if this choice is somewhat arbitrary, and another choice could have been made, why then does a photon's energy specifically rely on the frequency of these sine and cosine waves?

 

[QuantumCuriosity42]

It implies no such thing. If the frequency spectrum is continuous, as it is for light traveling in free space, so is the energy spectrum.

Yes, I apologize for my miscommunication. I should have said that it's quantized in terms of the frequency of multiple sine waves. However, my underlying question remains: why do we view the spectrum in frequencies based on harmonic functions? And why does this somewhat arbitrary decision to use such a basis align with the experimental frequency upon which a photon's energy depends?

 

[QuantumCuriosity42]

Sinusoidal waves serve as bases of energy in free space, V=0, which is familiar in many optical systems. If we are lucky enough, we can find energy eigenfunction for specific V, e.g. harmonic oscillator, square well, hydrogen atom,etc. In most practical cases we cannot find explicit eigenfunction so instead use method of perturbation. Sinusoidal waves, i.e. free motion in free space, are often used as base of perturbation in elementary particle phyisics as you see in Feynman diagrams.

Just to be clear, are you suggesting that Planck's energy-frequency relation (E=h f) is not universally true, but rather potential-dependent? Is it only true for V=0?

 

[QuantumCuriosity42]

Perhaps that shows that your chosen method to learn physics is unsound.

The main issue is that photons are the quanta of the quantized EM field. That is part of QED, the quantum theory of light. Whereas, electromagnetic waves are part of the classical theory of light, as described by Maxwell's equation.

Your question is really this:

If we have a quantized EM field with large number of photons of a given energy:

A) how do we show that this is approximated by a classical EM field of monochromatic light waves?

B) how is the frequency of those light waves related to the energy of the photons?

Thank you for reframing my question. It does help in clarifying the core of my confusion. Could you please elaborate on the answers to the questions A and B you've reformulated? Or provide some references.

 

[QuantumCuriosity42]

A) from a QED point of view classical electromagnetic waves are entirely different states of the electromagnetic field, so-called coherent states. They are states with an undetermined number of photons, and in no way you can understand them as some "stream" of classical particles.

The coherent states for a single mode $(\omega,\vec{k},\lambda)$ are defined as eigenstates of the corresponding annihilation operator for this mode. The eigenvalues, $\alpha$, are complex. For $|\alpha| \gg 1$ this state can be very well approximated as a classical em. plane wave. The probability to find a given number of photons is described by a Poisson distribution.

B) The energy density is described by the operator $\hat{\mathcal{u}}=:\hat{\vec{E}}^2/2 + \hat{\vec{B}}^2/2:$, analogous to classical electrodynamics.

If expressed in terms of plane-wave modes this implies that each corresponding photon has an energy $\hbar \omega$ and momentum $\hbar \vec{k}$, as in "old quantum theory" a la Planck, Einstein, and de Broglie.

Thanks for your detailed explanation. I must admit that I'm not familiar with many of the concepts you've mentioned. It seems like this might be more advanced quantum mechanics than I've been exposed to. Would you be able to provide some resources where I can delve deeper into these topics?

Regarding your statement "For |α| >> 1 this state can be very well approximated as a classical em. plane wave", does this imply that the Planck's energy–frequency relation, E=h*f, is just an approximation? Can it be expressed in terms of functions other than sinusoids (could they even be non-periodic)? Does the correspondence of energy shift from frequency to some other property of the waves in that basis?

So, is the photon's energy as described by E=h*f just an approximation based on what you've said?

I appreciate your patience and assistance.

 

[PeroK]

Thank you for reframing my question. It does help in clarifying the core of my confusion. Could you please elaborate on the answers to the questions A and B you've reformulated? Or provide some references.

I was leaving that to @vanhees71, who has provided the expert answer above.

 

[QuantumCuriosity42]

I was leaving that to @vanhees71, who has provided the expert answer above.

I hadn't seen vanhees71's message when I replied to you. My apologies for the oversight. Thanks.

 

[PeroK]

I hadn't seen vanhees71's message when I replied to you. My apologies for the oversight. Thanks.

There's no need to apologise. I've studied classical electromagnetism and some quantum field theory. But, my knowledge doesn't extend to describing the classical EM field in terms of the quantized EM field.

One thing I do know and is that photons and EM waves are not part of the same theory of light. In particular, there are no photons in the theory of classical EM. And it's the classical theory that is the approximation of the quantum theory.

It won't answer your question but Feynman's book The Strange Theory of Light and Matter is an accessible introduction to QED. It explains the quantum nature of light and how things like reflection, refraction and diffraction are described and explained by quantum theory.

But, to my recollection, it won't describe how QED explains EM waves. You'll need @vanhees71 for that!

 

[QuantumCuriosity42]

There's no need to apologise. I've studied classical electromagnetism and some quantum field theory. But, my knowledge doesn't extend to describing the classical EM field in terms of the quantized EM field.

One thing I do know and is that photons and EM waves are not part of the same theory of light. In particular, there are no photons in the theory of classical EM. And it's the classical theory that is the approximation of the quantum theory.

It won't answer your question but Feynman's book The Strange Theory of Light and Matter is an accessible introduction to QED. It explains the quantum nature of light and how things like reflection, refraction and diffraction are described and explained by quantum theory.

But, to my recollection, it won't describe how QED explains EM waves. You'll need @vanhees71 for that!

I've glanced through Feynman's book "The Strange Theory of Light and Matter", but as you say, it didn't provide an answer to my original question. I appreciate your help nonetheless.

 

[Vanadium 50]

sines and cosines are the 1-d solutions of \Box^2 u = 0.

It is true that you can decompose other functions into sines and cosines, but that constant energy solutions are sines and cosines. And photons are states of constant energy.

 

[PeterDonis]

I should have said that it's quantized in terms of the frequency of multiple sine waves.

That doesn't help. "Quantized" implies a discrete spectrum. If the spectrum is continuous, it is not "quantized", no matter how you gerrymander terms.

 

[QuantumCuriosity42]

That doesn't help. "Quantized" implies a discrete spectrum. If the spectrum is continuous, it is not "quantized", no matter how you gerrymander terms.

You are correct, but my original doubt remains. Why energy increases in relation to harmonic frequency.

 

[QuantumCuriosity42]

sines and cosines are the 1-d solutions of \Box^2 u = 0.

It is true that you can decompose other functions into sines and cosines, but that constant energy solutions are sines and cosines. And photons are states of constant energy.

Could you explain that more, or provide some references to read please. I don't understand "photon are states of constant energy".

 

[PeterDonis]

Why energy increases in relation to harmonic frequency.

Before you can even pose this question, you need to ask: energy of what? And harmonic frequency of what?

In other words, you need to specify what kind of state of the quantum electromagnetic field you are talking about. And you need to not mix together different types of states.

For example, you could say: energy of a single-photon Fock state of the field. But then the so-called "harmonic frequency" of this state has nothing whatever to do with any actual electromagnetic wave, because a Fock state does not describe an electromagnetic wave. It describes something that has no classical analogue at all.

Or you could say: harmonic frequency of an electromagnetic wave. But then the state of the quantum electromagnetic field you are talking about is a coherent state, which is not an eigenstate of the Hamiltonian and therefore has no definite energy.

In neither of these cases will the claim of yours that I quoted above be true in any useful sense. So your question is based on a misconception, that there are actual quantum field states to which your description even applies.

 

[anuttarasammyak]

Just to be clear, are you suggesting that Planck's energy-frequency relation (E=h f) is not universally true, but rather potential-dependent? Is it only true for V=0?

I confess that I do not have an established idea of "potential energy of photon". Potential energy V(x) is function of coordinate but photon has no wave function $\psi(x)$ as particles with mass have it. I do not think the formula $E=\hbar \omega$ is compatible with concept of position.

Sinusoidal waves correspnding to a $E=\hbar \omega$ should have infinite length. Wave trains of finite length contains various $\omega$ around its main value as shown by Fourier decomposition. So we are dealing infinite space for photon energy.

 

[PeterDonis]

I confess that I do not have an established idea of "potential energy of photon".

Potential energy isn't a property of the photon (or the particle in general). It's a property of the Hamiltonian. Or the Lagrangian, if you are using that formulation of quantum field theory (which is generally easier to use for the quantum electromagnetic field).

 

[Vanadium 50]

If you wish to talk about "the energy of a photon", it myst have an energy - i.e. be in a state of defined energy. Those states are not the ones represented by Besel functions or anything else - just sines of a single frequency.

 

[QuantumCuriosity42]

Before you can even pose this question, you need to ask: energy of what? And harmonic frequency of what?

In other words, you need to specify what kind of state of the quantum electromagnetic field you are talking about. And you need to not mix together different types of states.

For example, you could say: energy of a single-photon Fock state of the field. But then the so-called "harmonic frequency" of this state has nothing whatever to do with any actual electromagnetic wave, because a Fock state does not describe an electromagnetic wave. It describes something that has no classical analogue at all.

Or you could say: harmonic frequency of an electromagnetic wave. But then the state of the quantum electromagnetic field you are talking about is a coherent state, which is not an eigenstate of the Hamiltonian and therefore has no definite energy.

In neither of these cases will the claim of yours that I quoted above be true in any useful sense. So your question is based on a misconception, that there are actual quantum field states to which your description even applies.

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.

 

[QuantumCuriosity42]

If you wish to talk about "the energy of a photon", it myst have an energy - i.e. be in a state of defined energy. Those states are not the ones represented by Besel functions or anything else - just sines of a single frequency.

But why sines of a single frequency?

 

[PeterDonis]

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

And my point is that "at a simpler level", your question is not valid because it is based on implicit assumptions that are not valid.

I don't think my question is ambigous?

And yet it is. And when you resolve the ambiguities, you find that you no longer have a valid question. That was the point of my post #28.

 

[QuantumCuriosity42]

And my point is that "at a simpler level", your question is not valid because it is based on implicit assumptions that are not valid.And yet it is. And when you resolve the ambiguities, you find that you no longer have a valid question. That was the point of my post #28.

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?

 

[PeterDonis]

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.

 

[hutchphd]

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. Absent this ansatz, there were some nasty infinities in the thermodynamics. E and f were the usual classical objects for atoms and light.

 

[PeroK]

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?)

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.

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?

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.

 

[PeroK]

PS to understand your question you must understand the difference between the quantum mechanical and classical theories of light, and the relationship between them.

 

[PeroK]

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.

 

[Fra]

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.

Sometimes different people ask similar questions, but seeks an answer of different types.
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.

I can just add a bit of how I reasoned if it may help. For me, I asked this question in a more general context of the foundations of QM, and why certain mathematics are more "fit" to describe nature? (ie it has nothing todo specifically with "photons")

It is perfectly fine to decompose functions in various bases, but from the perspective of information processing and retention of information under constraints of limited memeory, it seems all the mathematically possible ways, still does not do the job equally good, in particular when you want to describe "periodic phenomena". Note that "periodic phenomena" just means "repeating", a phenomena can be periodic without beeing a harmonic etc. The problem is how do you represent the transform? In limited memory you need to truncate? (ie discard data). Then you want to of course not discard data at random, but discard data that is LEAST important (which we often call "noise"). Ie. we should not occupy scarce resources such as memory with noise, so the task is - which transforms gives us the best outlook to "compress data", so we can choose to dismiss the noise and keep the rest.

Periodic phenomena are almost I would say something we actively look for, when repeating experiments with a given duration. One could even say that our method, "truncates" our observations so that we simply register short periods. We can not see slow phenomena (because it never reaches sufficient statistical confidence levels), we can also not see extremly fast pheonmena as it requires extremely fast sample rates. And for a given mesaurement devices, we have limts to all this. (a general note, without getting into any details).

In QM the answer to your question is supeficially I think related to how we DEFINE say, momentum or energy. Ie. as conjugate variables. And these conjugate variables indeed singles out the fourier transform as the natural partner, in the way that invariance with respect to one varibla, means that the other variable is constant. and this is exactlty the properties you want from a stable retention of compressed information. IF the retained part is "constant" then we have a stability. (And stability of say atoms, was one of the original mysteries of atomic physics: WHY is it stable, well it's because of quantization, but WHY is it quantized? Then we throw in the conjugate relation. But why? is there a deeper reeason?

https://en.wikipedia.org/wiki/Conjugate_variables

This also relates to the HUP, and to actions, but I think to elaborate on this it becomes more interpretational and possibly speculative so I will pass. But this was just a short encouragement to you to say that I don't think your question is the least confused. But the answer lies in the foundational area I think. It's not specific to electromagnetism (at least not for me, the quyestion is bigger).

/Fredrik

 

[Fra]

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?

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?

/Fredrik

 

[vanhees71]

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.

Any free electromagnetic field can be decomposed into plane-wave modes, and so you can for the field operators of the quantized theory.

The corresponding mode functions are all $u(t,\vec{x})=\propto \exp(-\mathrm{i} \omega t + \mathrm{i} \vec{k} \cdot \vec{x})$. Also they must fulfill the wave equation,
$$\frac{1}{c^2} \partial_t^2 u - \Delta u=0,$$
from which you get
$$\omega=c |\vec{k}|.$$
The free electromagnetic field is then represented by an infinite set of harmonic oscillators, labelled by $\vec{k}$ and the helicity $\lambda \in \{1,-1\}$ to describe the polarization states too. $\lambda=1$ corresponds to right-circular and $\lambda=-1$ to left-circular polarized em. waves.

Each harmonic oscillator then is described by creation and annihilation operators $\hat{a}^{\dagger}(\vec{k},\lambda)$ and $\hat{a}(\vec{k},\lambda)$. They obey bosonic commutation relations,
$$[\hat{a}(\vec{k},\lambda),\hat{a}^{\dagger}(\vec{k}',\lambda')]=(2 \pi)^3 \delta^{(3)}(\vec{k}-\vec{k}') \delta_{\lambda \lambda'}.$$
A basis is then formed by the occupation-number (Fock) basis, starting from the ground state, the "vacuum", $|\Omega \rangle$, for which
$$\hat{a}(\vec{k},\lambda) |\Omega \rangle=0$$
for all $(\vec{k},\lambda)$ and the states $|N(\vec{k},\lambda) \rangle$, which are simultaneous eigenvectors of the number-density operators
$$\hat{N}(\vec{k},\lambda)=\hat{a}^{\dagger}(\vec{k},\lambda) \hat{a}(\vec{k},\lambda).$$
The energy and momentum can then be derived from the operators given as in classical electrodynamics by the corresponding energy-momentum tensor of the electromagnetic field. In terms of the number operators it results
$$\hat{H}=\sum_{\lambda} \int_{\mathbb{R}^3} \frac{\mathrm{d}^3 k}{(2 \pi)^3} \omega \hat{N}(\vec{k},\lambda),$$
$$\hat{\vec{P}}=\sum_{\lambda} \int_{\mathbb{R}^3} \frac{\mathrm{d}^3 k}{(2 \pi)^3} \vec{k} \hat{N}(\vec{k},\lambda).$$
This can be interpreted that the mode $(\vec{k},\lambda)$ comes in discrete "quanta" of energy and momentum, $E=\omega$ and $\vec{p}=\vec{k}$ (note that I use natural units, where $\hbar=1$.

 

[vanhees71]

There is no EM wave associated with an individual photon. You're still mixing up two different theories of light. Consider diffraction:

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".

Of course you can try to do this by considering the em. field in a finite cavity. Then again you get field modes, and here only certain $\vec{k}$ are allowed due to the boundary conditions, but each field mode is a standing wave. The "intensity", i.e., the energy density of the em. field, spreads over the entire cavity. You cannot localize the "photon" any better.

 

[gentzen]

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.

please don't

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?

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?

This was a simple question. I don't get your answer. I guess an answer appropriate for the level of QuantumCuriosity42 could start like

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.

(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.

 

[Fra]

This was a simple question.

Was it?

I was wondering if it was, as it seemed like the answers given was not satisfactory the OP. And as I recognize the question from myself, so I thought I'll add another perspective, and just leave it there. And the OP can pick the sort of answer that was appropriate.

/Fredrik

 

[PeroK]

Of course there is an em wave associated with an "individual photon"

But, not in the sense of a classical EM wave, satisfying Maxwell's equations. Otherwise, a single photon would exhibit classical EM behaviour.

 

[hutchphd]

At the risk of exposing my profound lack of understanding of quantum field theory, I will attempt to answer your questions. If you are truly curious you must do the hard work to learn advanced quantum field theory from, say, Bogoliubov and Shirkov

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

Yes of course there is.

Why not in terms of a square wave, or any other waveform for that matter

They are not eigensolutions for the free fields.

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.

Study enough to do field theory. This does not mean watch more videos

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?

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 $$E_{k,n}=\hbar \omega_k(n+\frac 1 2)$$

Would you be able to provide some resources where I can delve deeper into these topics?

Books about advanced quantum mechanics, preceeded by books about elementary quantum mechanics. See top

So, is the photon's energy as described by E=h*f just an approximation

no these are exact eigensolutions for the quantized free field

You are correct, but my original doubt remains. Why energy increases in relation to harmonic frequency.

Study tbooks. These are eigenmodes of harmonic oscillators
Your questions are all good ones but the answers are not simple. To really understand requires much concerted effort ( frankly more than I have been willing to invest ..... I presume any blunders in this colloquy on my part will be corrected by wiser hands)

 

[vanhees71]

But, not in the sense of a classical EM wave, satisfying Maxwell's equations. Otherwise, a single photon would exhibit classical EM behaviour.

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).

 

[PeterDonis]

However, when the pattern has built up we see that the photons collectively can be associated with a classical frequency $f$.

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.

The term "photon" here really refers to the fact that the detections are quantized--at low enough intensity we see discrete dots appear on the detector screen. But that sense of the term "photon" has nothing to do with the Planck relation.