# B Quantum theory for high-school students

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#### vanhees71

Science Advisor
Gold Member
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 text books 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 highschool 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 highschool 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

Science Advisor
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

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#### Peter Morgan

Gold Member
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

Science Advisor
Gold Member
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

Science Advisor
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?

#### 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?

#### Dadface

This thread seems to be going further and further beyond basic high school level. Just saying.

#### PeterDonis

Mentor
Thread closed for moderation.

#### PeterDonis

Mentor
This thread seems to be going further and further beyond basic high school level. Just saying.
And as a result, several technical discussions have been moved to new threads.

This thread is now reopened. Please keep discussion here limited to the specific topic of the teaching of quantum theory at the high school level.

#### bhobba

Mentor
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?
I doubt that eg the Feynman Lectures On Gravitation. He didn't like going to gravity conferences but do not confuse dislike with lack of understanding. And even then guys like Kip Thone claimed he had some rather non-trivial discussions with Feynman about GR. You can find out exactly what he did not like about gravity conferences here:
https://www.amazon.com/dp/0393340651/?tag=pfamazon01-20

BTW Feynman always claimed given what Einstein knew he could never have discovered relativity. I think he was referring to both the Special and General.

Of relevance here however is what should be taught at HS. IMHO its done all wrong here in Australia and the IB program - these are the two I know best.

You need a calculus based general physics course not only because the physics is explained better, but it reinforces what you learned/are learning in calculus. If you want to torture students you could use the Feynman Lectures - but most students are not in the class to get the most out of those three volumes at HS - a few could - but not the majority. Something like the following would be best for them:
http://www.physics2000.com/Pages/About.html

I know that book - its not too bad - but the QM bit needs to be supplemented by the teacher explaining, like most books about basic QM, its semi-historical. They should mention it will be changed later to something more modern as your physics education progresses.

Thanks
Bill

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#### bhobba

Mentor
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?
This is off - topic - please start a new thread to further discuss it if interested. But just a comment here - Newtons first law of motion actually follows from symmetry considerations - see - Landau - Mechanics - and the modern basis of classical physics - the principle of least action which follows from QM. Actually both the first and second law, as usually stated, are vacuous - but again a new thread is required.

But please, please if you want to discuss that start a new thread - and to answer your question - yes it would still be true - but explainig the details - please - not in this tread.

Thanks
Bill

#### bhobba

Mentor
So read the Neother theorem pdf and it is way above my head. Math was never a strong point.
Start a new thread at the B level about Noether. Me and others can explain it to you at that level, plus the very interesting history behind it.

It is one of the most important theorems of modern physics, and needs to be more widely known - especially by philosophers who by and large seem unaware of it.

Thanks
Bill

#### thierrykauf

Gold Member
The only reason, in my mind, why people start teaching quantum mechanics in high school, is not, sadly, because high school students suddenly became brighter, but because the amount of material needed to bring a student to the level of string theory is so large that you would need 48 hour days if you started in college :) I got a taste of this when I took a 4 semester graduate course on particle physics. The instructor told us "Your physics education stopped at 1926. I'm going to bring it to 1994 (the year I took the course). Fasten your seatbelt." Just trying to keep up with string theory papers (Witten's monthly 100 page articles for instance) was a full time job. I can't imagine facing a college student who only knows classical physics! And the thing is, not only is there more to teach but it's much harder material. So it requires either brighter students or teacher, and probably both.

#### cratylus

One thing the lectures do not seem to explain is complex numbers; they start out by assuming the students already know about those. Is that a valid assumption for high school students?

Not just this but vectors of complex numbers and various notations for the same mathematical entity - so I doubt that pedagogically it would succeed.

#### romsofia

I think it's a pretty cool idea, but that section on Eigenvalues/vectors is uh... leaves a lot to be desired.

#### QuantStart

Maybe we should speak of "quarticles", a contraction of quantum and particles.
Sometimes in the beginning quantum class I encourage students to think of quantum particles as "quantons" , peculiar objects from the quantum world, which all share peculiar non-classical features. I saw this term in the book "Quantics" by Levy-Leblond & Balibar.

#### vanhees71

Science Advisor
Gold Member
I doubt that eg the Feynman Lectures On Gravitation. He didn't like going to gravity conferences but do not confuse dislike with lack of understanding. And even then guys like Kip Thone claimed he had some rather non-trivial discussions with Feynman about GR. You can find out exactly what he did not like about gravity conferences here:
https://www.amazon.com/dp/0393340651/?tag=pfamazon01-20

BTW Feynman always claimed given what Einstein knew he could never have discovered relativity. I think he was referring to both the Special and General.

Of relevance here however is what should be taught at HS. IMHO its done all wrong here in Australia and the IB program - these are the two I know best.

You need a calculus based general physics course not only because the physics is explained better, but it reinforces what you learned/are learning in calculus. If you want to torture students you could use the Feynman Lectures - but most students are not in the class to get the most out of those three volumes at HS - a few could - but not the majority. Something like the following would be best for them:
http://www.physics2000.com/Pages/About.html

I know that book - its not too bad - but the QM bit needs to be supplemented by the teacher explaining, like most books about basic QM, its semi-historical. They should mention it will be changed later to something more modern as your physics education progresses.

Thanks
Bill
I don't think that you can use the Feynman Lectures in highschool. They are full-fledged introductory physics books at the university level, and I'd say they are rather theory than experimental books.

Nevertheless I agree with you that physics in the final classes of highschool should be taught calculus based, and I consider calculus a mandatory subject for any high-school student. Calculus must be consider a topic of general education for anybody at a high school, and indeed applications in physics are very nice examples for its applicability in real-world problems. In fact math is the key to almost everything in the modern world from the natural sciences and technology, including informatics to economy, sociology etc.

In other words any didactics, which tries to "avoid mathematics" in any STEM subject is flawed to begin with.

#### bhobba

Mentor
Calculus must be consider a topic of general education for anybody at a high school, and indeed applications in physics are very nice examples for its applicability in real-world problems. In fact math is the key to almost everything in the modern world from the natural sciences and technology, including informatics to economy, sociology etc.
Here in Aus, fortunately, those in power recognize it. But we still have some, you see them every now and then - it's abominable that you can graduate university without a foreign language etc

But I am heartened most reject it as non-sense. Whenever I hear such I ask people I know from a wide spectrum of occupations - chefs to engineers and every single one just laughs their head off - how ridicules. But most - well while not quite the exact opposite recognize math as an absolute necessity - and that includes calculus. What do they think, by and large, is THE career of the future? Big Data. I explain the central limit theorem to them - they see its importance straight away - also those that do not know it are a little surprised - but I tell then - it's true - but you need calculus to prove it - and not the simple stuff at HS either. They get my drift and see the necessity of advanced math at university.

In fact on a discussion panel show called Q&A here in Aus they had a number of educators and I thought - we will get more of this language/humanities type stuff. To my total surprise and amazement we got complete agreement all degrees in the future must include a significant amount of math and the declining number of people here in Aus not doing calculus based math at HS has to be stopped - its a matter of the utmost urgency.

It both gladdened and surprised the bejesus out of me.

Thanks
Bill

#### Dadface

I think that one thing which is relevant to this discussion and which seems to have been overlooked is illustrated by the question:

What percentage of high school students move on to study physics in greater detail?

I don't know the answer but I'm confident in assuming that the percentage is very small, even for those who choose physics as one of their specialist subjects, (A level in the UK). We have to be mindful of this when planning curricula and what we teach must not be too specialised and biased in favour of one small group of students.

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#### bhobba

Mentor
I don't know the answer but I'm confident in assuming that the percentage is very small
Actually I think its not that small - remember you have to do it for other subjects like engineering, biophysics, computational physics etc.

Here is Aus combined physics/math degrees are popular. Where I went to they have a specific strand in the math degree for engineers and physics double degrees - it's called applied and computational and the subjects are particularly valuable to those type of majors eg they must do the following 4 subjects:
MXB321 Applied Transport Theory, MXB322 Partial Differential Equations, MXB323 Dynamical Systems, MXB324 Computational Fluid Dynamics

A double degree in physics and math is excellent preparation for many post graduate degrees eg engineering which is moving towards masters as the basic qualification, scientific computing etc. The real action these days is not undergraduate - its graduate. You get a feel for what you enjoy/are good at and can make a wiser choice of career/qualification with a good background that will help in many areas.

Thanks
Bill

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