Is the energy of a burst of light the sum of the energy of each photon?

[Aaron121]

In A.P. French's Special relativity the author said,

We suppose that an amount $E$ of radiant energy (a burst of photons) is emitted from one end of a box of mass $M$ and length $L$ that is isolated from its surroundings and is initially stationary. The radiation carries momentum $E/c$.

The mass and length of the box are irrelevant here.

He said the momentum of the radiation is $E_{radiation}/c$. We know that the momentum of a single photon with energy $E_{photon}$ is $p_{photon}=E_{photon}/c$.

So is $E_{radiation}$ the sum of the energy of each photon, $E_{photon}$?

 

[PeroK]

In A.P. French's Special relativity the author said,
The mass and length of the box are irrelevant here.

He said the momentum of the radiation is $E_{radiation}/c$. We know that the momentum of a single photon with energy $E_{photon}$ is $p_{photon}=E_{photon}/c$.

So is $E_{radiation}$ the sum of the energy of each photon, $E_{photon}$?

If you take a photon to be the basic "packet" of energy associated with the radiation, then why wouldn't the total energy be the sum of all those packets?

 

[Orodruin]

In A.P. French's Special relativity the author said,

Based on your earlier thread, this is not a good textbook and is quite outdated.

He said the momentum of the radiation is $E_{radiation}/c$.

This is true only if all of the photons are emitted in the same direction.

 

[Aaron121]

@Orodruin I wouldn't qualify the book as "not good", it is even still used today at MIT.

@PeroK

then why wouldn't the total energy be the sum of all those packets?

The total energy $E_{radiant}$ is the energy of the electromagnetic wave (light pulse for example). We know for example the energy of a photon depends only on the frequency, so varying the power of the source without changing the frequency won't change the photon's energy. That's essentially why I wasn't sure we could consider the energy of the electromagnetic pulse (viewed as disturbance)to be the sum of the energy of each photon in the pulse. I don't really have a background in quantum mechanics.

 

[Orodruin]

@Orodruin I wouldn't qualify the book as "not good", it is even still used today at MIT.

This is an argument from authority. As such it should mean less to you than the fact that it has been shown to contain directly false and misleading claims.

That's essentially why I wasn't sure we could consider the energy of the electromagnetic pulse (viewed as disturbance)to be the sum of the energy of each photon in the pulse. I don't really have a background in quantum mechanics.

In SR, either you consider electromagnetic radiation as a wave or as a set of "classical photons" (if I remember correctly, it was pointed out in the other thread that this notion does not really make sense). Do not try to do both. Actually describing photons is one of the more complicated stories in introductory quantum field theory. A classical electromagnetic wave does not consist of a fixed number of photons.

Instead, it is cleaner to consider a "light pulse", which in itself has energy and momentum but is not a classical particle.

 

[Aaron121]

@Orodruin

A classical electromagnetic wave does not consist of a fixed number of photons.

Small side question, does this mean the EM wave has an infinite number of photons? If this is the case, and we assume that EM wave's energy is the sum of the energy of the photons, that would mean the EM wave has infinite energy, wouldn't it?

a "light pulse", which in itself has energy and momentum but is not a classical particle.

So the relation $E=pc$ applies to both photons and light pulses?

 

[Orodruin]

Small side question, does this mean the EM wave has an infinite number of photons?

No. It means exactly what I wrote. It does not have a definite number of photons. It is not an eigenstate of the photon number operator.

So the relation E=pcE=pcE=pc applies to both photons and light pulses?

It is relatively easy to confirm that the plane EM wave has that relationship between energy and momentum, so yes.

 

[PeroK]

I wouldn't qualify the book as "not good", it is even still used today at MIT.

That is very interesting. I did, however, find this on the MIT website:

http://web.mit.edu/8.701/www/Lecture Notes/8.701originsOfMass04FA13.pdf

"The early literature of relativity employed some compromise definitions of mass – specifically, velocity-dependent mass, in both longitudinal and transverse varieties. Those notions have proved to be more confusing than useful. They do not appear in modern texts or research work, but they persist in some popularizations, and of course in old books. "

At the very least, if you want to continue your studies using relativistic mass, then I suggest you adopt the notation:

$m_0$ for the rest mass

$m_R = \gamma m$ for the relativistic mass.

The convention on PF is that $m = m_0$. If you use $m$ instead of $m_R$ then you will generally cause confusion and ambiguity.

 

[Orodruin]

it is even still used today at MIT.

The "today" is also not really clear from that page, which is a course page from 2006 by the looks of it.

 

[vanhees71]

This is an argument from authority. As such it should mean less to you than the fact that it has been shown to contain directly false and misleading claims.In SR, either you consider electromagnetic radiation as a wave or as a set of "classical photons" (if I remember correctly, it was pointed out in the other thread that this notion does not really make sense). Do not try to do both. Actually describing photons is one of the more complicated stories in introductory quantum field theory. A classical electromagnetic wave does not consist of a fixed number of photons.

Instead, it is cleaner to consider a "light pulse", which in itself has energy and momentum but is not a classical particle.

I also do not understand, why one frequently uses arguments involving "photons", where all is in fact within classical physics. Usually this mixture of quantum and classical thinking leads to confusion and even wrong conclusions. I don't know this particular book by French very well, but I also find it suspicious to use a quantum-fieldtheoretical argument in a completely classical situation.

Of course it depends very much on the context (i.e., the quantum state), whether the total radiation energy is the sum of single-photon energies. That's the case for Fock states of non-interacting photons. Already for thermal radiation or coherent states it's no longer the case, because these are not states of a definite photon number, and it doesn't even make sense to consider such a state as a "stream of particle-like photons".

Classical electrodynamics is the paradigmatic example for a classical relativistic field theory, and everything within the classical realm (and that's quite a lot!) can be derived very well from it, including the (admittedly somewhat subtle) issue of the fundamental conservation laws for energy, momentum, and angular momentum.

 

[Ibix]

@Orodruin I wouldn't qualify the book as "not good", it is even still used today at MIT.

I've certainly seen professors with enough seniority to choose what course they teach (whatever anyone else thought) teaching things the way they wanted (whatever anyone else thought). Politics...

Edit: I wasn't at MIT and I don't know the prof there. But argument from authority is weak, as Orodruin says.

 

[Aaron121]

@PeroK For French's defense, the distinction between relativistic mass and rest mass is clearly laid out when he said, in page 23,

Many physicists prefer to reserve the word mass to describe the rest mass $m_{0}$, a uniquely defined property of a given particle. But this is essentially a matter of taste. Whatever words one elects to use, there is no disagreement on the fact that $\textbf{p}=m(v) \textbf{v}$ and $E=m(v)c^{2}$ describe the momentum and total energy of a particle, where $m(v)$ is given $m(v)= \frac{m_{0}}{(1-v^{2}/c^{2})^{1/2}}$.

The denominator $(1 — v^{2}/c^{2})^{1/2}$ appears so often in special relativity, and is so awkward to write, that nearly all discussions of relativity make use of a single symbol, $\gamma$, defined as follows.
Put
$\tag{1-22} \gamma(v)=\frac{1}{(1 — v^{2}/c^{2})^{1/2}}$
Then we have
$\tag{1-23}m = \gamma m_{0}$
$\tag{1-24}\textbf{p} = \gamma m_{0}\textbf{v}$
$\tag{1-25} E = \gamma m_{0}c^{2}$
where in using Eqs. $(1-23)$ to $(1-25)$ we must remember that $\gamma$ depends on the speed $v$ according to Eq. $(1-22)$.

 

[Vanadium 50]

@Aaron121, why did you ask us for advice if you didn't want to listen to it?

French is confusing you. Don't want to be confused? Stop using it. Same with popularizations.

 

[Orodruin]

I wasn't at MIT and I don't know the prof there.

Browsing the course page, the lecturer seems to have been Max Tegmark.

I also do not understand, why one frequently uses arguments involving "photons", where all is in fact within classical physics.

I understand it from the point of view that it is an easy illusion to create in students’ minds. In many respects, it is easier to talk about classical zero-mass particles and put that mental image in the students’ heads than to do a full field theory treatment. I usually substitute ”photon” for ”light pulse”, but admittedly I slip sometimes because of the ”easy” narrative.

 

[PeroK]

@PeroK For French's defense, the distinction between relativistic mass and rest mass is clearly laid out when he said, in page 23,

In French's defence, his book is 50 years old. The issue for you is that if you use an old book and things have moved on then you need to be aware of this.

 

[Dale]

For French's defense, the distinction between relativistic mass and rest mass is clearly laid out

Then why are you so confused?

Regardless of how MIT feels about the book the evidence here is that it is not working well for you personally. And it doesn’t work well with our conventions here. If we are to continue serving as your backup (which we are eager to do) then you will be better served with a book that matches our conventions. If you wish to continue using French then perhaps someone at MIT would be a better choice of backup.

 

[Dale]

In French's defence, his book is 50 years old.

I have seen some places actually try to teach directly from the seminal papers, which is even worse. I think they must rely on truly gifted teachers and outstanding students to compensate for the inherent handicap of such an outdated approach.

 

[kent davidge]

Despite being an old book, it is peer-reviwed. If it contains misleading concepts/teachings, does that mean that those very knowledgeable men who reviewed the book at the time were also misled by it? after all they approved it

ps: this is really a question; I'm not intending to mean anything else.

 

[PeterDonis]

Despite being an old book, it is peer-reviwed. If it contains misleading concepts/teachings, does that mean that those very knowledgeable men who reviewed the book at the time were also misled by it?

Short answer: yes.

A longer answer would be way off topic for this thread; if you really want to go into how scientific understanding evolves over time and how that eventually gets reflected in textbooks, you should start a separate thread (and probably in a different subforum of PF).

 

[Orodruin]

Despite being an old book, it is peer-reviwed.

What makes you think that? Published does not equate to peer reviewed.

My book had a couple of chapter drafts reviewed before comissioning. There was no review except proof-reading of the language after that.

 

[atyy]

Based on your earlier thread, this is not a good textbook and is quite outdated.

Where is this earlier thread?

This is an argument from authority. As such it should mean less to you than the fact that it has been shown to contain directly false and misleading claims.

It should be major errors. There are good books that contain errors, eg. Feynman lectures.

 

[Orodruin]

It should be major errors. There are good books that contain errors, eg. Feynman lectures.

All books contain errors. It is practically impossible to write a book about errors, but a book on SR should not state that p = mc for photons.

 

[Vanadium 50]

There are some very good French texts. This is not one of them. While there are plenty of good things in it, it is an exercise in teaching SR without teaching any of the mathematical mechanics behind SR. Four-vectors are not discussed until the very end, and if invariants are discussed at all, it's only in passing - I certainly don't remember any of that.

 

[Mister T]

I seem to recall that French' textbook was at one time in widespread use and has taken a lot of the blame for perpetuating the use of relativistic mass.

 

[atyy]

I seem to recall that French' textbook was at one time in widespread use and has taken a lot of the blame for perpetuating the use of relativistic mass.

The problem is not relativistic mass, it's the discussion of inertial mass for a photon.

 

[vanhees71]

Despite being an old book, it is peer-reviwed. If it contains misleading concepts/teachings, does that mean that those very knowledgeable men who reviewed the book at the time were also misled by it? after all they approved it

ps: this is really a question; I'm not intending to mean anything else.

As I said, I don't now this book, but it seems as if it is somewhat outdated in using ideas that were outdated almost immediately after the 1st two to three papers on the subject by Einstein. Einstein himself abandoned the idea of the various speed and direction-dependent "relativistic masses" of his earliest papers about relativistic dynamics. Today we aim at expressing everything in quantities which have simple well-defined transformation properties under Poincare transformations.

Last but not least this simplifies also the generalization to the general-relativistic physics, where anything non-covariant is simply not physically interpretable.

 

[pervect]

In A.P. French's Special relativity the author said,
The mass and length of the box are irrelevant here.

He said the momentum of the radiation is $E_{radiation}/c$. We know that the momentum of a single photon with energy $E_{photon}$ is $p_{photon}=E_{photon}/c$.

So is $E_{radiation}$ the sum of the energy of each photon, $E_{photon}$?

In this context, pretty much yes. I assume that these photons aren't diffracting, so they're being treated as classical point particles. As a result, the "light" in the box will obey geometric optics, and not diffract. Which is probably the intent and not going to cause any issues in this application, though it might in others.

I do have some other concerns that are more likely to cause issues, though.

I'm not sure where the author is going with this example, since I don't have that book, A good covariant treatment of the "box of light" problem is going to need to take into account the stress in the walls of the box, something that hasn't even been mentioned. The mass and momentum of the box aren't such a concern for the covariance issue, , but the stress in the walls of the box turns out to be important in non-obvious ways. Did French ever get around to this, or wasn't it a concern of his?

 

[vanhees71]

If you deal with "light" in the usual sense you deal in terms of QED with coherent states, for which the naive photon-as-a-particle picture doesn't make the slightest sense. The photon-as-a-particle picture is almost always misleading and should not be taught anymore today. It's as bad as the confusing idea of various kinds of "relativistic masses".

 

[f95toli]

Of course it depends very much on the context (i.e., the quantum state), whether the total radiation energy is the sum of single-photon energies. That's the case for Fock states of non-interacting photons. Already for thermal radiation or coherent states it's no longer the case, because these are not states of a definite photon number, and it doesn't even make sense to consider such a state as a "stream of particle-like photons".

It is perhaps worth mentioning that the average photon number can still be a useful measure of "classical" energy in a system; at least of that system is in turn interacting with some type of "quantum system". A good example would be the energy in a resonator which is interacting with one or more two-level systems. Even if the energy in the resonator is coming from a "classical" (coherent, not very coherent, thermal...) source you can say quite a lot about the effect this will have on one or more TLS by simply converting the energy to an average photon number by dividing by hf; if <n> << 1 resonant TLS will end up in their ground state. It is essentially just classical (pre-p1920) statistical physics even if we are dealing with photons.

This might sound obscure but turns out to be quite common even in some "applied" areas (detector physics) as well as for example solid-state quantum computing where energy is almost always given as average photon number.