The mathematical formalism is required to understand the concepts.
That is exactly the process taken in my favourite QM book, Ballentine, and is much more rational than the semi historical approach usually taken.
The only problem with Ballentine is it is at graduate level.
I have always thought a book like Ballentine, but accessible to undergraduate students, would be the ideal introduction.
In particular it would have a 'watered' down version of the very important chapter 3 that explains the dynamics of QM from symmetry. Its a long hard slog even for math graduates like me - definitely not for undergraduates. But the key results and theorems can be stated, and their importance explained, without the proofs. I think its very important for beginning students to understand the correct foundation of Schroedinger's equation etc from the start - if the not the mathematical detail.
I don't think there's one best way. Some learn better using the approach advocated here. Others learn better the other way around.
Honestly I think at the undergraduate level QM is the easiest physics class one has to take. It is just a cookbook on calculations. Every book is uninspired and my class was certainly uninspired. It is an incredibly boring subject at this level. So I don't think difficulty is the issue. It is simply the lack of physical concepts and a healthy dose of philosophy that is avoided when teaching QM at the undergraduate level. Indeed one of the professors I know basically called Griffiths' book a cookbook in differential equations. A good book can go a long way. For me the saving grace was Landau and Lifshitz. It is the sole reason I started liking QM. Seriously the way undergrad QM is taught really isn't fun for the students. Boredom from a lack of intellectusl stimulation really isn't how a physics class should be.
Hmmm, I still can't derive the Stefan-Boltzmann whatever - chills down my spine. How is that easy?
Is it easy?
Im actually not sure what youre referring to. Are you talking about the Stefan Boltzmann law of radiation? Im not sure what that has to do with undergrad QM apart from historical impetus but there is a particularly lucid derivation in section 9.13 of Reif if youre interested. It's more of a statistical mechanics derivation. Which is good because statistical mechanics, both classical and quantum, is actually extremely interesting at the undergrad level.
That I agree with.
I gave it away for health reasons no need to go into here. But I did enrol in a Masters in Applied Math at my old alma mater that included a good dose of QM. When mapping out the course structure with my adviser he said forget the intro QM course - since you have taken courses on advanced linear algebra, Hilbert spaces, partial differential equations etc it's completely redundant. Other students with a similar background to mine were totally bored. He suggested I start on the advanced course right away.
Really I think it points to doing a math of QM course before the actual QM course where you study the Dirac notation etc - basically the first and a bit of the second chapter of Ballentine. You can then get stuck into the actual QM.
And yes - I like Landau and Lifshitz too. Their Mechanics book was a revelation; QM, while good and better than most, wasn't quite as impressive to me as Ballintine. But like all books in that series it's, how to put it, terse, and the problems are, again how to put it, challenging, but to compensate actually relevant.
Actually, I only dimly remember what it is, although it was very exciting. It sounds right that it should be in a stat mech book, because the whole point IIRC was that classical thermodynamics was able to derive all sorts of completely correct things about blackbody radiation, yet classical stat mech could not. Then miraculously when one switched to quantum stat mech everything fell in place with classical thermo. I remember the narrative, but none of the calculations except Planck's. The text we used was Gasiorowicz, and I think his chapter 1 is all about this.
Apart from the Stefan-Boltzmann law, the other amazing derivation was Wien's displacement law. IIRC, these were all from classical thermodynamics, with no quantum mechanics, yet they are correct!
Well, the historic approach is bad. You are taught "old quantum mechanics" a la Einstein and Bohr only to be adviced to forget all this right away when doing "new quantum mechanics". I've never heard that it is a good didactical approach to teach something you want the students to forget. They always forget inevitably most important things you try to teach them anyway, but in a kind of Murphy's Law they remember all the wrong things being taught in the introductory QM lecture.
You see it in this forum: Most people remember the utmost wrong picture about photons, and it is very difficult to make them forget these ideas, because they are apparently simple. The only trouble is they are also very wrong. As Einstein said, you should explain things as simple as possible but not simpler.
Concerning philosophy, I think the healthy dose is 0! Nobody tends to introduce some philosophy in the introductory mechanics or electrodynamics lecture. Why should one need to do so in introdutory QM?
If you want to rise interpretational problems at all, you shouldn't do this in QM 1 or at least not too early. First you should understand the pure physics, and that's done with the minimal statistical interpretation. If you like Landau/Lifshits (all volumes are among the most excellent textbooks ever written, but they are for sure not for undergrads; this holds also true for the also very excellent Feynman lectures which are clearly not a freshmen course but benefit advanced students a lot), I don't understand why you like to introduce philosophy into a QM lecture. This book is totally void of it, and that's partially what it makes so good ;-)).
And to make matters worse they do not go back and show exactly how the correct theory accounts for the historical stuff and students are left with a sort of hodge podge, not knowing whats been replaced and what changed or the why of things like the double slit experiment.
May be QM is primarily predictive. Quantum mechanics construed as a predictive structure. After we try to interpret it with épistemic or ontological human sense.
For example "The debate on the interpretation of quantum mechanics has been dominated by a lasting controversy between realists and empiricists" : http://michel.bitbol.pagesperso-orange.fr/transcendental.html
I think philosophers worry more about that sort of thing more than physicists or mathematicians.
An axiomatic development similar to what Ballentine does is all that's really required, with perhaps a bit of interpretational stuff thrown in just to keep the key idea behind the principles clear.
And I really do mean IDEA - not ideas - see post 137:
It always amazes me exactly the minimal assumptions that goes into QM and what needs 'interpreting'.
When discussing the best approach to teaching something like quantum mechanics, I think you really have to consider the purpose in teaching it. Some of the people studying quantum mechanics are going to go on to become physics researchers, but my guess is that that is a tiny, tiny fraction. A small fraction of those who learn QM go on to get undergrad physics degrees, and a small fraction of them go on to get postgraduate physics degrees, and a small fraction of them go on to get jobs as physics researchers. So for the majority (I'm pretty sure it's a majority) who are not going to become physics researchers, what do we want them to know about quantum mechanics?
I'm not asking these as rhetorical questions, I really don't know. But I think that if we want people to be able to solve problems in QM, there might be a best way to teach it to get them up to speed in solving problems. If we want them to understand the mathematical foundations, there might be a different way to teach it. If we want them to be able to apply QM to problems arising in other fields--say chemistry or biology or electronics--there might be another best way to teach it.
So when people say things like "You shouldn't bring up X, because that will just confuse the student" or "The historical approach, with all of its false starts and blunders, is just not relevant to today's students", they need to get clear what, exactly, they want the student to get out of their course in QM. And I think that the answer to that question isn't always the same for all students.
I assume you mean the idea expressed by the sentence:
I would say that that's a single sentence, but I'm not sure I would call it a single idea. There are many other ideas involved in understanding why we would want basis-independence, why we are looking for probabilities in the first place, why we want the outcome probabilities to be determined by [itex]E_i[/itex] (as opposed to depending on both the system being measured and the device doing the measurement), what is an "observation" or "measurement", why should it have a discrete set of possible results, etc.
probably not theory, but the people :
Erwin Schrodinger : Mind and matter - What Is Life? - My View of the World - ...
Werner Heisenberg : Physics and Philosophy: The Revolution in Modern Science - Mind and Matter - The physicist's conception of nature - ...
Know both those books - but they are old mate.
These days the following is much better at that sort of level:
But of relevance to this thread you will get a lot more out of that book if you know some of the real deal detail.
To understand the quantum theory in terms of mathematical language, we have in "France" some good free lecture like this one from "Ecole polytechnique" : http://www.phys.ens.fr/~dalibard/Notes_de_cours/X_MQ_2003.pdf
on the other side there is not a unique look on its interpretation.
In fact Landau and Lifshitz introduce philosophy early and correctly in their QM book, which is what makes it so wonderful.
I just wanted to add that, whether or not the student is going to go on to become a physicist, there are certain ways to teach quantum mechanics that I think are just bad. There might be ways to teach a little bit of the feel of what quantum mechanics is about without getting into the mathematics that would be necessary to solve actual problems. But what is worse than useless is to skip the actual facts about quantum mechanics and instead teach people sound bites about how "Quantum mechanics teaches us that the mind creates its own reality" or whatever Deepak Chopra might say about it. However, the goal of giving the layman a flavor of quantum mechanics without being misleading is very difficult to pull off.
I start my undergrad QM course with Quantum Mechanics and Experience, David Z. Albert, Harvard Univ Press, 1992, ISBN 0-674-74113-7. It's not the dry math start that you find in, say, Principles of Quantum Mechanics, 2nd Ed., R. Shankar, Plenum Press, 1994, ISBN 0-306-44790-8. Don't get me wrong, I like Shankar and use it after the students do the calculations in Albert and some AJP papers cited below. I choose this intro because it involves some interesting phenomena that we can easily model mathematically. The phenomena is electron spin to include entanglement, so its "weirdness" tends to motivate the students to work on the matrix algebra needed to model it. And, the parameters in the matrix algebra correspond directly to Stern-Gerlach orientations and spatial locations of detector outcomes which are easy to visualize. Thus, while the outcomes are "mysterious," the modeling of the experiment is intuitive. I then have them reproduce the quantum calculations for each of Mermin's AJP papers on "no instruction sets":
"Bringing home the quantum world: Quantum mysteries for anybody," N.D. Mermin, Am. J. Phys. 49, Oct 1981, 940-943.
“Quantum mysteries revisited,” N.D. Mermin, Am. J. Phys. 58, Aug 1990, 731-734.
“Quantum mysteries refined,” N.D. Mermin, Am. J. Phys. 62, Oct 1994, 880-887.
Again, in each case, there is an easy-to-understand counterintuitive outcome that motivates the students to work with the simple, intuitive matrix modeling. We finish this intro by reproducing all the calculations in:
“Entangled photons, nonlocality, and Bell inequalities in the undergraduate laboratory,” D. Dehlinger and M.W. Mitchell, Am. J. Phys. 70, Sep 2002, 903-910
to include the error analysis. That gives them a grounding in an actual experiment. Only after all that do we proceed to Shankar.
However; There seems to be a Platonic trend among the "speculative" types including all the string theorists,
i.e.-no observables, no predictions....sounds like an elegant theory of pure mathematics.
Multiverses, "anthropic principle, demanding multiverses, Maldecena's conjecture ADS/cft also elegant
but lacking physical relevance. His holographic universe came about because he felt that information is
conserved in two dimensions inside black holes! Have these people no humility?
Q.M. requires more than analysis unless your limited to applied physics and just don't care.
The power of QM of course lies in its mathematical formalism but it is a physical theory and requires
interpretation. At this time, however (I'll say it again) all interpretation is premature. but even the
extraordinarily inelegant interpretations are better than a strictly analytical i.e.-Platonic approach.
Pure math is the last thing QM is.
At the axiomatic level the primitive of the theory is an observation eg see post 137:
The fundamental axiom is:
An observation/measurement with possible outcomes i = 1, 2, 3 ..... is described by a POVM Ei such that the probability of outcome i is determined by Ei, and only by Ei, in particular it does not depend on what POVM it is part of.
When you get right down to it much of the difficulty of QM boils down to exactly what is an observation? Its generally taken to be something that occurs here in an assumed classical common-sense world. But QM is supposed to be the theory that explains that world - yet assumes its existence from the get-go.
Much of the modern research into the foundations of QM has been how to resolve that tricky issue - with decoherence playing a prominent role.
A lot of progress has been made - but issues still remain - although opinions vary as to how serious they are.
The outcome of single quantun outcomes does not yield a meaningful outcome. Iteration of "identical
experiments" yields probability densities. P-densities do not predict where a quantum event will occur
only statistical weightings
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