# Why Is Quantum Mechanics So Difficult?

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Strangely enough, QM’s formalism isn’t any more difficult than other areas of physics. The mathematics of the “standard” QM isn’t any worse than, let’s say, electromagnetism. Yet, to many people, especially non-physicists, QM presents a very daunting effort to understand.

I strongly believe that it all comes down to how we understand things and how we expand our knowledge. Typically, when we teach students new things, what we do is build upon their existing understanding. We hope that a student already has a foundation of knowledge in certain areas, such as basic mathematics, etc., so that we can use that to teach them about forces, motion, energy, and other fun stuff in intro physics. Then, after they understand the basic ideas, we show them the same thing, but with more complications added to it.

The same thing occurs when we try to help a student doing a homework problem. We always try to ask what the student know already, such as the basic principle being tested in that question. Does the student know where to start? What about the most general form of the equation that is relevant to the problem? Once we know a starting point, we then build on that to tackle that problem.

The common thread in both cases is that there exists a STARTING point as a reference foundation on which, other “new” stuff are built upon. We learn new and unknown subject based upon what we have already understood. This is something crucial to keep in mind because in the study of QM, this part is missing! I am certain that for most non-physicists, this is the most common reason why QM is puzzling, and why quacks and other people who are trying to use QM into other areas such as “metaphysics” or mysticism, are using it in a completely hilarious fashion.

There is a complete disconnect between our “existing” understanding of the universe based on classical understanding, and QM. There is nothing about our understanding of classical mechanics that we can build on to understand QM. We use the identical words such as particle, wave, spin, energy, position, momentum, etc… but in QM, they attain a very different nature. You can’t explain these using existing classical concepts. The line between these two is not continuous, at least, not as of now. How does one use classical idea of a “spin” to explain a spin 1/2 particle in which one only regains the identical symmetry only upon two complete revolutions? We simply have to accept that we use the same word, but to ONLY mean that it produces a magnetic moment. It has nothing to do with anything that’s spinning classically. We can’t build the understanding of the QM spin using existing classical spin that we have already understood.

Now interestingly enough, the MATHEMATICAL FORMULATION of QM is quite familiar! The time-dependent Schrodinger equation has the same structure as a standard wave equation. We call the energy operator as the Hamiltonian not for nothing since it looks very familiar with the hamiltonian approach to classical mechanics. The matrix formulation also isn’t anything new. What this means is that while the conceptual foundation of QM is completely disconnected with our traditional conceptual understanding, the mathematical formulation of QM completely follows from our existing understanding! Mathematically, there is no discontinuity. We build the formalism of QM based on our existing understanding!

This is why, in previous threads in PF, I disagree that we should teach students the concepts of QM FIRST, rather than the mathematical formulation straightaway. There is nothing to “build on” in terms of conceptual understanding. We end up telling the students what they are out of thin air. The postulates of QM did not come out of our classical understanding of our world. Instead, the mathematical formalism is the only thing that saves us from dangling in mid air. It is the only thing in which our existing understanding can be built on.

What this implies clearly is that, if one lacks the understanding of the mathematical formalism of QM, one really hasn’t understood QM at all! One ends up with all these weird, unexplained, unfamiliar, and frankly, rather strange ideas on how the world works. These conceptual description QM may even appear “mystical”. It is not surprising that such connections are being made between QM and various forms of mysticism. One lacks any connection with the existing reality that one has understood. So somehow, since QM can do this, it seems as if it’s a license to simply invent stuff weely neely.

The mathematical formalism of QM is what defines the QM description. The “conceptual description” is secondary, and is only present because we desire some physical description based on what we already have classically. It is why people can disagree on the interpretation of QM, yet they all agree on the source, the mathematical formalism of QM.

This, however, does not mean that QM is nothing more than “just mathematics”. This is no more true than saying the musical notes on a sheet of paper are just scribbles. The notes are not the important object. Rather, it is the sound that it represents that’s the main point. The musical notes are simply a means to convey that point clearly and unambiguously. Similarly, the mathematics that is inherent in QM and in all of physics, is a means to convey an idea or principle. It is a form of communication, and so far it is the ONLY form of communication accurate and unambiguous enough to describe our universe. It reflects completely our understanding of a phenomena. So a mathematical formulation isn’t “just math”.

You cannot use your existing understanding of the universe to try to understand the various concepts of QM. There is a discontinuity between the two. It is only via the mathematical continuity of the description can there be a smooth transition to build upon. Without this, QM will not make “sense”.

Originally posted on Jun16-14

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251 replies
1. bhobba says:

This is why, in previous threads in PF, I disagree that we should teach students the concepts of QM FIRST, rather than the mathematical formulation straightaway.

Agree entirely.

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.

Thanks

Bill

2. bohm2 says:

I don't think there's one best way. Some learn better using the approach advocated here. Others learn better the other way around.

3. WannabeNewton says:

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.

4. atyy says:

Hmmm, I still can't derive the Stefan-Boltzmann whatever – chills down my spine.  How is that easy?

5. WannabeNewton says:

Hmmm, I still can't derive the Stefan-Boltzmann whatever – chills down my spine.  How is that easy?

What o.O

6. WannabeNewton says:

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.

7. bhobba says:

Indeed one of the professors I know basically called Griffiths' book a cookbook in differential equations.

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.

Thanks

Bill

8. atyy says:

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.

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!

9. vanhees71 says:

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.

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 ;-)).

10. bhobba says:

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.

Abso-friggen-lutely.

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.

Thanks

Bill

11. microsansfil says:

It is just a cookbook on calculations.

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

Patrick

12. bhobba says:

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'.

Thanks

Bill

13. stevendaryl says:

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.

14. stevendaryl says:

And I really do mean IDEA – not ideas – see post 137:

I assume you mean the idea expressed by the sentence:

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.

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 $E_i$ (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.

15. microsansfil says:

I think philosophers worry more about that sort of thing more than physicists or mathematicians.

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 – …

Patrick

16. bhobba says:

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:

http://www.amazon.com/Understanding-Quantum-Mechanics-Roland-Omnès/dp/0691004358

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.

Thanks

Bill

17. microsansfil says:

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.

Patrick

18. atyy says:

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 ;-)).

In fact Landau and Lifshitz introduce philosophy early and correctly in their QM book, which is what makes it so wonderful.

19. stevendaryl says:

When discussing the best approach to teaching something like quantum mechanics, I think you really have to consider the purpose in teaching it…

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.