Interpretation of the photoelectric effect

In summary: Anyway, this thought is based on the following idea:In summary, using a simple 1-dim. model with a single electron plus time-dependent perturbation theory, we obtain the following picture: the energy of the photoelectron is proportional to the work-function (the initial energy), and the number of photoelectrons is proportional to the probability of the transition. This means that time-dependent perturbation theory of non-relativistic quantum mechanics is able to reproduce the essential characteristics of the photoelectric effect without ever mentioning light quanta.
  • #1
tom.stoer
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The photoelectric effect is usually presented as an example disproving classical electromagnetism as viable model for interaction of light with matter and as evidence of quantization of energy in the electromagnetic field, i.e. the existence of photons. I would like to discuss a thought based on non-relativistic quantum mechanics w/o and relation to Planck, Einstein etc. showing - imho - why this is not so straightforward.

Using a simple 1-dim. model with a single electron plus time-dependent perturbation theory - in the same manner as used in the derivation of the spectrum of the hydrogen atom including selection rules - we obtain the following picture:

We have
- a quantum well of finite depth -V
- a discrete spectrum for [itex]\epsilon_i < 0[/itex]
- a continuous spectrum for [itex]\epsilon_f > 0[/itex]
- a classical electromagnetic field [itex]E(x,t) = E_0\,\sin(kx - \omega t)[/itex]

Calculating the transition matrix element

[tex]M_{fi} = \langle \epsilon_f|E(x,t)| \epsilon_i\rangle \sim E_0\,\delta(\epsilon_{fi} - \hbar\omega)[/tex]

and using Fermi's golden rule we find that
- the probability is proportional to [itex]|E_0|^2[/itex]
- therefore the number of photoelectrons is proportional to that probability
- the energy of the photoelectron is [itex]\epsilon_f = \hbar\omega - |\epsilon_i|[/itex]
- therefore the initial energy plays the role of the so-called "work-function", i.e. [itex]W = |\epsilon_i|[/itex]

That means that time-dependent perturbation theory of non-relativistic quantum mechanics with a classical electromagnetic field is able to reproduce the essential characteristics of the photoelectric effect w/o ever mentioning light quanta. Therefore this effects shows clearly that the interaction of light with matter cannot be obtained from purely classical reasoning, but it also shows that Einstein's hypotheses of light quanta cannot be derived in a straightforward manner.

Replies welcome ...
 
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  • #2
Exactly! It's among the greatest sins of physics didactics to tell students this old tale about "photons". Indeed the photoelectric effect does not prove the quantization of the em. field, but it is sufficient to quantize the electrons and calculate the transition amplitudes to scattering states by irradiating the bound electron with classical em. waves, using the dipole approximation, which is sufficient for the usual experiments with visible light and some metal plate; it leads to the same results as Einstein's "heuristics paper". It's ironic that Einstein got the Nobel prize for the only work he did that's really outdated today instead for General Relativity or his work on fluctuations in statistical physics (Brownian motion and all that) which lead to the direct proof of the atomistic structure of matter.

For details concerning the photoelectric effect, see

https://www.physicsforums.com/insights/sins-physics-didactics/
 
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  • #3
Thanks for your reply!
 
  • #4
tom.stoer said:
time-dependent perturbation theory of non-relativistic quantum mechanics with a classical electromagnetic field is able to reproduce the essential characteristics of the photoelectric effect w/o ever mentioning light quanta.
Yes. This has been known since around 1965. For more see the references given here.
 
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  • #5
It was not that late. In Sommerfeld's "Atombau und Spektrallinien, Bd. II" you find an Article by Bethe as citation for the standard calculation about the photoelectric effect on atoms:

H. Bethe, Über die nichtstationäre Behandlung des Photoeffekts, Ann. d. Phys. 4, 443 (1930)
 
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  • #6
Lamb and Scully's work appeared almost simultaneously with the work by the (in)famous E.T. Jaynes (otherwise highly praised for his work on statistical thermodynamics) on the semi-relativistic theory of electrodynamics, which was perceived as a rebuttal of QED, or better stated: there's no need to quantize the electromagnetic field. Thus there's a certain reluctance in accepting and propagating semi-classical models. Therefore, textbooks still propagate some incorrect ideas.
 
  • #7
I don't think that semiclassical methods are in some way discredited. They have their application in the realm of their validity, and that's a pretty large realm. Dispersion theory is an example, which was treated very early. For those interested in the history, I can only recommend to read Sommerfeld's already mentioned marvelous textbook, which has been translated to English under the title "Wave Mechanics". It's out of print, but you can find it online ;-). It's not only historically interesting, but provides also very elegant mathematical methods in the typical Sommerfeld style (it's also not making much philosophical interpretation gibberish ;-)).

Of course, there is no doubt that the complete theory is quantum field theory (i.e., for atomic physics mostly QED), but the photoelectric effect is not the prime empirical proof for the necessity to quantize the em. field. That's rather the Lambshift of atomic spectral lines, spontaneous emission, quantum beats, etc. etc.
 
  • #8
tom.stoer said:
The photoelectric effect is usually presented as an example disproving classical electromagnetism as viable model for interaction of light with matter and as evidence of quantization of energy in the electromagnetic field, i.e. the existence of photons. I would like to discuss a thought based on non-relativistic quantum mechanics w/o and relation to Planck, Einstein etc. showing - imho - why this is not so straightforward.

Yeah. Since experiments involving light typically involve matter as well (you can't really do much with just light), it seems that it's really hard to show the quantum nature of the electromagnetic field independently of the quantum nature of matter.

But historically, Einstein was trying to explain the photoelectric effect before there was a quantum theory of electrons, so your analysis would not have occurred to him, of course.
 
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  • #9
stevendaryl said:
But historically, Einstein was trying to explain the photoelectric effect before there was a quantum theory of electrons, so your analysis would not have occurred to him, of course.
of course :-)
 
  • #10
Well, but Planck was insisting (also wrongly in view of modern QED) that the quantum nature in the sense of quantization of energy ("##E=h \nu=\hbar \omega##") is only in the absorption and emission of em. radiation with matter. Ironically that's in a very good approximation correct for the photoelectric effect but not for the black-body radiation, he has discovered, because for the latter you need spontaneous emission, which cannot be explained without quantizing the em. field. That's why Einstein discovered spontaneous emission in 1917 in his famous analysis of the Planck Law in terms of kinetic theory.
 
  • #11
My statement ist not that the quantum interpretation is not correct or does not answer these questions; it's simply the fact that the quantum hypothesis for light is sufficient, but not necessary.

The questions is, if our educational approch shall follow the historical course - including all detours - or if we should (carefully) depart and find a more objective perspective.
 
  • #12
I think the semi classical approach is good pedagogy, new students should be exposed to QED, perhaps for some elite students?

BTW, great analysis.
 
  • #13
I find the historical approach confusing from my own experience learning QT. In high school we got the naive photon picture, applied to the photoelectric effect and Compton scattering, then (worst of all) the Bohr-Sommerfeld model of atomic structure. Fortunately we had a very good physics teacher, who then told us that all this is outdated and now substituted by modern quantum mechanics, and then she taught us the basics of the wave-mechanics approach, including solving the Schrödinger equation for the rigid box, the harmonic oscillator, and (qualitatively) the hydrogen atom. For me the greatest obstacle for at least a beginning of an understanding what all this means was to unlearn the "old quantum mechanics", which we had to learn before. There's no merit in learning outdated "old QT".

Of course, one has to start with non-relativistic QT ("1st quantization"), treating the em. field classically. You come very far with that, and it's mandatory to get an understanding of this approximation first, before one can indulge into learning relativistic QFT. Again, I'd not teach oldfashioned relativistic QM a la Bjorken&Drell vol. I, because it's more confusing than helpful; relativistic QT today is relativistic local QFT as needed to understand the Standard Model. I once gave the lecture QM2, which usually includes non-relativistic many-body theory and an introduction to relativistic QM. I didn't do that, but taught non-relativistic many-body theory already as quantum field theory ("2nd quantization") and then introduced relativistic QT as QFT, up to the canonical quantization of the em. field and QED including the usual tree-level calculations for scattering processes like electron-electron, electron-positron scattering, pair annihilation, Compton effect. According to the evaluation of the course, the students liked that very much. Their only criticism was, that I hadn't started with relativistic QFT earlier :-).
 
  • #14
Your approach suggests QT should possibly not be taught at all in high school. The way you suggest is a bit above the average 16yo.
 
  • #15
No, that's not what I'm saying. Our physics teacher could teach us this at the 12th or 13th grade at a level we could understand. What I wrote is of course more concerning physics majors at universities. I think, if you don't teach "modern physics" (QT, relativity, cosmology) at high schools you don't get the pupils interested in physics at all.
 
  • #16
12th or 13th grade in German system is 18-19 yo, so you'd normally already have the basic analysis (differential and integral calculus in 1 real variable) under your belt.
 
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  • #17
excuse my ignorance, but I just do not see where you're going. You're demonstrating that light is a wave and you do not need its quantization for the photoelectric effect?
But assume that the electrons are quantized ... and apply the golden rule of Fermi. But the Planck relationship appears, and this is not covered in classical connotation of the electromagnetic field, besides the fact that you are using a quantum approach to some objects and a classic for others ... in short, a quantum mechanics to "patchy" ...
 
  • #18
I think there is a misunderstanding here that we are teaching the wrong physics. We may be teaching the wrong logic in arriving at a conclusion, but I do not see that we are teaching the wrong physics.

I've taught the photoelectric effect topic many times. And in each of my lessons, I've approached this topic the same way J.J. Thorn et al. approached it, by indicating in the beginning that:

Students often believe that the photoelectric effect, and Einstein’s explanation of it, proves that light is made of photons. This is simply not true; while the photoelectric effect strongly suggests the existence of photons, it does not demand it.

But I continue using the photon picture in the description of this phenomenon because (i) there have been other experiments that have a clearer indication that this photon picture is valid (see the J.J. Thorn et al. references and other references on which-way experiments, plus photon anti-bunching experiments), and (ii) the photon picture is exclusively used in the entire photoemission community without encountering any inconsistencies. So for all practical purposes, this IS the valid picture!

So it is one thing that we're teaching something that seems to make a stronger claim than it should, especially within the historical context. It is another to teach something that doesn't really exist. Nothing here is indicating that we're teaching the latter. So sure, clarify to the students that the photoelectric effect as we know it now actually doesn't disqualify the wave-picture of light the way we thought back then. But we can easily indicate, as done in the J.J. Thorn et al. paper that there are other experiments that are now considered more convincing in favor of this "photon" picture.

Zz.
 
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  • #19
I believe that the photoelectric effect, is extremely understandable, even on an intuitive level, considering the light made of "particles" (photons), which, as "billiard balls" collide with the electrons, and ousting them from their bound states. With this assumption, it becomes really clear and simple to explain the photoelectric effect. After all, the idea is Einstein. So I find it is the most logical and easiest way to teach quantum mechanics, which is extremely counter-intuitive.
 
  • #20
Karolus said:
I believe that the photoelectric effect, is extremely understandable, even on an intuitive level, considering the light made of "particles" (photons), which, as "billiard balls" collide with the electrons, and ousting them from their bound states.

Actually, I would not teach it using such a phrase, because "billiard ball collision" is the wrong picture for this. Such a collision implies that the photon bumped into an electron, but then that photon careened off to somewhere else. This is certainly not true in the photoelectric effect (it may be a more suitable picture in Compton scattering, but even that is dubious). This is in addition to the wrong idea that we might unintentionally instill into the students' heads that photons may resemble billiard balls. And trust me, I know how difficult it is to get a wrong picture out of a student's head.

I try not to make any visual representation of photons other than indicate that each photon carries a specific amount of energy and warn them from thinking that a photon is similar to a classical particle.

Zz.
 
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  • #21
tom.stoer said:
The photoelectric effect is usually presented as an example disproving classical electromagnetism as viable model for interaction of light with matter and as evidence of quantization of energy in the electromagnetic field, i.e. the existence of photons. I would like to discuss a thought based on non-relativistic quantum mechanics w/o and relation to Planck, Einstein etc. showing - imho - why this is not so straightforward.
[]

Replies welcome ...
Nice. In Einstein's 1916 paper he appears to show that the exchange of momentum between atom and EM field requires quanta. But I'm not sure.

The 1916 paper is reviewed briefly in this publication https://arxiv.org/ftp/arxiv/papers/1412/1412.2060.pdf
 
  • #22
tom.stoer said:
The photoelectric effect is usually presented as an example disproving classical electromagnetism as viable model for interaction of light with matter and as evidence of quantization of energy in the electromagnetic field, i.e. the existence of photons. I would like to discuss a thought based on non-relativistic quantum mechanics w/o and relation to Planck, Einstein etc. showing - imho - why this is not so straightforward.

Using a simple 1-dim. model with a single electron plus time-dependent perturbation theory - in the same manner as used in the derivation of the spectrum of the hydrogen atom including selection rules - we obtain the following picture:

We have
- a quantum well of finite depth -V
- a discrete spectrum for [itex]\epsilon_i < 0[/itex]
- a continuous spectrum for [itex]\epsilon_f > 0[/itex]
- a classical electromagnetic field [itex]E(x,t) = E_0\,\sin(kx - \omega t)[/itex]

Calculating the transition matrix element

[tex]M_{fi} = \langle \epsilon_f|E(x,t)| \epsilon_i\rangle \sim E_0\,\delta(\epsilon_{fi} - \hbar\omega)[/tex]

and using Fermi's golden rule we find that
- the probability is proportional to [itex]|E_0|^2[/itex]
- therefore the number of photoelectrons is proportional to that probability
- the energy of the photoelectron is [itex]\epsilon_f = \hbar\omega - |\epsilon_i|[/itex]
- therefore the initial energy plays the role of the so-called "work-function", i.e. [itex]W = |\epsilon_i|[/itex]

That means that time-dependent perturbation theory of non-relativistic quantum mechanics with a classical electromagnetic field is able to reproduce the essential characteristics of the photoelectric effect w/o ever mentioning light quanta. Therefore this effects shows clearly that the interaction of light with matter cannot be obtained from purely classical reasoning, but it also shows that Einstein's hypotheses of light quanta cannot be derived in a straightforward manner.

Replies welcome ...

Now, since this whole thread is trying to nit-pick (in the best sense) of how we interpret and describe the photoelectric effect, I feel that it should also be fair to carefully examine this counter claim.

Here, εi is designated as the initial "bound" state of the electrons, correct? In this post, it is considered to be quantized or discrete.

But is this really what is going on in the standard photoelectric effect that the Einstein equation is describing, and the one done experimentally by Millikan? The photon energy that was used in this type of experiment usually goes up to as high as the UV range, and they are usually performed on metals. What this means is that the photoelectrons are emitted from the conduction band of the metal. These electrons are not in discrete energy states! The E vs. k band structure of metals is a continuous, "parabolic" dispersion curve.

I've always heard this argument that the photoelectric effect can be simulated if the material has discrete energy states, and I've never understood why this accepted. For photoionization experiments, sure, since these are done on atoms of gasses. But in a solid, especially the first few eV from the Fermi energy, there are no "quantized" or "discrete" energy states in the band structure. And yet, most photoelectric effect experiments are done in this energy range.

Zz.
 
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  • #23
ZapperZ said:
Actually, I would not teach it using such a phrase, because "billiard ball collision" is the wrong picture for this. Such a collision implies that the photon bumped into an electron, but then that photon careened off to somewhere else. ...

Zz.

Yes, indeed ..., comparing the photons in billiard balls can be misleading, then the photon does not "bounce", if anything, is absorbed ... we could say it's finished "in the hole" ... but the lesson quantum would turn into a game of pool! :smile:
However, joking aside, I refuse, within certain limits, to teach quantum mechanics in too abstract and purely mathematical. In my opinion, when Einstein had the idea of photon, I do not think he thought only in mathematical terms. As Einstein himself asserted, (far be it from me from being in the head of Einstein), he always did "mental images" of physical phenomena.
I think it's very important to find the "mental images" of physical processes, even if these images have be confronted with mathematical.
For example, in the same mathematics, even in the most abstract questions, come into play commonly used concepts, such as "continuity", "smoothness of a function", "sets", "holes in the plane" or "holes in a straight" etc.
 
  • #24
Karolus said:
I believe that the photoelectric effect, is extremely understandable, even on an intuitive level, considering the light made of "particles" (photons), which, as "billiard balls" collide with the electrons, and ousting them from their bound states. With this assumption, it becomes really clear and simple to explain the photoelectric effect. After all, the idea is Einstein. So I find it is the most logical and easiest way to teach quantum mechanics, which is extremely counter-intuitive.
An exactly this conclusion is what I fight against. It's simply wrong. Photons are as far away from anything like billiard balls can be. They don't even admit the definition of a position observable!
 
  • #25
ZapperZ said:
Now, since this whole thread is trying to nit-pick (in the best sense) of how we interpret and describe the photoelectric effect, I feel that it should also be fair to carefully examine this counter claim.

Here, εi is designated as the initial "bound" state of the electrons, correct? In this post, it is considered to be quantized or discrete.

But is this really what is going on in the standard photoelectric effect that the Einstein equation is describing, and the one done experimentally by Millikan? The photon energy that was used in this type of experiment usually goes up to as high as the UV range, and they are usually performed on metals. What this means is that the photoelectrons are emitted from the conduction band of the metal. These electrons are not in discrete energy states! The E vs. k band structure of metals is a continuous, "parabolic" dispersion curve.

I've always heard this argument that the photoelectric effect can be simulated if the material has discrete energy states, and I've never understood why this accepted. For photoionization experiments, sure, since these are done on atoms of gasses. But in a solid, especially the first few eV from the Fermi energy, there are no "quantized" or "discrete" energy states in the band structure. And yet, most photoelectric effect experiments are done in this energy range.

Zz.
The photoeffect describes the transition of an electron from a bound state into the continuum of scattering states through irradiation with a classical electromagnetic wave (or, if you want to express it in terms of QED, a coherent state). Although in the conduction band the electrons move freely within the metal they are still bound to the metal. That's why the photo electron has the kinetic energy of ##E=\hbar \omega-W##, where ##\omega## is the frequency of the em. field and ##W>0## the binding energy (!) of the electron in the metal. That basic equation from Einstein's 1905 paper stays true as the standard calculation with "Fermi's Golden Rule" shows.
 
  • #26
vanhees71 said:
The photoeffect describes the transition of an electron from a bound state into the continuum of scattering states through irradiation with a classical electromagnetic wave (or, if you want to express it in terms of QED, a coherent state). Although in the conduction band the electrons move freely within the metal they are still bound to the metal. That's why the photo electron has the kinetic energy of ##E=\hbar \omega-W##, where ##\omega## is the frequency of the em. field and ##W>0## the binding energy (!) of the electron in the metal. That basic equation from Einstein's 1905 paper stays true as the standard calculation with "Fermi's Golden Rule" shows.

I'm not arguing that the conduction electrons are in a bound state to the bulk metal. I'm arguing that they are not in a discrete energy state, which is the premise in the original post.

Zz.
 
  • #27
vanhees71 said:
An exactly this conclusion is what I fight against. It's simply wrong. Photons are as far away from anything like billiard balls can be. They don't even admit the definition of a position observable!
Even worse!

The billard ball picture does not provide the correct statistics for the transition rate, whereas a detailed calculation in non-rel. QM including the matrix element for the interaction term with the el.-mag. field does.
 
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  • #28
ZapperZ said:
I'm not arguing that the conduction electrons are in a bound state to the bulk metal. I'm arguing that they are not in a discrete energy state, which is the premise in the original post.
The fact that the initial state is a discrete energy state is irrelevant. All what matters is that we can calculate the transition matrix element in principle. Using Bloch waves - approximating the quasi-discrete states for finitely many electrons (!) - will improve the result, but will not change the overall picture.

What I am saying is not that the light quanta hypothesis is wrong; I am saying that it is sufficient, more far reaching than necessary, but not necessary by itself. So presenting the photoelectric effect as a hint towards the quantization of light is essentially correct, but presenting it as a evidence or even proof is not.

What is required to explain this effect is quantized energy transfer from the el.-mag. field to the electrons; this is reproduced correctly w/o using quanta of the el.-mag. field.
 
  • #29
tom.stoer said:
The fact that the initial state is a discrete energy state is irrelevant. All what matters is that we can calculate the transition matrix element in principle. Using Bloch waves - approximating the quasi-discrete states for finitely many electrons (!) - will improve the result, but will not change the overall picture.

But again, the whole issue with this is that you and a few others here are being particular about "accuracy" of the statement that we're using, so I will nitpick at the details here, because that is the current "playing field".

I only picked one issue I had with your derivation, but even that is a bit vague to me. For example, it appears that your εi is not only stated to be discrete in your derivation (that you claim is irrelevant here), but it is also at one particular value for ALL electrons involved in the photoemission, since you claim that it plays the same role as the work function.

In Einstein's photoelectric effect model, the form that you are comparing to is only valid for the most energetic photoelectrons, not for ALL photoelectrons. There is a continuous spectrum of energy of the photoelectrons being emitted (and a continuous in-plane momentum as well, if we want to be complete). So the energies of the photoelectrons that we obtain is dependent upon not only the work function, but also the binding energy. It was never stated if there is a running sum/integral over all initial states. Equating εi to the work function is equivalent to photoemitting only from the Fermi energy level, which never happen in real life.

Am I being overly critical? I don't know, but as I've stated, if we want to be all-out accurate, then your model must also obey the same set of criteria and rules that you are applying to the usage and teaching of the photoelectric effect. It is only fair, no?

What I am saying is not that the light quanta hypothesis is wrong; I am saying that it is sufficient, more far reaching than necessary, but not necessary by itself. So presenting the photoelectric effect as a hint towards the quantization of light is essentially correct, but presenting it as a evidence or even proof is not.

What is required to explain this effect is quantized energy transfer from the el.-mag. field to the electrons; this is reproduced correctly w/o using quanta of the el.-mag. field.

Again, as I've stated, I understand the inaccuracy here, but I've also stated that I find this to be a trivial inaccuracy simply because it reflects an accurate physics. It is why I've never understood @vanhees71 vehement admonishment about such inaccuracy, as if this is such a big-deal error that will ruin the student's understanding of this phenomenon. It certainly didn't ruin mine. And when I learned about it even more and discovered this inaccuracy in the logic, I chalked it up to another item in the learning process in which what I've been told isn't exactly what it is. This issue is neither uncommon nor unexpected.

And oh, just to prove the point, in case anyone missed it, I've described how you can "violate" the photoelectric effect description. Yet you won't see me demanding we change the textbooks and the way we teach this phenomenon.

Zz.
 
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  • #30
Do you expect me to go through all details of Bloch waves in order to show that again non-rel. QM w/ class. el.-mag. field will provide the correct delta-function in the energy and a factor containing the field strength squared?

To save your argument that this framework is not sufficient and that photons are required, the more sophisticated methods would have to spoil all results presented so far. Is this what you expect?
 
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  • #31
tom.stoer said:
Do you expect me to go through all details of Bloch waves in order to show that again non-rel. QM w/ class. el.-mag. field will provide the correct delta-function in the energy and a factor containing the field strength squared?

To save your argument that this framework is not sufficient and that photons are required, the more sophisticated methods would have to spoil all results presented so far. Is this what you expect?

Please note that in my criticism of your derivation, I never argued that the photon picture is required. I am pointing out that your derivation also contained inaccuracies. I pointed out one clear example where your claim that εi represents the work function is wrong, because you do not get photoelectrons JUST from one single energy state, and not JUST from the Fermi energy.

I am putting your derivation under the same microscope that we seem to be putting the photon picture for the photoelectric effect under. Whether photon picture is necessary or not is now irrelevant here. We are examining the derivation in Post #1, which you had solicited feedback for, did you not?

Zz.
 
  • #32
I think it is clear that using Bloch waves, continuum states for both initial and final states, etc. can improve the derivation considerably.

Please forgive me not to post numerous equations hiding the essential conclusion - that a simply toy model already provides a sufficient explanation of all characteristic features w/o inventing photons. I never claimed to provide an accurate derivation of all details, so your criticism sounds a little bit like a straw man.
 
  • #33
Do any of you have a source (textbook or article) where the full continuum case is treated? Thank you!
 
  • #34
The continuum case is treated in all textbooks on scattering theory, i.e., basically any QT (and of course QFT) textbook. Of course, as usual, the continuous part of the spectrum makes the most trouble mathematically, and you have to properly deal with the distributions, like how to square the S-matrix element, containing an energy-momentum conserving ##\delta## distribution (answer: you have to use true normalizable states (aka "wave packets") or introduce a finite box or whatever to regularize the ##\delta## distribution and then take the appropriate limit).
 
  • #35
vanhees71 said:
An exactly this conclusion is what I fight against. It's simply wrong. Photons are as far away from anything like billiard balls can be. They don't even admit the definition of a position observable!

Perhaps you did not read carefully my post. I said that comparing the photons in billiard balls is a simple and intuitive way to frame the understanding of the effect photo-electric. I did not say that the photons behave like billiard balls (which would be a classic description and deterministic, as even beginners know, is completely contradicted by quantum mechanics, which is probabilistic, indeterministic, not causal, and not local)
However, in the photoelectric effect frame, billiard balls undermine electrons, is an image effective and not too wrong
 
<h2>1. What is the photoelectric effect?</h2><p>The photoelectric effect is a phenomenon in which electrons are emitted from a material when it is exposed to light of a certain frequency or higher.</p><h2>2. How does the photoelectric effect support the particle theory of light?</h2><p>The photoelectric effect demonstrates that light behaves as a stream of particles, known as photons, rather than as a wave. This is because the energy of the emitted electrons is directly proportional to the frequency of the light, rather than its intensity.</p><h2>3. What is the work function of a material in relation to the photoelectric effect?</h2><p>The work function is the minimum amount of energy required to remove an electron from the surface of a material. In the photoelectric effect, the energy of the incident photons must be equal to or greater than the work function in order for electrons to be emitted.</p><h2>4. How does the photoelectric effect relate to the development of quantum mechanics?</h2><p>The photoelectric effect was one of the key experimental observations that led to the development of quantum mechanics. It demonstrated that energy is transferred in discrete packets, rather than continuously, which challenged the classical wave theory of light.</p><h2>5. What are some real-world applications of the photoelectric effect?</h2><p>The photoelectric effect has many practical applications, including solar panels, photodiodes, and photocells. It is also used in devices such as cameras, barcode scanners, and smoke detectors.</p>

1. What is the photoelectric effect?

The photoelectric effect is a phenomenon in which electrons are emitted from a material when it is exposed to light of a certain frequency or higher.

2. How does the photoelectric effect support the particle theory of light?

The photoelectric effect demonstrates that light behaves as a stream of particles, known as photons, rather than as a wave. This is because the energy of the emitted electrons is directly proportional to the frequency of the light, rather than its intensity.

3. What is the work function of a material in relation to the photoelectric effect?

The work function is the minimum amount of energy required to remove an electron from the surface of a material. In the photoelectric effect, the energy of the incident photons must be equal to or greater than the work function in order for electrons to be emitted.

4. How does the photoelectric effect relate to the development of quantum mechanics?

The photoelectric effect was one of the key experimental observations that led to the development of quantum mechanics. It demonstrated that energy is transferred in discrete packets, rather than continuously, which challenged the classical wave theory of light.

5. What are some real-world applications of the photoelectric effect?

The photoelectric effect has many practical applications, including solar panels, photodiodes, and photocells. It is also used in devices such as cameras, barcode scanners, and smoke detectors.

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