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

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!

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

QuantumCuriosity42 said:

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.

Vanadium 50 said:

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.

Drakkith said:

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.

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.

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".

anuttarasammyak said:

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.

Drakkith said:

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.

QuantumCuriosity42 said:

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.

QuantumCuriosity42 said:

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.

QuantumCuriosity42 said:

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.

QuantumCuriosity42 said:

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?

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.

Vanadium 50 said:

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?

PeterDonis said:

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?

anuttarasammyak said:

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?

PeroK said:

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.

vanhees71 said:

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.

QuantumCuriosity42 said:

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.

PeroK said:

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.

QuantumCuriosity42 said:

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!

PeroK said:

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.

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

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.

QuantumCuriosity42 said:

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.

PeterDonis said:

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.

Vanadium 50 said:

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

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".

QuantumCuriosity42 said:

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.

QuantumCuriosity42 said:

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.

anuttarasammyak said:

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).

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.

PeterDonis said:

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.

Vanadium 50 said:

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?

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

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.

QuantumCuriosity42 said:

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.

PeterDonis said:

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?