Quantum theory for high-school students

[Stephen Tashi]

I was shocked that Susskind as unaware that "The Law of Large Numbers" is a theorem, especially the weak law.

It's a theorem in probability theory, but saying that it (or any other aspect of probability theory) governs a physical situation is an assumption.

 

[PeterDonis]

Feynman as well as many other luminaries () call it a particle.

Yes, but they don't mean the same thing by "particle" that @diPoleMoment means. They mean something like "a discrete detection event like a little dot on a screen". They don't mean "a little billiard ball".

Suppose a precocious 12 year old passes by (or even a famous biologist) and asks you, Mr. Donis, what's a photon? You wouldn't say a particle of light?

Not without a considerable explanation of what "particle" means in this context. I would prefer to use a word like "quantum" that does not have a lot of misleading connotations.

I certainly hope you don't tell her to read Ballentine.

No, but I might tell her that "photon" is a complicated concept, and unless and until you're ready to tackle the complications it's better not to think of light as made of photons.

 

[jtbell]

Feynman as well as many other luminaries () call it a particle.

Yes, but they don't mean the same thing by "particle" that @diPoleMoment means. They mean something like "a discrete detection event like a little dot on a screen". They don't mean "a little billiard ball".

Once upon a time, physicists did mean something like a little speck of dust or billiard ball. The introduction to Einstein's 1905 paper on the photoelectric effect makes this clear:

Nach der hier ins Auge zu fassenden Annahme ist bei Ausbreitung eines von einem Punkte ausgehenden Lichtstrahles die Energie nicht kontinuierlich auf größer und größer werdende Räume vertelit, sondern es besteht diesselbe aus einer endlichen Zahl von in Raumpunkten lokalisierten Energiequanten, welche sich bewegen, ohne sich zu teilen und nur als Ganze absorbiert und erzeugt werden können.

My attempt at a translation: "According to the assumption to be considered here, when a light beam spreads out from a point, the energy is not distributed continuously over regions that becoms larger and larger, instead it consists of a finite number of energy quanta localized at spatial points, which move without dividing and can be absorbed and created only in their entirety."

With that kind of picture, the word "Teilchen" (German) or "particle" is inescapable. Eventually it became clear that photons aren't really "localized at spatial points", at least while propagating. However, by then physicists were so accustomed to referring to them as "particles" that they in effect redefined the word "particle" instead of trying to get everybody to agree on a new word, and figuring out how to deal with the use of "particle" in previously-written articles and textbooks.

 

[Nugatory]

Suppose a precocious 12 year old passes by (or even a famous biologist) and asks you, Mr. Donis, what's a photon? You wouldn't say a particle of light?
I certainly hope you don't tell her to read Ballentine. I also hope it is something short, she's in a hurry.

"When light interacts with matter, it always delivers its energy in discrete lumps landing at at a single point. Whenever this happens, we say 'a photon appeared at that point'".

Then I hand them my paperback copy of Feynman's "QED: The strange theory of light and matter", which is quite appropriate for a precocious twelve-year-old.

 

[vanhees71]

Once upon a time, physicists did mean something like a little speck of dust or billiard ball. The introduction to Einstein's 1905 paper on the photoelectric effect makes this clear:
My attempt at a translation: "According to the assumption to be considered here, when a light beam spreads out from a point, the energy is not distributed continuously over regions that becoms larger and larger, instead it consists of a finite number of energy quanta localized at spatial points, which move without dividing and can be absorbed and created only in their entirety."

With that kind of picture, the word "Teilchen" (German) or "particle" is inescapable. Eventually it became clear that photons aren't really "localized at spatial points", at least while propagating. However, by then physicists were so accustomed to referring to them as "particles" that they in effect redefined the word "particle" instead of trying to get everybody to agree on a new word, and figuring out how to deal with the use of "particle" in previously-written articles and textbooks.

The difference between Einstein and popular-science and unfortunately too many textbook writers, Einstein didn't know about the concept of photons in the sense of modern QED (which was first formulated 21 years later by Jordan in the famous "Dreimännerarbeit" for the first time, and at that time was not appreciated by the community; only Dirac's famous article one year later using the creation-annihilation-operator formalism for light quanta brought QED to the masses).

The even more important difference between Einstein and those writers, who just copy age-old wrong physics and didactics just to sell their books, was that Einstein didn't believe he had the correct picture yet. He even didn't believe that QED is the final answer. Given the fact that QED is strictly speaking not well-defined mathematically even today, he might be finally right. However, on the other hand, renormalized perturbative QED (and the entire Standard Model, which Maiani thinks it should be renamed to Standard Theory, of elementary particle physics) is among the most accurate physical theories ever, being in accordance with experiments with an accuracy of 12-13 digits for quantities like the anomalous magnetic moment of the electron and the Lamb shift of the hydrogen spectrum.

For sure, it is not responsible to teach students, no matter at which level of their education, these wrong and outdated pictures. It's not that I claim, one can teach them QED in high school, but one should at least not teach them wrong intuitions that are even wrong in a qualitative heuristic sense, and to provide a picture as if photons are like little lumps or billiard balls (i.e., localizable particle-like objects) is wrong in this qualitative sense!

It's much better to teach high-school students some elements of Maxwell theory, which is done for decades at high schools (I guess not only in Germany but around the world) and then qualitatively explain the photon concept in the right way and strictly remaining in the realm of established observable facts rather then 118-year-old wrong concepts of Einstein, of which Einstein himself was of course very clear to provide just a "heuristic aspect" not a complete consistent theory! He considered the "puzzle of radiation" the much more difficult problem than even his General Theory of Relativity!

 

[Let'sthink]

In Holland it sure isn't, as far as I can tell :D

That should not be a big issue. Complex numbers can be introduced as ordered pairs and operations of sum, multiplication can be defined on them and sq rt(-1) can be taken as some device or technique for converting the definitions into simple algebra with i and its powers predefined.

 

[martinbn]

Looks like the intro chapter of a quantum computation textbook.

 

[bhobba]

So I ask, what would you call it?

It's a quantum particle. But mostly physicists are lazy and call it a particle.

If a precocious 12 year old asks what is a quantum particle hand them Feynman's QED book and say its just the start - what it really is will gradually emerge as you study more. Why can't I tell you now? - as Feynman knew - you need to build up to it and your math needs to develop.

Thanks
Bill

 

[haushofer]

That should not be a big issue. Complex numbers can be introduced as ordered pairs and operations of sum, multiplication can be defined on them and sq rt(-1) can be taken as some device or technique for converting the definitions into simple algebra with i and its powers predefined.

Yes, but I know from experience that students also take a conceptual leap in understanding complex numbers.

 

[A. Neumaier]

The photon is not a "particle".

There is no doubt that physicists consider the photon to be a particle.

The problem is that ''particle'' means for them something very different from the intuitive layman's notion of a tiny little bullet. Rather it means - as discussed in more detail in my Insight article The Physics of Virtual Particles - a collective, elementary excitation of a quantum field, described by an irreducible unitary representation of the Poincare group.

The 2017 Review of Particle Physics (issued by the Particle Data Group) has ''Particle Listings'' which may be taken as an authoritative definition of which objects are currently regarded as (existing or hypothetical) particles. The very first on the listings is the photon (gamma, as part of the ''Gauge & Higgs bosons'' listing). You'll find there upper bounds on its mass and charge, with references to corresponding experiments.

From the discussion in the introduction, one can see that a particle is something whose existence is inferred indirectly from a lot of statistics. But I was unable to find on their site a more precise definition of what the Particle Data Group means by a particle. It is obvious that they didn't think of it as a little bullet, but neither is it defined in terms of the standard model (which would render the photon to have mass and charge exactly zero, so that experiments about their value would be pointless). The 20 page text on Online Particle Physics Information consists primarily of references to useful information, but does not seem to have a reference to an authoritative glossary from which one could glean a concise explanation of what it means for leading edge experimental physicists to be a particle.

 

[haushofer]

Maybe we should speak of "quarticles", a contraction of quantum and particles.

 

[Stephen Tashi]

How can there be agreement on teaching an interpretation of QM to high school students when there is no agreement on an interpretation of QM among those with advanced degrees?

Perhaps the "shut up and calculate" approach is appropriate. To make an analogy, friction is (according to Feynman) a poorly understood phenomena, yet there is a standard pedagogy for teaching it. The approximation that friction is proportional to the normal force between surfaces is stated. Then it is possible to shut up about the exact nature of friction and expand the repertoire of practical (or practical sounding) problems that can be assigned to students.

If we take that approach to teaching QM (which I suspect the majority of forum members won't) then what is the repertoire of QM problems that will be assigned to students and what would they have to understand in order to work those problems?

 

[Zafa Pi]

Maybe we should speak of "quarticles", a contraction of quantum and particles.

That won't do at all.
1. The quart is too voluminous for a photon.
2. The quart has been supplanted by the liter (litre) in physics.
If quarks can have beauty or charm why not call photons cuticles?

Joking aside, after reading the above interesting posts and watching Feynman, where he uses particle over and over, I would answer the youngster/biologist with:
A photon is a particle of light. It obeys the unintuitive laws of QM rather than the usual classical laws like bullets. If you want an example I'll you tell about the double slit experiment.

 

[Dadface]

This thread has got rather complicated and gone well beyond the original intent of the opening post which was about "quantum theory for high school students". In the UK the QM content for school students is a tiny part of the overall syllabus and introduces topics such as spectra and energy levels, De-Broglie waves and photo electricity, as per Einsteins analysis of the subject.
Some experts here may dislike the syllabus requirements but the following facts should be remembered:

1. Quite rightly a major aim of the syllabus can be summarised as follows: "The content should be such that it helps students to develop an interest in physics". I think the QM content is at a good level to help achieve that aim.

2. People may object that Einsteins treatment of photo electricity has been superseded. But I don't think that matters provided that students are informed of that and that the syllabus requirements are such that they give a good introduction to the subject.

3. Only a tiny fraction of students will go on to study physics and the syllabus should try to cater for everybody. Again I think the QM content is such that it can spark an interest in many students including those who go on to study law, engineering, medicine etc.

 

[Peter Morgan]

In the absence of anything better, as far as I know, I would point someone to my YouTube video,

as a systematic alternative to thinking in terms of particles. Events are caused by someone placing an "event apparatus" as much as because someone turned on the power to another piece of apparatus on the other side of the room; it's as plausible for the "stuff" between to be a field as for there to be particles. QM/QFT only describes the statistics of events, it doesn't describe how those events happen. I want 3blue1brown to do a good YouTube video for that.
The subject of the original post, "Quantum theory for high-school students" might do well to formulate the whole construction in terms of Fourier analysis instead of in terms of differentiation and integration. More kids are familiar with frequency analysis, and only a subset of high-school students need to be able to actually do the transformations from time-domain to frequency-domain. We have 3blue1brown's good examples of how much can be done with the basic idea, most recently in his

Linear algebra is overkill for quantum field theory, because everything can be done in terms of addition and composition of operator actions, which can be said in simpler language as "applying successive modulations" to the vacuum state (I hesitate to mention my recent very rough attempt on YouTube to work with that, but the link is

and, again, there's no alternative that I know of).

 

[naysc]

I believe this could be interesting to many people here who are interested in quantum theory but are not (yet) professional physicists:
http://lanl.arxiv.org/abs/1803.07098

Could u pls help me to find the lectures coz I am not getting them!

<< Mentor Note -- Poster has been reminded not to use text speak at the PF >>

 

[vanhees71]

Well, it's soooooo hopeless :-(. I'm preparing for my theoretical-physics lectures for high-school-teacher students, which is my new (now permanent :-))))) job now. I'm just preparing my own manuscript, because I cannot with confidence recommend any of the textbooks labeled "Theoretical Physics for teachers" or something like that. Yesterday I got a pretty new one. It's called "Physics - understandable" (in German "Physik verständlich"), and it left a very ambition impression on me. On the one hand the aim of the book according to the foreword is to bring forward the intuition about theoretical physics, avoiding mathematical formalism. That's usually a warning sign, but on the other hand particularly for future teachers to get good intuition about theory is more important than to get the full formalism of mathematical subtleties, but the emphasis must be "good intuition" and not "some intuition".

The author is pretty aware of the many shortcomings of the standard literature for this audience, which is amazing since these shortcomings are often (unfortunately not always) overcome in even not too new textbooks for physics majors (undergrad students). He discusses all kinds of issues with these typical problematic topics. Of course, I immediately flipped to the two major obstacles in the textbook literature: relativistic (velocity-dependent) mass and intro QM. The joke is that he pleads strongly for the use of the velocity dependent mass, even with the wrong statement it's the gravitational mass as well as the inertial mass (and then in a later chapter telling it in the right way when summarizing the foundations of GR), then he gives the arguments against its use but says, he's of other opinion. He doesn't even mention the important point that one should not use coordinate dependent quantities and that energy together with momentum is the right thing to use and leave the mass a scalar (in the sense of Poincare/Lorentz transformations).

The QT part is even sadder. He starts with the Planck spectrum of black-body radiation which rightfully needs field quantization, i.e., the photon picture. Then he uses the naive billiard-ball photon picture all the time although in a very beautiful section he writes all the arguments against it, including the point that both Compton and photoelectric effects are explainable through the semiclassical approximation and explicitly (and rightly!) stating that both effects do not necessarily prove the necessity of field quantization. So, why the heck is he using the wrong intuitions although obviously knowing much better?

If even people who know their physics still write wrong books only because it's tradition in the didactics community, there's no hope for improvement :-(((.

 

[vanhees71]

Could u pls help me to find the lectures coz I am not getting them!

Just click on the pdf link, and you get the paper from the arXiv, or what are you looking for?

 

[Demystifier]

Could u pls help me to find the lectures coz I am not getting them!

Click on PDF in the up-right corner.

EDIT: Damn, @vanhees71 was faster.

 

[Dadface]

If even people who know their physics still write wrong books only because it's tradition in the didactics community, there's no hope for improvement :-(((.

I think possibly you may be in danger planning lectures that are more complicated than they need to be. I'm not familiar with the educational system in Germany but here in the UK all topics should be taught mainly as the curriculum demands.

I think you can easily deal with any shortcomings, for example you could explain that the syllabus requires that the QM course is an introductory course only which considers some of the historical developments of the subject . You could point out that QM has advanced greatly and continues to be developed and you could give references to any students who want to study the subject in greater detail. If you are required to teach relativistic mass then do so but point out that it's a concept that has gone out of favour with a majority of physicists.

I'm guessing that most of your your student teachers will go on to teach physics in high school and if that's the case I suggest that you look at the physics specifications of the exam boards used in Germany. It may also be helpful to look at the textbooks used by the school students.

 

[vanhees71]

I think possibly you may be in danger planning lectures that are more complicated than they need to be. I'm not familiar with the educational system in Germany but here in the UK all topics should be taught mainly as the curriculum demands.

I think you can easily deal with any shortcomings, for example you could explain that the syllabus requires that the QM course is an introductory course only which considers some of the historical developments of the subject . You could point out that QM has advanced greatly and continues to be developed and you could give references to any students who want to study the subject in greater detail. If you are required to teach relativistic mass then do so but point out that it's a concept that has gone out of favour with a majority of physicists.

I'm guessing that most of your your student teachers will go on to teach physics in high school and if that's the case I suggest that you look at the physics specifications of the exam boards used in Germany. It may also be helpful to look at the textbooks used by the school students.

Of course, I've to teach the curriculum, but this fortunately doesn't say that one should teach wrong things. When it comes to these issues, of course, I'll discuss them but also explain to them, why it's considered incorrect for decades now.

I've also studied a high-school textbook (see my posting on it in this thread), which was not developed much further from the textbook we had at school 28 years ago and which also contained the questionable idea of relativistic mass. As I said, of course, I've to discuss this with my students, as they will have to teach it to the poor high school students. In Germany the schools are subject to the federal states (which is another nuissance, because that implies we have 16 different curricula, which are mostly incompatible; so if parents have to move from one state to the other there's big trouble for the children at school). In Hessen we have what's called "Zentralabitur", i.e., all students have to take the same exam, implying that the teachers have to stick to the curriculum, and if they ask for the relativistic mass, they have to teach it, if you want it or not.

The photon issue is much easier to solve. You just say that photons are no point-like particle but field quanta that exchange energy and momentum with charged particles, where the energy-momentum relations ("on-shell conditions") as well as energy-momentum conservation hold in each process. If you check the books on photons, at the level of high school that's the only thing that is really used, and all is fine. No need for wrong intuitions at all! That's why I do not understand, why still the old wrong conceptions of before 1925 are taught today.

The rest of the QM curriculum at school discusses elementary Schrödinger-wave mechanics, and I also do not see any problem there to explain to them the Born rule (probabilistic interpretation) and problemetize the Copenhagen interpretation and old-fashioned remnants of the old quantum theory like the wave-particle dualism. It shouldn't also too difficult to understand that the uncertainty relation is a general proper of the quantum state and thus the preparation procedure rather than any impossible to accurately measure position or (sic!) momentum, no matter in which state the particle is prepared in.

Of course, also the history of sciences should be covered to a certain extent. To understand how the notions of today were developed, can help a lot to the understanding of the subject. Particularly it helps to clarify why the intuitive pictures provided by theoretial physics change all the time and why, e.g., nowadays mass is considered a Lorentz scalar and not velocity dependent anymore or why we believe in a much more abstract photon picture after about 70 years of modern QED and the tremendous progress of quantum optics (or generally AMO) during the last 2-3 decades.

Last but not least, I have two sets of manuscripts from professors who have given the course before, and there's nothing in these manuscript I wouldn't teach myself in this way. So I don't think that my views are too incompatible with what should be taught in these lectures.

 

[atyy]

Of course, also the history of sciences should be covered to a certain extent. To understand how the notions of today were developed, can help a lot to the understanding of the subject. Particularly it helps to clarify why the intuitive pictures provided by theoretial physics change all the time and why, e.g., nowadays mass is considered a Lorentz scalar and not velocity dependent anymore or why we believe in a much more abstract photon picture after about 70 years of modern QED and the tremendous progress of quantum optics (or generally AMO) during the last 2-3 decades.

I don't think it is helpful. If basic physics is changing all the time, then we can just not learn it, since by the time we learn it, it will change again.
https://www.lhc-closer.es/taking_a_closer_look_at_lhc/0.relativity
https://lhc-machine-outreach.web.cern.ch/lhc-machine-outreach/lhc-machine-outreach-faq.htm
http://www.einstein-online.info/dictionary/relativistic-mass.html

 

[Peter Morgan]

One terrible line in the paper referenced to by the original post, in section 3.5, " Why they actually represent probabilities is a question that has no good answer except that this is how quantum theory works and it can be verified experimentally." What?
The reason is pretty clear if one talks about states as normalized linear maps from operators to expected values. A state ρ, say, maps an algebra to ℂ, a hermitian operator A to an expected value, the identity to 1, and positive operators to positive values: $\rho:\mathcal{A}\rightarrow\mathbb{C};A\mapsto\rho(A), \underline{1}\mapsto 1, \rho(A^\dagger A)\ge 0$. $\rho_0(A)=\langle 0|A|0\rangle$ is the prototypical elementary state, from which we can construct other states such as $\langle\psi|A\psi\rangle$, assuming normalization. A natural projection operator is $|\phi\rangle\langle\phi|$, again assuming normalization, for which the expected value in the state $\langle\psi|A\psi\rangle$ is $\langle\psi|\phi\rangle\langle\phi|\psi\rangle=|\langle\phi|\psi\rangle|^2$. It's surely clear enough that probabilities emerge naturally as expected values associated with projection operators? Thinking of states as linear maps, which is a commonplace in mathematics, is far preferable to thinking of vectors in the Hilbert space as states, which is too much the default in physics. In the latter way of thinking, it seems that we have to use what looks like a quadratic expression, which is unhelpful.

 

[vanhees71]

I don't think it is helpful. If basic physics is changing all the time, then we can just not learn it, since by the time we learn it, it will change again.
https://www.lhc-closer.es/taking_a_closer_look_at_lhc/0.relativity
https://lhc-machine-outreach.web.cern.ch/lhc-machine-outreach/lhc-machine-outreach-faq.htm
http://www.einstein-online.info/dictionary/relativistic-mass.html

Hm, are you saying one cannot learn physics or any other natural science, because there is progress made in research? That's ridiculous and disproven by the many very good students working already on research topics (often leading to publishable results!) already in their BSc thesis in the universities around the world.

Thanks for pointing me to the nice first link from CERN. The 2nd one is already bad again. Why do they speak about relativistic mass when you can as well use energy? The 3rd link is an abuse of Einstein's signature, and the poor guy is dead and cannot fight against this abuse. Einstein clearly had the modern view against velcity/speed-dependent mass although the idea occurs in his 1905 paper and was used by Planck and others a few years later too. For a very good historical study on that question, see

https://doi.org/10.1063/1.881171

 

[atyy]

Hm, are you saying one cannot learn physics or any other natural science, because there is progress made in research? That's ridiculous and disproven by the many very good students working already on research topics (often leading to publishable results!) already in their BSc thesis in the universities around the world.

Thanks for pointing me to the nice first link from CERN. The 2nd one is already bad again. Why do they speak about relativistic mass when you can as well use energy? The 3rd link is an abuse of Einstein's signature, and the poor guy is dead and cannot fight against this abuse. Einstein clearly had the modern view against velcity/speed-dependent mass although the idea occurs in his 1905 paper and was used by Planck and others a few years later too. For a very good historical study on that question, see

https://doi.org/10.1063/1.881171

Well, apparently even Purcell and Feynman didn't understand relativity, years after Einstein and Minkowski established it, and after QED was already successful. So if they didn't understand it, why should we bother now?

 

[Demystifier]

If basic physics is changing all the time, then we can just not learn it, since by the time we learn it, it will change again.

https://www.goodreads.com/quotes/2397-there-is-a-theory-which-states-that-if-ever-anyone

 

[phywithAK]

Thanks Demystifier for this link, it was really something needed to get a little more interest in learning quantum mechanics. I too am studying quantum physics 101 here and something like this is what I wanted.

Please keep sharing such interesting links.

 

[diPoleMoment]

They aren't. The photon is not a "particle".

So I know they are not, yet if there are smaller quantum particles and every electron has a photon cloud, is it possible electrons are a condensed form of photons?

 

[diPoleMoment]

So read the Neother theorem pdf and it is way above my head. Math was never a strong point.

Euclidean rotations. If we make the further assumption that the potential only depends of the mutual distances |~ra−~rb| between the particles, and not on the orientation of the relative position vectors ~ra −~rb, V = V (|~r1 −~r2|,...,|~r1 −~rN|,|~r2 −~r3|,...) ,

So with the summations of the different position vectors, then centripetal force is not taken into consideration? Just out of curiosity,

 

[diPoleMoment]

In post 82, I said that it help to motivate why energy is the source of gravity in GR.

Here are examples:

Blau, Lecture notes on gravity http://www.blau.itp.unibe.ch/newlecturesGR.pdf, gives heuristic motivation for relativistic gravity (p20) with statements like: "We already know (from Special Relativity) that ρ is not a scalar but rather the 00-component of a tensor, the energy-momentum tensor".

Schutz, Gravity from the Ground up http://www.gravityfromthegroundup.org/ also makes use of the notion of relativistic mass.
p190 "As an object moves faster, its of an object increases with its speed. We noted above that no force, inertial mass increases, so it is harder to accelerate it. This enforces the speed of light as a limiting speed: as the object gets closer to the speed of light, its mass increases without bound"

On why rest mass is not the correct generalization for the source of gravity in GR:
p242 "What would happen to a gravitational field created by rest-mass when rest-mass is turned into energy by nuclear reactions? Would gravity disappear? This seems unreasonable. Rest mass is a dead end."

p242 "the active gravitational mass generates the curvature of time, which is the most important part of the geometry of gravity. Its density is defined as the density of ordinary mass-energy, plus three times the average pressure divided by c2."

So I am going simplistic, so pardon if it is way off. Once an object is in motion it will remain in motion until an equal and opposite force stops it. If there is no mass that is calculatingly significant, wouldn't this still be true?