Some sins in physics didactics - comments

[vanhees71]

vanhees71 submitted a new PF Insights post

Some Sins in Physics Didactics

Continue reading the Original PF Insights Post.

 

[Greg Bernhardt]

Great first entry vanhees71!

 

[Ken G]

Fascinating, so the photoelectric effect did not really demonstrate light was a particle, it merely showed that the electron cannot resonate with the radiation field unless there are frequency components present that can lift the electron past the work function. IIRC, Planck derived his famous function using similar thinking, he didn't imagine the high frequencies were underoccupied because of light quanta, only because electrons could only give energy to the field in quantized bits.

 

[Drakkith]

Nice article!

 

[vanhees71]

Fascinating, so the photoelectric effect did not really demonstrate light was a particle, it merely showed that the electron cannot resonate with the radiation field unless there are frequency components present that can lift the electron past the work function. IIRC, Planck derived his famous function using similar thinking, he didn't imagine the high frequencies were underoccupied because of light quanta, only because electrons could only give energy to the field in quantized bits.

Exactly! Planck didn't like Einstein's "light quanta hypothesis". In contradistinction to that he was an immediate follower of Einstein's special relativity resolution of the puzzle concerning the lack of Galilei invariance of Maxwell electrodynamics, and he wanted to get Einstein to Berlin very much. Together with von Laue and other Berlin physicist he made Einstein an irresistable job offer, including the post of a director of the Kaiser-Wilhelm-Institut für Theoretische Physik, which consisted only of Einstein himself at the time, which meant minimal effort of time for him. In addition, and this was the most attractive feature of the offer for Einstein, he was free from any teaching duties but still being a professor at the University. For this, of course, Planck needed the agreement of the faculty, and in his letter of recommendation, he stated that Einstein was a genius, and one should not take it against him that he sometimes got over the line into speculation, particularly concerning his "light-quanta hypothesis".

Ironically the opposite was true for the Nobel-prize committee. For them (both spatial and general) relativity was too speculative to ground his nomination for the prize, and they rather gave it for the light-quanta hypothesis. He got the prize for 1921 in 1922, and I guess the main reason was the discovery of the Compton effect, which convinced many physicists of the time about the reality of light quanta, then also dubbed with the modern name "photon". That's the more ironic, because at this time there was neither non-relativistic quantum theory nor quantum-field theory, which latter was introduced only in 1927/28 by Dirac and in 1929 by Jordan et al.

So, in some sense you can say that Einstein got his Nobel for the only theory he discovered that has not survived (completely) the development of modern quantum theory. In my opinion if you have to name only one achievement of Einstein's to theoretical physics to justify his Nobel prize, then it's General Relativity. You could have awarded him for many other things, including his tremendous capability in statistical physics (already the 1905 Brownian Motion paper would have deserved the prize). Einstein, of course, well deserved the prize (if not him, who else?), but that it was given for his light quanta, is really funny ;-).

 

[Ken G]

It was as though they had given him the Nobel prize for general relativity including a built-in cosmological constant, then regretted it when universal expansion was discovered, then been vindicated when dark energy was inferred! Of course, if we ever discover a need for a lumineferous ether, we'll be glad they gave it to him for the light-quantum hypothesis over special relativity...

 

[vanhees71]

Interesting, where have you heard that the Nobel committee first wanted to give it for GR? I've never heard this, but only that they hesitated to give the prize for relativity at all. So there's no Nobel for the discovery of GR at all!

It's pretty funny with Nobel prizes anyway. A said case of negligence is Lise Meitner, who for sure should have gotten the prize together with Otto Hahn since she was the one who gave the correct interpretation of Hahn's results in terms of fission of Uranium nuclei. Hahn didn't have a clue! The reason seems to be that Siegbahn's influence in the Nobel-prize decisions prevented the Nobel prize for Meitner, whom he didn't like due to his antisemitic attitude.

 

[Ken G]

Interesting, where have you heard that the Nobel committee first wanted to give it for GR? I've never heard this, but only that they hesitated to give the prize for relativity at all. So there's no Nobel for the discovery of GR at all!

I don't know what deliberations they had, I just mean that giving him the Nobel for the interpretation of the photoelectric effect could have proved disastrous if it had not turned out that light was quantized, merely the process of adding energy to the electromagnetic field inherited the required resonances from quantum mechanics. Then they might have felt they had made a mistake-- only to be vindicated later by quantum field theory! I was commenting that something quite similar to that might have happened had they given him the Nobel for GR with a cosmological constant in it, since then Hubble's observations would have made it look like they had been premature-- only to be vindicated later by dark energy. It just shows our many ups and downs with all of Einstein's great ideas.

It's pretty funny with Nobel prizes anyway. A said case of negligence is Lise Meitner, who for sure should have gotten the prize together with Otto Hahn since she was the one who gave the correct interpretation of Hahn's results in terms of fission of Uranium nuclei. Hahn didn't have a clue! The reason seems to be that Siegbahn's influence in the Nobel-prize decisions prevented the Nobel prize for Meitner, whom he didn't like due to his antisemitic attitude.

Yes, she tops the list of Nobel snubs: http://www.scientificamerican.com/slideshow/10-nobel-snubs/

 

[zonde]

I don't think that you show what you promise here:
"In the next section we shall use this modern theory to show, what’s wrong with Einstein’s original picture and why it is a didactical sin to claim the photoelectric effect proves the quantization of the electromagnetic field and the existence of “light particles”, now dubbed photons."
What you show (I assume your mathematical argument is correct) is that modern wave model can accommodate quantized energy transfer in photoelectric effect.

But Einstein's model is certainly good for didactical purposes because - in science it is important that proposed model gives testable prediction, that this prediction is tested and it is confirmed. In that sense explanation of photoelectric effect from perspective of photons is good example.
But of course claiming that such confirmation "proves" particular model can totally spoil positive side of such example. But this is very general objection and is not very specific to particular case.

 

[Ken G]

I can give an example of what I think vanhees71 is talking about, because I've taught students about the photoelectric effect, and this is what I used to say. I said that if light was just an electromagnetic wave, and not a particle, then you should be able to crank up the intensity of a red light until it is knocking off electrons out of the metal. The idea is, if it's just the strength of the electric field that is jostling the electrons around, you should be able to compensate for low frequency by having a high intensity. But if you have to knock the electron out in a single "quantum event," then you need enough energy per quantum of light, since you only get to use one such quantum before the metal has in some sense reabsorbed the electron.

vanhees71 is saying my explanation was a didactic sin-- first of all, if you have a strong enough field, it could be a DC field and still get electrons out, so it's just not true that low frequency couldn't work. But what is really going on is that the field amplitude is always way too low to knock the electron out in a single period of the oscillation, so you need a kind of resonant accumulation of the effect, and that can be completely accommodated by a wave picture for the light. The need for a resonance, comes from the quantum mechanics of the electron, so is a "first quantization" issue, it does not require the light come in quanta, so is not a "second quantization" issue.

I see his argument as correct, so much so in fact that I am smacking my head and saying "doh" for ever repeating the everyday argument that the photoelectric effect proved that light had to come in quanta. It was basically a coincidence stemming from the existence of a time period in which we did not understand the quantum mechanics of the electron, that we ever thought that way, so we don't need to re-enter a mistaken mindset every time we bring up the photoelectric effect! vanhees71 is saying that once we understood the quantum mechanics of the electron, we had cause to reject Einstein's explanation of the photoelectric effect, but since quantum electrodynamics came along in short order, that rejection never actually happened. It's a bit like Einstein's cosmological constant, which did encounter a period of rejection, but it was not long lived! I think the case can be made that two of Einstein's most famous suggestions, that light is quantized and that there is a cosmological constant, both turned out to be true for reasons other than the ones that motivated his suggestions! So not to take too much away from the Great One, but it could be concluded that on both those counts, he got lucky.

 

[ShayanJ]

I just see one loophole here. Its true that when we go to QM, it turns out that some phenomena that are impossible in CM, become possible. But we often find out that those strange phenomena have a relatively low probability to happen and that explains why we weren't measuring them before. So I think someone should actually do some calculations using the semi-classical theory of the photoelectric effect and find the amplitude for the immediate emission of electrons(with some proper definition of immediate) and compare that with experiments. I think it may be too low to account for the experimental value. I'm not saying it will be, but I'm just thinking that only because such an explanation is possible, doesn't give us the conclusion. Maybe such explanation is still inadequate!

 

[strangerep]

[...] So I think someone should actually do some calculations using the semi-classical theory of the photoelectric effect [...]

Have you studied the quantum optics textbook of Mandel & Wolf? They perform careful calculations along these lines, both for the semi-classical case, and also the full quantum case.

 

[ShayanJ]

Have you studied the quantum optics textbook of Mandel & Wolf? They perform careful calculations along these lines, both for the semi-classical case, and also the full quantum case.

Oh...So I guess the amount of agreement is satisfactory!

But I still can argue that this isn't a sin in education. Because in a QM course, photoelectric effect is described as a step in the historical development of QM. Historical development means what phenomena inspired scientists to suggest a particular theory. So as far as historical development is concerned, it doesn't matter photoelectric effect actually proves the existence of photons or not, it just matters that Einstein thought as such. In fact no one could predict such a semi-classical description! So I think its not a sin.

 

[Ken G]

All I can say is, after reading this, I will from now on say that the photoelectric effect was incorrectly interpreted as evidence that the radiation field comes in quanta, when in fact it was merely evidence that getting an energy E into an electron often requires resonant coupling to some electromagnetic power at frequency E/h. A radiation field that doesn't oscillate at that frequency is therefore not good at doing it. However, it turns out that radiation is regarded as quantized anyway.

Incidentally, I'm not even sure you need to quantize the radiation field to get spontaneous emission. It seems to me a classical treatment of the radiation field can work for that as well, if you simply let the Fourier mode that perturbs the electron be the electromagnetic field that the electron itself creates, in the spirit of the bootstrap effect sometimes used to analyze the radiative reaction force. Which leaves us with the question-- what is the best observational evidence that the radiation field needs to be quantized? The Compton effect? Even photon shot noise could conceivably be modeled as stochastic amplitude variations in a classical field, I would think. Maybe there's even some way to get the Compton effect with a classical field, if such stochastic amplitude variations are included?

 

[ShayanJ]

All I can say is, after reading this, I will from now on say that the photoelectric effect was incorrectly interpreted as evidence that the radiation field comes in quanta, when in fact it was merely evidence that getting an energy E into an electron often requires resonant coupling to some electromagnetic power at frequency E/h. A radiation field that doesn't oscillate at that frequency is therefore not good at doing it. However, it turns out that radiation is regarded as quantized anyway.

Incidentally, I'm not even sure you need to quantize the radiation field to get spontaneous emission. It seems to me a classical treatment of the radiation field can work for that as well, if you simply let the Fourier mode that perturbs the electron be the electromagnetic field that the electron itself creates, in the spirit of the bootstrap effect sometimes used to analyze the radiative reaction force. Which leaves us with the question-- what is the best observational evidence that the radiation field needs to be quantized? The Compton effect? Even photon shot noise could conceivably be modeled as stochastic amplitude variations in a classical field, I would think. Maybe there's even some way to get the Compton effect with a classical field, if such stochastic amplitude variations are included?

I remember @ZapperZ once said that multiphoton photoemission and angle-resolved photoemission can only be explained in terms of photons.

 

[zonde]

I see his argument as correct, so much so in fact that I am smacking my head and saying "doh" for ever repeating the everyday argument that the photoelectric effect proved that light had to come in quanta.

Simply ban the word "proved" from your lexicon whenever you are talking about science. However when teaching something you want to present your subject as solid as possible so there is sort of conflict with inconclusive statements that science can make.

It was basically a coincidence stemming from the existence of a time period in which we did not understand the quantum mechanics of the electron, that we ever thought that way, so we don't need to re-enter a mistaken mindset every time we bring up the photoelectric effect! vanhees71 is saying that once we understood the quantum mechanics of the electron, we had cause to reject Einstein's explanation of the photoelectric effect, but since quantum electrodynamics came along in short order, that rejection never actually happened.

In science cause for rejecting some model is falsification of it's predictions. You could rather say that:
Once we understood the quantum mechanics of the electron, we had no cause to reject (pathced) wave model because of the photoelectric effect.

 

[vanhees71]

What I wanted to say is that one must not teach students "old quantum theory" as if it was still considered correct. The photoelectric effect, at the level of accuracy described in Einstein's paper, does not show that the electromagnetic field is quantized, as shown by the standard calculation provided in my Insights article (the only thing, I've never found is the argument given there, why one can omit the interference term between the two modes with $\pm \omega$ of the em. field, which are necessarily there, because the em. field is real).

I've not calculated the cross section to the end, because I thought that's an unnecessary complication not adding to the point at the level of the (in my opinion false) treatment in introductory parts of many QM1 textbooks. You can do this quite easily yourself, using as an example the analytically known hydrogen wavefunctions for the bound state and a plane-wave free momentum eigenstate for the continuum state. Then you integrate out the angles and rewrite everything in terms of energy instead of $\vec{p}$. You can find the resul in many textbooks, e.g., Sakurai, where this example is nicely treated.

Of course, what I've calculated is the leading-order dipole approximation. Perhaps one should ad a paragraph showing this explicitly, but I don't know, whether one can add something to a puglished insight's article. There are also some typos :-(.

So here is the derivation. What we need is the right-hand side of Eq. (15), i.e., the matrix element in the Schrödinger picture (which coincides by assumption with the interaction picture at $t=t_0$). First of all we note that in the interaction picture
$$\dot{\hat{\vec{x}}}=\frac{1}{\mathrm{i} \hbar} [\hat{\vec{x}},\hat{H}_0]=\frac{1}{m} \hat{\vec{p}}.$$
Thus we have
$$\langle E (t_0)|\hat{\vec{p}}(t_0)|E_n(t_0) \rangle=\frac{m}{\mathrm{i} \hbar} (E-E_n) \langle E(t_0)|\hat{\vec{x}}|E \rangle.$$
Now if you plug this into (20) then due to the energy-conserving $\delta$ distribution and making use of the fact that this piece relevant for the absorption (photoeffect) transition rate only comes from the positive-frequency piece $\propto \exp(-\mathrm{i} \omega t)$ in $\vec{A}$, you find that what enters is in fact
$$\alpha^2 \propto |\vec{E}_0 \cdot \langle E(t_0)|\hat{\vec{x}}(t_0) E_n(t_0) \rangle|^2,$$
and this is nothing else than the electric-field amplitude times the dipole-matrix transition matrix element.

The whole calculation also shows that there's no absorption of frequency modes of the em. field if $\hbar \omega$ is smaller than the binding energy of the initial state of the electron and that the rate of absorption processes is proportional to the intensity of the external field (for small fields so that perturbuation theory is still applicable).

For those who like to print the article, I've put it on a new website, I've just created:

http://fias.uni-frankfurt.de/~hees/pf-faq/

 

[Ken G]

Simply ban the word "proved" from your lexicon whenever you are talking about science. However when teaching something you want to present your subject as solid as possible so there is sort of conflict with inconclusive statements that science can make.


In science cause for rejecting some model is falsification of it's predictions. You could rather say that:
Once we understood the quantum mechanics of the electron, we had no cause to reject (pathced) wave model because of the photoelectric effect.

I agree with all of your more careful restatements, yet you are saying the same thing. We have taken vanhees71's points here.

 

[atyy]

As zonde says, the issue is general. An observation cannot prove a theory. At best, it can prove a theory within a well-defined model class, eg. in classical Mendelian genetics or in Wilsonian renormalization where one considers the "space of all possible theories".

However, although the photoelectric effect does not prove that the electromagnetic field is quantized, now that we do know the electromagnetic field is quantized, can Einstein's explanation be considered correct?

 

[vanhees71]

I'd say no, although you can doubt this in some sense: Of course, the photoeffect must also be describable in terms of QED. The setup, most similar to the semiclassical one in my article, is to use a free atom and a coherent state of the em. field as "initial state" and a free electron of momentum $\vec{p}$, another coherent state of the em. field, and a free proton in the final (asymptotic) state. You should get the same, or a very similar, result as in the semiclassical treatment. In some sense you can indeed say, Einstein's picture is not that wrong, because the corresponding transition-matrix element describes the processes as absorption of one photon out of the coherent field (and even more, because it includes the change of the state of the em. field due to the interaction with the atom in 1st-order perturbation theory).

Nevertheless, at this level of accuracy of the description and just making a measurement to demonstrate the validity of Eq. (1) from Einstein's paper, does not "prove" the necessity of a quantization of the em. field, because there is this semiclassical calculation, leading to this formula (1).

 

[Ken G]

And if you assert a different historical sequence, in which the quantum mechanics of an electron is discovered prior to the photoelectric effect, there is no Nobel prize there-- it's more like "ho hum, yes quantum mechanics works in other situations than just atoms." No one even imagines the radiation is quantized, there's just no need for it from that experiment. Einstein was right about the cosmological constant too, it seems, but we don't teach students "scientists concluded there is a cosmological constant because it is needed to make the universe static, and modern observations of dark energy confirm that there is indeed a cosmological constant." Instead we call it Einstein's greatest blunder-- even though he was right!

So is the quanta of photons Einstein's second greatest blunder, on the same grounds? In many ways, it kind of is. What if he had used the photoelectric effect to deduce the quantum mechanics in vanhees71's calculation, instead of hypothesizing photons? The latter takes away the need to explain why those frequency modes need to be present to get the necessary energy coupling to the electron, so can be viewed as an opportunity lost, akin to the Hubble law, if we are postulating 20-20 hindsight.

 

[atyy]

And if you assert a different historical sequence, in which the quantum mechanics of an electron is discovered prior to the photoelectric effect, there is no Nobel prize there-- it's more like "ho hum, yes quantum mechanics works in other situations than just atoms." No one even imagines the radiation is quantized, there's just no need for it from that experiment. Einstein was right about the cosmological constant too, it seems, but we don't teach students "scientists concluded there is a cosmological constant because it is needed to make the universe static, and modern observations of dark energy confirm that there is indeed a cosmological constant." Instead we call it Einstein's greatest blunder-- even though he was right!

It was his greatest blunder because he did not realize he was right!

 

[Ken G]

Don't you just love irony?

 

[atyy]

I guess a better (?) reason for including the cosmological constant is not so much the actual measurement, but the Wilsonian viewpoint? Include all terms consistent with the basic assumptions. For some reason that seems to have become known as Gell-Mann's totalitarian principle http://en.wikipedia.org/wiki/Totalitarian_principle.

Edit: Hmmm, reading the Wikipedia article, it may be different. In Gell-Mann's case, the motivation seems to have been basic probability, whereas in the Wilsonian case, the terms are generated automatically by the renormalization flow.

 

[ddd123]

It's pretty funny with Nobel prizes anyway. A said case of negligence is Lise Meitner, who for sure should have gotten the prize together with Otto Hahn since she was the one who gave the correct interpretation of Hahn's results in terms of fission of Uranium nuclei. Hahn didn't have a clue! The reason seems to be that Siegbahn's influence in the Nobel-prize decisions prevented the Nobel prize for Meitner, whom he didn't like due to his antisemitic attitude.

On that note, a more recent example is Cabibbo having been neglected for the CKM matrix work while awarding the Nobel to the KM part. I couldn't find any explanation for that, political or otherwise, and the only hypothesis I could think of that it was because Cabibbo was catholic seems far fetched :)

 

[LsT]

I, as a layman, can confirm that the fact that photoelectric effect is presented as if it is obvious that it prooves the quantization of light, has done no good for me. In my mind I was envisioning for some time now, if it is possible to "bounce out" losely fitted balls from a wall by resonance with sound waves, irrespective of intensity. And probably this is possible. So if it was not of vanhees71 post, maybe sometime I would post another incosistent question in this forum. Now I understand at least that there is a MUCH bigger story behind this. And it creates a motive for me to learn the math so I can "get it". Of course if it is on the context of the history of QM in an academic course, it's another matter.

 

[atyy]

I'd say no, although you can doubt this in some sense: Of course, the photoeffect must also be describable in terms of QED. The setup, most similar to the semiclassical one in my article, is to use a free atom and a coherent state of the em. field as "initial state" and a free electron of momentum $\vec{p}$, another coherent state of the em. field, and a free proton in the final (asymptotic) state. You should get the same, or a very similar, result as in the semiclassical treatment. In some sense you can indeed say, Einstein's picture is not that wrong, because the corresponding transition-matrix element describes the processes as absorption of one photon out of the coherent field (and even more, because it includes the change of the state of the em. field due to the interaction with the atom in 1st-order perturbation theory).

Nevertheless, at this level of accuracy of the description and just making a measurement to demonstrate the validity of Eq. (1) from Einstein's paper, does not "prove" the necessity of a quantization of the em. field, because there is this semiclassical calculation, leading to this formula (1).

So in QM, Einstein is wrong and in QED Einstein is right. In old quantum theory, Einstein clearly can be right. But is there a way to make Einstein wrong in old quantum theory, along the lines of what Planck considered, where the quantization is in the energy levels of the electrons, not the electromagnetic field?

 

[vanhees71]

This is again a somewhat more philosophical than physical question. Strictly speaking both Einstein and Planck where wrong, but Einstein got the black-body law right around 1917. Concerning the photoeffect at the level of sophistication discussed in his 1905 paper you cannot distinguish between the light-quantum picture, i.e., that light consists of light particles in Einstein's sense and knock out electrons in collision-like events, and Planck's view that there's a classical electromagnetic wave, but the absorption of em. field energy can only be in energy quanta of size $\hbar \omega$.

Concerning black-body radiation, Planck's derivation is interesting, because he used a then ad-hoc counting method of the microstates. Of course, nowadays we recognize this counting method as the correct one for bosons. As we also know today, the foundation of this counting rule is neither justifyable with classical fields nor with classical particles. Here Einstein in 1917 had the right insight by discovering spontaneous emission, which only about 10 years later could be derived from fundamental principles by Dirac in terms of modern quantum theory by introducing creation and annihilation operators for modes of em. fields, which is nothing else than field quantization, and indeed field quantization is the only way to make full sense of electromagnetic phenomena within relativistic quantum theory today.

It's also true that Einstein was well aware that the "old quantum theory" was far from satisfactory, and his long struggle with the "radiation problem" finally lead to the development of modern quantum theory. So one should not diminish Einstein's and Planck's achievements in "old quantum theory", but I still consider it a sin to start quantum physics lectures with this old quantum theory, which introduces wrong pictures already on the qualitative level, which then you have to "unlearn" again. It's unnecessary and confusing for the students. On the other hand, it's also very important to have some insight into the historical developments to fully appreciate the modern picture of contemporary physics. I guess that 100 years later also this status of science may be seen as just a historical step to a better understanding of nature. That's just the way science (hopefully!) works.

 

[atyy]

This is again a somewhat more philosophical than physical question. Strictly speaking both Einstein and Planck where wrong, but Einstein got the black-body law right around 1917. Concerning the photoeffect at the level of sophistication discussed in his 1905 paper you cannot distinguish between the light-quantum picture, i.e., that light consists of light particles in Einstein's sense and knock out electrons in collision-like events, and Planck's view that there's a classical electromagnetic wave, but the absorption of em. field energy can only be in energy quanta of size $\hbar \omega$.

But why should only quanta of a certain size be absorbed? Can't the qunatized system take a chunk out of a classical EM wave that has a frequency below the critical frequency? In the old quantum theory view, I do understand Einstein's model, but I find it quite hard to understand Planck's alternative.

It's also true that Einstein was well aware that the "old quantum theory" was far from satisfactory, and his long struggle with the "radiation problem" finally lead to the development of modern quantum theory. So one should not diminish Einstein's and Planck's achievements in "old quantum theory", but I still consider it a sin to start quantum physics lectures with this old quantum theory, which introduces wrong pictures already on the qualitative level, which then you have to "unlearn" again. It's unnecessary and confusing for the students. On the other hand, it's also very important to have some insight into the historical developments to fully appreciate the modern picture of contemporary physics. I guess that 100 years later also this status of science may be seen as just a historical step to a better understanding of nature. That's just the way science (hopefully!) works.

But there are curious things about old quantum theory that make it seem "right". For example, the de Broglie relations are relativistic. Many intuitions of old quantum theory are preserved in non-relativistic QM or relativistic QFT.

Bohr model: stationary waves and boundary conditions determining discrete energy levels - still true in the Schroedinger equation

Wave particle duality: Still true in QM where the Hilbert space is particles, and the Schroedinger equation is waves. Also true in non-rigourous QFT where there are Fock space particles and waves via the Heisenberg picture equations of motion for the operators.

de Broglie relations: Still true for the relativistic free quantum fields

 

[bhobba]

The thing to realize about Einstein isn't that he got things right or wrong - they are important of course - its that he could see deeper into things than anyone else.

There is another great thinker - Von Neumann. Although called a mathematician, and he was one of the greatest that ever lived, he really was much much more than that - he was a polymath. He had, like Feynman, the mind of a magician, but even Feynman said he was above him. Those exposed to him said he was the only human ever that was fully awake. His technical mathematical brilliance was very great - well above Einstein, who, while a competent mathematician, was nowhere close to that class.

The thing though is this, as great as Von Neumann was, as good as he was in seeing to the heart of a problem, and he was good, Einstein was better. He couldn't match von Neumanns technical brilliance, but his ability to hone in on the essential issues was without peer. And that is the important thing, not mathematical brilliance, being fully awake, or any of the other attributes someone like Von Neumann had, and its what is required to make progress.

Einstein got a number of things wrong, but really that's just by the by, he was still able to see to the heart of things and hone in on what was important. He got the photoelectric effect wrong - but got right understanding it was essential to future progress. Here is what Poincare (he was also a noted polymath), along with Madam Curie, said of Einstein:

'Herr Einstein is one of the most original minds that we have ever met. In spite of his youth he already occupies a very honorable position among the foremost savants of his time. What we marvel at him, above all, is the ease with which he adjusts himself to new conceptions and draws all possible deductions from them. He does not cling to classic principles, but sees all conceivable possibilities when he is confronted with a physical problem. In his mind this becomes transformed into an anticipation of new phenomena that may some day be verified in actual experience...The future will give more and more proofs of the merits of Herr Einstein, and the University that succeeds in attaching him to itself may be certain that it will derive honor from its connection with the young master'

Thanks
Bill

 

[vanhees71]

Wave particle duality: Still true in QM where the Hilbert space is particles, and the Schroedinger equation is waves. Also true in non-rigourous QFT where there are Fock space particles and waves via the Heisenberg picture equations of motion for the operators.
de Broglie relations: Still true for the relativistic free quantum fields

This is exactly, why I consider the teaching of old quantum theory in the beginning of quantum physics courses as a sin. First of all in quantum theory there is no wave-paticle duality. In my opinion it's the greatest overall achievement of modern quantum theory to have get rid of such contradictory pictures about nature. Formally you are right, and the Schrödinger equation has some features of a classical (wave) field, but that's only a mathematical analogy. That any partial differential equation in a space-time with the symmetries and admitting a causality principle (orientation of time) we usually assume (Galilei or Poincare symmetry for Newtonian and special relativistic physics, respectively) has similar forms than any other is very natural, because these symmetries are restrictive enough to determine at least a large part of the form such equations should take. Nevertheless, the Schrödinger-wave function is not a classical field and thus does not describe any kind of classical waves. You don't observe waves, when you consider a single particle but a single particle being detected somewhere in space, and these particles are not "smeared" out in a continuum like way, as a superficial interpretation of the Schrödinger wave function as a classical field would suggest. The right interpretation was given by Born: It describes the probability amplitudes to find a particle at a place, i.e., its modulus squared is the position-probability distribution for particles.

Also I don't see, why you say the Hilbert-space formalism describes particles. The abstract Hilbert-space formalism consists of abstract mathematical objects, which have the advantage that you cannot mix them up with the hand-waving associations of the old quantum theory. That's why I think, the ideal way to teach quantum theory is to start right away with the basis-free formulation in (rigged) Hilbert space. Sakurai's textbook shows a clever way, how to do this without getting into all mathematical subtleties of unbound operators. The aim in teaching quantum theory must be to introduce the students to the fact that the micro-world is "unintuitive", and that's so, because our senses and brains are not made primarily to comprehend or describe the microcosm but to survive in a macroscopic world, which behaves quite "classically", although the underlying "mechanism" is of course quantum.

To teach this, you don't need old-fashioned precursor models, of which the worst is the Bohr-Sommerfeld model of the atom, because that's not even right in any sense of the modern theory. I never figured out, why the model gets the hydrogen energy levels right. The reason must ly in the very high symmetry of the hydrogen-atom hamiltonian (an SO(4) for the bound energy eigenstates, a Galilei symmetry for the zero-modes, and a SO(1,3) for the other scattering states). The same holds, of course true, for the N-dimensional harmonic oscillator which always has a SU(N) symmetry. For the harmonic oscillator it's clear, why it looks so classical also in quantum theory: The equations of motion of the operators representing observables in the Heisenberg picture are linear and thus the expectation values obey exactly the classical equations of motion.

 

[Dadface]

Nice article but I would like to point out a practical difficulty in presenting the photoelectric effect as described here.

In the UK photoelectricity is first introduced to AS level students who are 16 to 17 years old. The AS specification is such that there will be about 16 hours of teaching time available to teach the whole of the quantum theory unit photoelectricity being just one of four topics studied. Subtract the time taken for lab sessions, class discussions settling in time and so on and the teacher will be left with a maximum of about two hours to cover the subject.

In addition to timing problems, students (and probably most teachers whose maths is rusty) would not have the necessary maths expertise to follow this more detailed approach to the subject. I would suggest carry on teaching it as it is and inform the students that there are more sophisticated approaches to the subject but these are beyond the scope of AS.

I would be grateful if someone could clarify the following:
It seems to be suggested that there is a better alternative to the equation E = hf . Have I been reading this correctly? If so I can't see that there is a better way of calculating, for example, the frequency of the photons produced in electron positron annihilation.

 

[atyy]

This is exactly, why I consider the teaching of old quantum theory in the beginning of quantum physics courses as a sin. First of all in quantum theory there is no wave-paticle duality. In my opinion it's the greatest overall achievement of modern quantum theory to have get rid of such contradictory pictures about nature. Formally you are right, and the Schrödinger equation has some features of a classical (wave) field, but that's only a mathematical analogy. That any partial differential equation in a space-time with the symmetries and admitting a causality principle (orientation of time) we usually assume (Galilei or Poincare symmetry for Newtonian and special relativistic physics, respectively) has similar forms than any other is very natural, because these symmetries are restrictive enough to determine at least a large part of the form such equations should take. Nevertheless, the Schrödinger-wave function is not a classical field and thus does not describe any kind of classical waves. You don't observe waves, when you consider a single particle but a single particle being detected somewhere in space, and these particles are not "smeared" out in a continuum like way, as a superficial interpretation of the Schrödinger wave function as a classical field would suggest. The right interpretation was given by Born: It describes the probability amplitudes to find a particle at a place, i.e., its modulus squared is the position-probability distribution for particles.

Well, it's a wave in Hilbert space, which is "particle space" (I'll explain that below).

Also I don't see, why you say the Hilbert-space formalism describes particles. The abstract Hilbert-space formalism consists of abstract mathematical objects, which have the advantage that you cannot mix them up with the hand-waving associations of the old quantum theory.

In QM, the Hilbert space is a particle space because if we have one particle, then we write ψ(x). If we have two particles then we write basis functions that are ψ_m(x₁)ψ_n(x₂). So we still have particles, it just so happens they don't have definite position and momentum at all times.

That's why I think, the ideal way to teach quantum theory is to start right away with the basis-free formulation in (rigged) Hilbert space. Sakurai's textbook shows a clever way, how to do this without getting into all mathematical subtleties of unbound operators. The aim in teaching quantum theory must be to introduce the students to the fact that the micro-world is "unintuitive", and that's so, because our senses and brains are not made primarily to comprehend or describe the microcosm but to survive in a macroscopic world, which behaves quite "classically", although the underlying "mechanism" is of course quantum.

The way I was taught had the quantum mechanical axioms quite early, but we still got the old quantum theory. Lecture 1 was dimensional analysis to motivate the introduction of Planck's constant. Lecture 2 was old quantum theory, including all the wonderful thermodynamics. Lecture 3 was the quantum mechanical axioms (state is a ray, etc).

To teach this, you don't need old-fashioned precursor models, of which the worst is the Bohr-Sommerfeld model of the atom, because that's not even right in any sense of the modern theory. I never figured out, why the model gets the hydrogen energy levels right. The reason must ly in the very high symmetry of the hydrogen-atom hamiltonian (an SO(4) for the bound energy eigenstates, a Galilei symmetry for the zero-modes, and a SO(1,3) for the other scattering states). The same holds, of course true, for the N-dimensional harmonic oscillator which always has a SU(N) symmetry. For the harmonic oscillator it's clear, why it looks so classical also in quantum theory: The equations of motion of the operators representing observables in the Heisenberg picture are linear and thus the expectation values obey exactly the classical equations of motion.

I'm not sure either, I too think it has to do with the symmetry. The Bohr-Sommerfeld quantization somehow has a notion of integrability in it, and the hydrogen atom is integrable in some sense. (Actually, I never know exactly what a quantum integrable system is, since integrability is really a classical concept). Anyway, the semiclassical quantization is still useful, for example, to explain phenomena like "scars" http://www.ericjhellergallery.com/index.pl?page=image;iid=22 [Broken].

So I would still like to know if the QM calculation you used has a simple "old quantum theory" interpretation without Einstein's photons, closer to Planck's view. Could we say that somehow the wave has to be of a certain frequency because of a resonance effect?

 

[vanhees71]

Nice article but I would like to point out a practical difficulty in presenting the photoelectric effect as described here.

In the UK photoelectricity is first introduced to AS level students who are 16 to 17 years old. The AS specification is such that there will be about 16 hours of teaching time available to teach the whole of the quantum theory unit photoelectricity being just one of four topics studied. Subtract the time taken for lab sessions, class discussions settling in time and so on and the teacher will be left with a maximum of about two hours to cover the subject.

In addition to timing problems, students (and probably most teachers whose maths is rusty) would not have the necessary maths expertise to follow this more detailed approach to the subject. I would suggest carry on teaching it as it is and inform the students that there are more sophisticated approaches to the subject but these are beyond the scope of AS.

I would be grateful if someone could clarify the following:
It seems to be suggested that there is a better alternative to the equation E = hf . Have I been reading this correctly? If so I can't see that there is a better way of calculating, for example, the frequency of the photons produced in electron positron annihilation.

NO, this I don't buy! You must not teach high school students misleading stuff (in fact, we were told "old quantum theory" also before the modern theory was taught in high school, and our (btw. really brillant) teacher said, before starting with the modern part that we should forget the quantum theory taught before, and she was right so.

Of course, in high school, you cannot teach the abstract Dirac/Hilbert-space notation and also not time-dependent perturbation theory, but you can completely omit misleading statements referring to the "old quantum theory". At high school we learned modern quantum theory in terms of wave mechanics. I don't know, how the schedule looks in the UK, but in Germany, usually one has a modul about classical waves before entering the discussion of quantum theory, and thus you can easily argue in the usual heuristic way to introduce first free-particle non-relativistic "Schrödinger waves", but telling right away the correct Born interpretation. This gains you time to teach the true stuff and not waste it for outdated misleading precursor theories that are important for the science historian only (although history of science makes a fascinating subject in itself, and to a certain extent it should also be covered in high school).

I don't understand the 2nd question. Of course, the energy eigenvalue $E$ and the frequency of the corresponding eigenmode of the Schrödinger field are related by $E=\hbar \omega=h f$, where $\omega=2 \pi f$ and $\hbar=h/(2 \pi)$. Usually nowadays one doesn't use the original Planck constant $h$ but $\hbar$, because you don't need to write some factors of $2 \pi$ when using $\omega$ instead of $f$.

 

[vanhees71]

Well, it's a wave in Hilbert space, which is "particle space" (I'll explain that below).



In QM, the Hilbert space is a particle space because if we have one particle, then we write ψ(x). If we have two particles then we write basis functions that are ψ_m(x₁)ψ_n(x₂). So we still have particles, it just so happens they don't have definite position and momentum at all times.



The way I was taught had the quantum mechanical axioms quite early, but we still got the old quantum theory. Lecture 1 was dimensional analysis to motivate the introduction of Planck's constant. Lecture 2 was old quantum theory, including all the wonderful thermodynamics. Lecture 3 was the quantum mechanical axioms (state is a ray, etc).



I'm not sure either, I too think it has to do with the symmetry. The Bohr-Sommerfeld quantization somehow has a notion of integrability in it, and the hydrogen atom is integrable in some sense. (Actually, I never know exactly what a quantum integrable system is, since integrability is really a classical concept). Anyway, the semiclassical quantization is still useful, for example, to explain phenomena like "scars" http://www.ericjhellergallery.com/index.pl?page=image;iid=22 [Broken].

So I would still like to know if the QM calculation you used as a simple "old quantum theory" interpretation without Einstein's photons, closer to Planck's view. Could we say that sonehow the wave has to be of a certain frequency because of a resonance effect?

It's a bit strange to me to say the Schrödinger waves are in Hilbert space. It's simply a scalar complex valued field which describes waves, but just in the mathematical sense. It's not like the waves of a measurable field quantity like, e.g., the density of air or water (sound waves) or the electromagnetic field, but its physical meaning is given by the Born rule.

What the quantum mechanics is supposed to describe is, of course, given by the choice of the problem. If you have a single particle, you start by defining observables by assuming some operator algebra, heuristically taken from some classical analgous situation. Of course, you cannot derive this in a mathematical sense from anything, but you have to more or less guess it. What helps you, are symmetries and (Lie-)group theory to guess the right operator algebra. The self-adjoint operators live on a Hilbert space, and the rays in this Hilbert space represent the (pure) states. Then, if $|\psi \rangle$ is a normalized representant of such a ray, a wave function is given wrt. a complete basis, related to the determination of a complete set of compatible observables, $|o_1,\ldots,o_n \rangle$, i.e.,
$$\psi(o_1,\ldots,o_n)=\langle o_1,\ldots o_n|\psi \rangle.$$
That's it. In my opinion there's no simpler way to express quantum theory than this. Admittedly it's very abstract und unintuitive, but that's the only way we have found so far to adequately describe (pretty comprehensively) the phenomena in terms of a pretty self-consistent mathematical scheme.