# FeaturedI Interpretation of the photoelectric effect

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1. Apr 2, 2017

### tom.stoer

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 $\epsilon_i < 0$
- a continuous spectrum for $\epsilon_f > 0$
- a classical electromagnetic field $E(x,t) = E_0\,\sin(kx - \omega t)$

Calculating the transition matrix element

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

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

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 ...

2. Apr 2, 2017

### vanhees71

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/

3. Apr 2, 2017

4. Apr 2, 2017

### A. Neumaier

Yes. This has been known since around 1965. For more see the references given here.

5. Apr 2, 2017

### vanhees71

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)

6. Apr 2, 2017

### dextercioby

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. Apr 2, 2017

### vanhees71

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. Apr 2, 2017

### stevendaryl

Staff Emeritus
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.

9. Apr 2, 2017

### tom.stoer

of course :-)

10. Apr 2, 2017

### vanhees71

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. Apr 2, 2017

### tom.stoer

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. Apr 2, 2017

### houlahound

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. Apr 3, 2017

### vanhees71

I find the historical approach confusing from my own experience learning QT. In highschool 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. Apr 3, 2017

### houlahound

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. Apr 3, 2017

### vanhees71

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. Apr 3, 2017

### dextercioby

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.

Last edited: Apr 3, 2017
17. Apr 3, 2017

### Karolus

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. Apr 3, 2017

### ZapperZ

Staff Emeritus
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:

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.

19. Apr 3, 2017

### Karolus

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. Apr 3, 2017

### ZapperZ

Staff Emeritus
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.