Concerning Planck's quantum theory

[01030312]

Considering Planck's assumption of quantized energy, was this idea proved later in quantum electrodynamics or any other theory of electromagnetic radiation? I have seen at places that Max Planck had an intuition about this idea, along with extensive research on the problem. Is intuition thing true?
And the final part, since classical electrodynamics suggests the energy density for a radiation to be proportional to square of frequency, if I have to study the motion of radiation along null geodesic in curved space, should I use energy to be proportional to frequency or square of frequency? Of course quantum nature is used in Minkowski space-time, still for complete classical nature, which of the two is correct form?

 

[andrewr]

Considering Planck's assumption of quantized energy, was this idea proved later in quantum electrodynamics or any other theory of electromagnetic radiation? I have seen at places that Max Planck had an intuition about this idea, along with extensive research on the problem. Is intuition thing true?
And the final part, since classical electrodynamics suggests the energy density for a radiation to be proportional to square of frequency, if I have to study the motion of radiation along null geodesic in curved space, should I use energy to be proportional to frequency or square of frequency? Of course quantum nature is used in Minkowski space-time, still for complete classical nature, which of the two is correct form?

I will leave the general relativity questions aside -- those are really for the other forum and I wouldn't know how to answer them in any event off the top of my head.

Planck developed his theory based on measurements of blackbody radiation. That is, physicists were studying objects which were as effective at adsorbing radiation as they are at emitting them. The name "black" is often associated with pigmented objects of such a color -- but the particular experiments being done at the time were associated with highly reflecting cavities. In particular, a very effective model of an ideal radiator/adsorber is a box with a pin-hole in the side to allow radiation in or out.

Computing the spectra of what is emitted from such a box (black - box) after receiving arbitrary energy input from outside -- depends on the number of states (wavelengths, harmonics) which can be fit into different paths of reflection inside the box. The words "ultraviolet catastrophe" are a revisionist history (fake) of why Max Plank attempted to solve the problem (he wasn't even aware of the problem when he "solved" it.) But in any event, in order to compute the spectra -- he developed a complicated argument which only predicted accurately if he assumed that energy was quantized.

As an aside:
Einstein in his paper on special relativity, shows that a consequence of the constancy of the speed of light, and including Maxwell's equations, is that energy must be proportional to frequency. There is nothing in the special theory of relativity which shows what value that constant of proportionality ought to have -- or why Planck's constant is the only one which works. (See my original thread on these forums for more information -- The Doppler radar paradox: sorry about all the noise from people who communicate poorly on that thread.) I also got into the question of the square law vs. linear nature of energy v. frequency in that thread as well -- but no one really answered the question "what is a photon" in terms of EM fields (Maxwell's equations).

The square law comes from amplitudes and lengths of waves -- which are not defined when speaking of E=hf.
Considering that the square law version is based on mathematics assuming distributed charges -- when in reality there are only "point" charges, realize that the square law is an average of some kind or an interpolation. The linear relationship between energy and frequency has to be the more fundamental law. as a suggestion: I would approach the problem of reconciling these two superficially disagreeing views as one of deriving the amplitude of a wave in volts (or V/m, A/m) based on the number and position of photons which make up the energy of the radiation in question.

Historically, Planck's constant was proved using blackbody radiation. Later formula and theories merely incorporate it.

For all the apparent intelligence on the forums, very few people really answer questions or even understand the subtleties of what is being asked.
Best wishes... and happy fishing...

 

[mathman]

One of the earliest (maybe the earliest) papers, showing that there is a reality to Planck's idea that radiation is quantized, was Einstein's 1905 paper explaining the photoelectric effect. Einstein got the Nobel prize for it.

 

[01030312]

'andrewr'
Thank you, Very helpful comments of yours. I looked back at Einstein's paper and my understanding of stress-energy tensor, and here is what I would like to say-
Einstein related proportionality of energy and frequency by considering that both are time components of four vector(transform similarly). So clearly Planck's constant could have been a scalar function instead of a constant. Well, I tried to understand this relation in classical electrodynamics and tried my best to remove quadratic term-

Consider a space without charges, with radiation; charges are far away. So A_z being solutions to d'Alembertian are equal to \sum C_zn exp(ink^sx_s), and n is integer.

When C are constant we get k^sk_s =0. So we take amplitudes to be functions of n only. We can easily calculate stress-energy tensors from this-
T_mn=k_mk_n \sum H_q exp(iqk^sx_s) . H is a quadratic function of C, q is integer
Now four-energy density of this system is T_mn uⁿ hence equal to k<sub>m *(kⁿun)*other terms. uⁿ is four velocity of observer.

This is still quadratic. We can assume radiation to be in localized form by choosing correct H. We take Fourier transform of H to be G(k^mxm), in short G(kx). Now we integrate the above equation along world line of observer. Differential element on right hand side is d(kx). Since G is localised, the integral with kx will be independent of boundary and of k, for sufficiently large region of integral, and hence a finite constant.

Now on left hand side, we have integral of energy density function along world line of observer. Now along the world line, the observer is carrying an infinitesimal 3-volume. Thus other that the point where photon world-line and observer world-line intersect, there is no contribution to integral. Thus integral is the energy observed by the observer, when photon is infinitesimally close to it.

Hence we finally get after integration, Em proportional to km. So photon energy could be as considered above. Comments expected for possible logical mistakes.</sub>

 

[andrewr]

'andrewr'
Thank you, Very helpful comments of yours. I looked back at Einstein's paper and my understanding of stress-energy tensor, and here is what I would like to say-
Einstein related proportionality of energy and frequency by considering that both are time components of four vector(transform similarly). So clearly Planck's constant could have been a scalar function instead of a constant. Well, I tried to understand this relation in classical electrodynamics and tried my best to remove quadratic term-

You're welcome.
I haven't pulled the equation apart by Einstein very well for study myself -- his notation isn't what I am used to in that paper; my background is an EE, and generally simple phasor notation is used in my field of study along with PDE/ODE formulations. Four vectors, tensors, and Minkowski space are not something I work with in general -- and so I won't really be able to critique what you are arguing from that perspective -- but will be rather slow and error prone in working with them. I am happy to show you how quickly I get lost...

Consider a space without charges, with radiation; charges are far away. So A_z being solutions to d'Alembertian are equal to \sum C_zn exp(ink^sx_s), and n is integer.

OK; so I think you mean...
A vector "potential" -- A_z -- which is equivalent to the combined E and B fields, but with extra information not contained in these two fields. (Guage invariance issue/see Aharonov-Bohm effect).

The d'Alembertian is the wave operator: \frac{\partial}{\partial t}+\nabla
C_zn, point potential vectors of A_z space -- is a Fourier like coefficient for the n^th wave in a group/packet, typically a constant as variation is taken into account by summation of differing wavelengths in a Fourier series. Since n is an integer, your approach appears to be a harmonic series rather than a continuous distribution (integral wave packet).

e^{\jmath n\left[\begin{array}{c}<br /> k^{0}\\<br /> k^{1}\\<br /> k^{2}\\<br /> k^{3}\end{array}\right] \cdot \left[\begin{array}{cccc}<br /> x_{0} &amp; x_{1} &amp; x_{2} &amp; x_{3}\end{array}\right]}

When C are constant we get k^sk_s =0. So we take amplitudes to be functions of n only. We can easily calculate stress-energy tensors from this-
T_mn=k_mk_n \sum H_q exp(iqk^sx_s) . H is a quadratic function of C, q is integer
Now four-energy density of this system is T_mn uⁿ hence equal to k_{m *(kⁿun)*other terms. uⁿ is four velocity of observer.}

... and then a miracle happened ...
You lost me; the notation's connection with the text escapes my brain ...

T^μν would be Einstein notation for an electromagnetic stress energy tensor.
http://en.wikipedia.org/wiki/Electromagnetic_stress-energy_tensor
T_mn seems to be the m'th and n'th element of a matrix named T -- presumably the stress energy tensor you mentioned at the start of the post.

hmmm...
k^sk_s =0
that looks like:
\left[\begin{array}{c}<br /> k^{0}\\<br /> k^{1}\\<br /> k^{2}\\<br /> k^{3}\end{array}\right] \cdot \left[\begin{array}{cccc}<br /> k_{0} &amp; k_{1} &amp; k_{2} &amp; k_{3}\end{array}\right]=0

So the sum of the individual k components squared is zero. Dot product is zero...
If I were to take an educated guess, I would think that the negative value assigned to the squared time element (element 0?) in each vector and the positive values of the positions might cancel;

I am missing the postulates needed to work out why a constant C_zn implies zero for what looks to me to be an inner product, and I don't have a general relativity background which is the usual home to tensor arguments. (I am not sure general relativity is even valid... but keeping an open mind.)

This is still quadratic. We can assume radiation to be in localized form by choosing correct H. We take Fourier transform of H to be G(k^mx_m), in short G(kx). Now we integrate the above equation along world line of observer. Differential element on right hand side is d(kx). Since G is localised, the integral with kx will be independent of boundary and of k, for sufficiently large region of integral, and hence a finite constant.

Now on left hand side, we have integral of energy density function along world line of observer. Now along the world line, the observer is carrying an infinitesimal 3-volume. Thus other that the point where photon world-line and observer world-line intersect, there is no contribution to integral. Thus integral is the energy observed by the observer, when photon is infinitesimally close to it.

Hence we finally get after integration, E_m proportional to k_m. So photon energy could be as considered above. Comments expected for possible logical mistakes.

I wish I could help more, but there is too much mathematical formalism to learn for me to be of any practical help at this particular level of argumentation. You would essentially have to explicitly write out all the things which your notation takes for granted wasting the shorthand value of the expressions as I am merely at the linear algebra and multivarirate calculus, PDE, and ODE level and would require baby-steps to follow your argument.

Special relativity is something I am able to grasp well, but general relativity requires me to accept certain coincidences as proof of identity -- and I am not sure I am ready to do that. As Dr. Young used to point out -- the gravitational field of any object is radially oriented and not linear like the elevator gedanken that Einstein bases the transition from acceleration to gravity upon; although one can warp space to linear-ize acceleration -- other possibilities exist.

I think he meant (but let me not put words in his mouth and realize this is my interpretation) that acceleration is not necessarily a linear phenomena in and of itself even if "forces" act on it in one dimension (or two if you count time) the acceleration is not necessarily restricted to those two dimensions;

One question I would have for you is whether you think that the proportionality constant has different values in different places based on your analysis, eg: that a spatial or temporal integration averages the variance out, but that it does exist in practice/possibility -- such that \hbar is not truly constant but an average of some kind.

Best wishes.
--Andrew.

 

[01030312]

OK; so I think you mean...
A vector "potential" -- A_z -- which is equivalent to the combined E and B fields, but with extra information not contained in these two fields. (Guage invariance issue/see Aharonov-Bohm effect).

The d'Alembertian is the wave operator: \frac{\partial}{\partial t}+\nabla
C_zn, point potential vectors of A_z space -- is a Fourier like coefficient for the n^th wave in a group/packet, typically a constant as variation is taken into account by summation of differing wavelengths in a Fourier series. Since n is an integer, your approach appears to be a harmonic series rather than a continuous distribution (integral wave packet).

d Alembertian should be \frac{\partial ²}{\partial t²} + \nabla ²
(superscript notation is not appearing in this code)
There is not much to say if the vector potential has continuous or discrete(in your terms harmonic) expansion, both work here well. Finally I am taking its fourieur transform G(kx).

e^{\jmath n\left[\begin{array}{c}<br /> k^{0}\\<br /> k^{1}\\<br /> k^{2}\\<br /> k^{3}\end{array}\right] \cdot \left[\begin{array}{cccc}<br /> x_{0} &amp; x_{1} &amp; x_{2} &amp; x_{3}\end{array}\right]}



... and then a miracle happened ...
You lost me; the notation's connection with the text escapes my brain ...

T^μν would be Einstein notation for an electromagnetic stress energy tensor.
http://en.wikipedia.org/wiki/Electromagnetic_stress-energy_tensor
T_mn seems to be the m'th and n'th element of a matrix named T -- presumably the stress energy tensor you mentioned at the start of the post.

hmmm...
k^sk_s =0

Last statement simply means that k_m is a component of null-vector, and hence represents an entity generally termed as photon.

I am missing the postulates needed to work out why a constant C_zn implies zero for what looks to me to be an inner product, and I don't have a general relativity background which is the usual home to tensor arguments. (I am not sure general relativity is even valid... but keeping an open mind.)

One question I would have for you is whether you think that the proportionality constant has different values in different places based on your analysis, eg: that a spatial or temporal integration averages the variance out, but that it does exist in practice/possibility -- such that \hbar is not truly constant but an average of some kind.

Its easy now, take flat space-time,so take C_n and k_n constants, and plug it in d' Alembertian of vector potential = 0 equation. Now rest part is just a bit of this and that, and also a simple argument that u can measure photon's energy when it is infinitesimally close to you (I am actually scared of using 'photon' here, since even you know that no one understands it well. But still hope you know what I mean). Final answer says that proportionality value is constant. Einstein in his paper showed that given a rest frame and a moving frame, energy of radiation and frequency both change in similar way. So they could be related. Sure they were proportional to each other, but I guess it did not settle very well if their ratio was a universal constant.
And finally the proof I have given above, even if it is true, should not be taken at all as an important construction. I am sure photon is far more mysterious and fundamental and an elegant theory is what is needed to identify its property, not a fixing-the-equations technique to get energy value.

Special relativity is something I am able to grasp well, but general relativity requires me to accept certain coincidences as proof of identity -- and I am not sure I am ready to do that. As Dr. Young used to point out -- the gravitational field of any object is radially oriented and not linear like the elevator gedanken that Einstein bases the transition from acceleration to gravity upon; although one can warp space to linear-ize acceleration -- other possibilities exist.

I think he meant (but let me not put words in his mouth and realize this is my interpretation) that acceleration is not necessarily a linear phenomena in and of itself even if "forces" act on it in one dimension (or two if you count time) the acceleration is not necessarily restricted to those two dimensions;


Best wishes.
--Andrew.

Here, I would like to say what I approximately get from general relativity. Einstein saw that since all bodies fall in same way in gravity as in accelerating frame, so gravity and acceleration are equivalent. Then he went further to derive his equations based on principle that all frames you choose for measurement are equivalent. The description of gravity in one frame should be present in other frames too. Now a freely falling person in uniform gravity finds gravity to vanish. And suppose there is a matter which can produce uniform gravity. This means uniform gravity does not exist, since the description of gravity in new freely falling frame has zero as its value. So such matter does not actually produce gravity.

Thus gravity is something else, some effect which is present even if we are freely falling. There Einstein might have realized that in case of a person falling towards earth, along with a ball, he would find an effect of gravity on ball- the effect being acceleration of ball towards him, since both are moving radially, so they can't be parallel. Now there is a different picture of gravity- gravity is the ability to cause two freely falling bodies to converge towards each other, the same way as on a curved surface, parallel 'straight' line come close to each other. Then came the story of curved surfaces and geometries. I humbly suggest you to go through the theory of manifolds, a person capable of perspective like yours deserves to know these ideas.

I am not very sure about the complete behaviour of acceleration, but I think Einstein's approach using geometry is really elegant. The question that which theory is completely applicable to nature seems quite a restriction on science, as every theory is mathematically complete in itself. A complete theory will identify every structure which is capable of existing( Not talking about parallel universes, that idea seems very weak to me, no offense).

 

[andrewr]

d Alembertian should be \frac{\partial ²}{\partial t²} + \nabla ²
(superscript notation is not appearing in this code)
There is not much to say if the vector potential has continuous or discrete(in your terms harmonic) expansion, both work here well. Finally I am taking its fourieur transform G(kx).

Ahh ... you're right -- I downloaded a program Lyx after trying several other Latex typesetting programs to speed up my attempts at easily readable Latex, OFFLINE (and finding them a waste of a whole day and night) I upgraded several things on my computer that have been irritating me for several months as well esp. in the area of scientific python bug fixes. In the excitement of finally getting everything to compile and at least pretend to run -- I forgot the square term in the wave equation...
The only thing now, is that I have to press the reset button and find out if I undermined the foundation of my OS or not -- and if I am going to be spending the next week trying to figure out where I went wrong. Its a rare thing for me to fail at an upgrade -- but if I do -- it is a big enough hassle that I am downright scared to reboot and am tempted to await the Ben-Franklin style power failure to find out the truth...

Its easy now,

take flat space-time,so take C_n and k_n constants, and plug it in d' Alembertian of vector potential = 0 equation.

Flat space time:
Given:\text{η=\ensuremath{\left[\begin{array}{cccc}<br /> -1 &amp; 0 &amp; 0 &amp; 0\\<br /> 0 &amp; 1 &amp; 0 &amp; 0\\<br /> 0 &amp; 0 &amp; 1 &amp; 0\\<br /> 0 &amp; 0 &amp; 0 &amp; 1\end{array}\right]}}

s^{2}=-c^{2}\left(\triangle t\right)^{2}+\left(\triangle x\right)^{2}+\left(\triangle y\right)^{2}+\left(\triangle z\right)^{2}
OK, that makes a bit more sense.

Now rest part is just a bit of this and that, and also a simple argument that u can measure photon's energy when it is infinitesimally close to you (I am actually scared of using 'photon' here, since even you know that no one understands it well. But still hope you know what I mean). Final answer says that proportionality value is constant. Einstein in his paper showed that given a rest frame and a moving frame, energy of radiation and frequency both change in similar way. So they could be related. Sure they were proportional to each other, but I guess it did not settle very well if their ratio was a universal constant.

Hmmm ... considering I don't know how wide a photon is -- exactly -- being infinitesimally close requires a bit of imagination... but that's the nature of calculus and belief in space divisions which pre-date Heisenberg's monkey wrench. (or homonid wrench for the true Darwinists...)

And finally the proof I have given above, even if it is true, should not be taken at all as an important construction. I am sure photon is far more mysterious and fundamental and an elegant theory is what is needed to identify its property, not a fixing-the-equations technique to get energy value.

Wisdom.

Don't underestimate, though, that the mathematics might hide clues as to how to interpret the units of Planck's constant. It is interesting that in my original attempt, I discovered that the interference of two waves of dissimilar wavelengths is what an outside observer would see when watching an wavelength measuring experiment that is moving relative to their observation point. It is also interesting to note, that the correct relatively shifted wavelength for Lorentz transformation, then, depends on at least two waves with different energy values and the "node" points (length contraction) are only correctly computed by using this difference.

When I think about energy density as you have brought it up -- I am immediately struck by the single direction of propagation which is involved in the equation as one difference. But for that energy to be realized/measured, a reflection or multi-reflection trapping of the wave energy must occur. I am also have vague thoughts starting to stir regarding the analogy between spatial frequency (wave number) and Energy, versus temporal density (frequency) and power. There is no such thing as conservation of "power" unless all the power is accounted for as "energy" -- yet I am beginning to wonder if a similar problem might lurk in the idea of Energy as (spatial) density. The though is ill formed yet.

Here, I would like to say what I approximately get from general relativity. Einstein saw that since all bodies fall in same way in gravity as in accelerating frame, so gravity and acceleration are equivalent.

But, that's where my prof Dr. Young of the progeny of the double slit would box my ears if I quoted Einstein's argument -- I observed him several times derailing the argument with a chuckle. His POV was that given a macroscopic body -- gravity angularly acts on different parts of it differently.
There is no plane gravitational field unless one believes in infinitesimally thin objects.
Place three objects linearly side by side in a real field and at least two will be at different radii from the item attracting them. It is at most a crude approximation to say that acceleration of three bodies side by side in the same field would be the same -- for although the magnitude may be the same (excluding or computing out their attraction to one another) the direction is not the *exactly* the same.

So although there may indeed be a partial equivalence between the two ideas in their effects -- linear acceleration and gravitation experience on Earth -- there is also a distinct and measurable difference which one must "choose" how to iron out mathematically.

Then he went further to derive his equations based on principle that all frames you choose for measurement are equivalent. The description of gravity in one frame should be present in other frames too. Now a freely falling person in uniform gravity finds gravity to vanish.

In a sense, a linear body oriented at 90 degrees to the center of gravity attraction point located R distance away would experience extra compression or expansion along its length because the vectors of attraction for the most extended points on the rod are at angles to each other and the attraction point:
In order to avoid the problem -- one needs to demand that the source of attraction be infinitely long and uniformly distributed such that there is no "center" of mass for the attracting object.
A infinite (and of course planar) line weight would suffice, one does not need a "sheet" gravitational source so that the gravitational field can decrease in any plane intersecting the line -- yet be totally uniform along the plane.

However, this type of creation is highly abnormal in actual space. The "blob" of the Earth and friends are discreet masses spread out on average, perhaps, but clearly not in a fashion capable of generating a truly planar attraction.

I would diagram the problem for a finite width rod being attracted in a real universe as:
\frac{rod}{\searrow\frac{\downarrow}{attractor}\swarrow}

Clearly, the stronger the gravitational pull of the attractor -- the more the rod will experience tangential forces to the average attraction vector.
Since the field from a point gravitational source (as planets are computed in Kepler motion, even though they are really finite width) is allowed for computation -- I would even expect the effect to be measured for arbitrarily large attracting masses, so long as they are not "infinitely" large and perfectly uniform.

And suppose there is a matter which can produce uniform gravity. This means uniform gravity does not exist, since the description of gravity in new freely falling frame has zero as its value. So such matter does not actually produce gravity.

From your statement (assuming free-fall of course!), you already see what I have stated and in your very statement you have cinched the problem -- gravity in a real universe always has effects which show up whereas the gedanken does not. So, I am looking forward to where you will go with it.

Thus gravity is something else, some effect which is present even if we are freely falling. There Einstein might have realized that in case of a person falling towards earth, along with a ball, he would find an effect of gravity on ball- the effect being acceleration of ball towards him, since both are moving radially, so they can't be parallel.

Correct, they obviously will drift towards each other in the tangent direction, and thus if they know not about the attractor in common to them both -- they may well assume that they have more mass than they actually do, or perhaps that the gravitational acceleration constant is slightly larger than it really is.
A mathematical analysis of deviation of the error v. time/distance would be interesting to determine to what extent one could experimentally separate the error from reality, and the risk assessment that our own measurements of gravity might be in slight error due to such an effect.

Now there is a different picture of gravity- gravity is the ability to cause two freely falling bodies to converge towards each other, the same way as on a curved surface, parallel 'straight' line come close to each other. Then came the story of curved surfaces and geometries. I humbly suggest you to go through the theory of manifolds, a person capable of perspective like yours deserves to know these ideas.

My brother would like you very much -- he's a math major... loves words like manifold, lie fields, galois, etc. I always am annoyed to learn I pronunciate these things wrong... darn French...
But thank you for not clobbering me, it seems most other threads I have managed to hit someone on a really bad day... rest assured, if my health improves I probably will eventually learn more about these things, it is just a matter of time and life being quantized...

I am not very sure about the complete behaviour of acceleration, but I think Einstein's approach using geometry is really elegant. The question that which theory is completely applicable to nature seems quite a restriction on science, as every theory is mathematically complete in itself. A complete theory will identify every structure which is capable of existing( Not talking about parallel universes, that idea seems very weak to me, no offense).

I am not totally certain what you mean, here, but I do recognize that there is beauty in correctly expressed mathematical ideas which even if they are applied wrong, hold much learning for how to solve similar problems which are analogical.

This post is going long, and I think perhaps a little dabbling tomorrow on an alternate path, eg: the Schrodinger eqn. non-relativistic, might also hold a few gems for the question of what the \hbar is quantization. Both the square law 1/2mV**2 (T or KE) and E=hf are found in the same equation -- and although it is a somewhat artificial equation, none the less it is the classical limit of relativity at slow speeds and as one can adjust classical theory for relativity by adjusting (eg: mass), perhaps that would be useful conceptually -- it's a gamble, but we'll see what comes of it.

--Andrew.

 

[andrewr]

Sorry about the slow second response; I've been thinking about what I wanted to say on the Schrodinger equation side of E->ν² vs. E=\hbarν . with the keep it simple philosophy -- and the more I looked and thought the more questions I had myself...

I finally realized that the Schrodinger equation gives the same result as classical physics; just in another form, and that is pretty messy to show here. So I am not going to distinguish a photon's dispersion from that of a neutron for what follows since basic Schrodingers is not relativistically corrected... but you can work out the details if it interests you for light to see if my concept still holds, and I think it must even if minor modifications are made.

Essentially, the Schrodinger equation evolves the probability of a state in time based on a phase velocity being proportional to momentum. The actual phase velocity is determined by frequency -- which is found from E=hf; f= E/h. however, the kinetic energy of such phase motion is simultaneously interpreted as square law -- E=p**2/2m; for the definition of energy in the Hamiltonian takes the momentum (a proportional velocity) of E=hf and squares it to compute T (KE) which is at first sight incompatible. However, phase velocity is not the same as particle velocity. In order to compute a particle velocity, at least two frequencies would be required (and most would argue a gaussian); so similar to my comment regarding wavelength in relativity, and the Lorentz length contraction of a moving object requiring DIFFERENT energies interfering with each other to produce the correct length -- and NOT a physical length change like one would think of a solid "stick"; Thus the two formulations arrive at the same conclusion; In order to call something to be "moving" more than one frequency MUST be involved for any photonic wavelength measurement to be made.

(Think about how one measures frequency and wavelength -- it's a chicken egg problem...)
Then consider how "wavelength" is used as a measuring stick of things in other inertial reference frames.

I don't think I totally understood your original question now that I have been looking things up, but I have some more observations that might be useful.

In classical physics, for example --doing wave on a string problems, the amplitude causes a stretching of the string which means energy is proportional to the square of the stretch. As one raises the frequency, it has the same effect of increasing the SLOPE dy/dx which stretches the string more -- and thus also is square law increase in energy *because* it requires doubling the stretching of the string to double the frequency. Consider the total stretched "length" of the string with a constant amplitude but varying frequency on a fixed length of string to understand what I mean.

So that matches with your original question; but there's a catch.

In space -- say in a Hertzian/photon wave -- the energy is given by the Poynting vector which is Voltage x Current analogy; eg: for orthogonal values -- V/m * A/m -> watts/m².
That's not exactly a "density" for the meter measure is misleading and the result is not watts per volume anyhow -- but since E and H are scalar multiples in free space -- and the Pointing vector is:

\overrightarrow{P}=\overrightarrow{E}x\overrightarrow{H}

there is no frequency dependency of energy predicted at all in electromagnetics; which is how the question usually came up in class several years ago -- although the teacher never answered it since it was out of scope.

Energy stored in either of these fields goes as square law with amplitude -- just as in the case of the string; Energy in an inductor is L/2I² {L=inductance/equiv. permeability in space}, and energy in a capacitor is C/2{U,V,E...}² (C=capacitence/equiv. permittivity ε₀). Yet when computing the energy in a wave, one usually multiplies the B by any orthogonal H which is still square law -- but which would give the wrong energy for a string if one were to equate analogically E with displacement and H with velocity/momentum. It is a change in EM wave voltage which matters to the correct analogy.

What is superficially disturbing is that although the individual elements share analogical characteristics for transmitting energy (square law in both terms); the results are different. To be sure, in an EM field the units are amps/meter and volts/meter, not amps and volts; nor joules(stored in tension) and joules(stored in momentum) as in the case of the string, so the final result is watts/square meter and not joules/string length. (As I mentioned in a previous post, one also has to be careful for there is no conservation of "Power" just energy.)

However, there does appear to be enough information to make an educated guess as to how the EM field and the physical string might relate. The potential energy in the string is the problem, for it is the difference in position and thus tension which is being measured.

On a string, the displacement dy/dx is squared and added to 1 to compute the "lengthening factor" of a stretching string beyond a "flat" one. There is no lengthening without a string to tension -- and since the amplitude of the string depends on this lengthening factor, the amplitude of the EM wave does not correspond to the amplitude on a string. If one were to compute an equivalent string amplitude for an EM wave, they would first have to look at the potential energy (E field or V/m) as proportional to the length of the string at that location; Since spring energy (hooke's law) goes as k*displace**2, where the rest length is taken as zero energy, so k* ( stretchedlen - normlen )² = energy stored in waveform of a string. But the differential deflection length is simply the differential hypotenuse of a dx/dx by dy/dx right triangle reduced by dx/dx.

Since the voltage field has to correspond to that deflection, an equivalent string waveform could be computed in theory; but the transformation of sine wave energy in one form does not correspond to a pure sine wave in the other form. I think the integral formed is called elliptic -- and it isn't worth solving analytically.

But, surprise, an integral of a E field corrected to be string-like is guaranteed to reduce the amplitude of the answer as frequency increases because area under the curve decreases. Since a string wave's energy goes as amplitude**2 times frequency**2, The drop in energy due to amplitude loss is offset by an increase in energy due to frequency.

The bottom line is that frequency squared energy dependence is a facet of the representation. Both methods agree in that the amplitude squared is equivalent to energy; it is just means a different amplitude -- in one form, the amplitude is frequency dependent -- in the other it is not.

The (coulomb field/displacement) current waveform produces another issue (B/H field in EM) -- for it is kinetic energy analogue; and in the string problem kinetic energy has nothing to do with string tension so it transforms without functional change and is a good analogy. however, that also means that the energy stored in momentum will have nothing to do with *frequency* even on a string -- which is something I had never noticed before and surprised me at first. The momentum picks up the frequency dependence of energy from the stretched string transferring energy -- not because there is any intrinsic square law dependency on frequency built into the momentum of a string, or EH field.

In some books an approximation is used for the string which ignores the rest tension (assumes it is zero), and avoids the elliptic integral that way; so look carefully when comparing notes.

Well, that exhausts the amount of effort I can apply to that problem for a while, unless I need to make some corrections... (hopefully not). It was fruitful for me to go through that -- definitely warms me up for a problem I am tackling.

--Andrew.