After the 'Theoretical minimum' series, what is essential to know about QM?

[vanhees71]

Again, I don't understand this. Boas's book is meant to introduce to you almost all the math you need to understand QM. You need SKILLS know how to use the math! That's why you need repeated drill exercises.

Only after you understand the math can you understand the "theory of QM". How do you think you'd expect to understand how to solve the quantum harmonic potential if you don't know what Hermite polynomials are, or how would you solve a spherical potential if you don't know how to find solutions that give you the Bessel function and the spherical harmonics? These are how the "math corresponds to the theory".

Zz.

I cannot agree more, and also I think you don't understand math, if you cannot use it. As a physicist, I found it very amusing when studying with the "pure mathematicians" mathematics (and I went to a lot of math lectures at the time, because I liked them, and it's also for physicists a good thing to know also the abstract side of maths with all its formal proofs and to think about things like the axiom of choice etc.) that they weren't able to solve (even not too complicated) integrals but were very eager to prove their existence. I was very proud, when my tutor, who was in his graduate studies in applied mathematics, asked for help to find the equations of motion in some continuum mechanics problem from Hamilton's principle. It was interesting, because the Lagrangian contained higher then first derivatives, and he couldn't figure out, how to do the variations and integrations by parts necessary. So I did it in my physicist's handwaving way, and it was clear after that that his action was right to derive the equation, which was known from the literature. Then he said, now he had to prove all my handwaving rigorously.

So you must keep in mind that math is different for physicists and pure mathematicians. I guess, you can know all in Bourbaki and still not be able to use it for the purpose of the natural sciences. Of course also the way scientists use math is sometimes not sufficient for a pure mathematician, where rigor in the formal proofs is the purpose and not so much the application in the sense of a calculational tool.

So, indeed, it seems that you must get for yourself clear what you want to study, before you buy books!

 

[entropy1]

Again, I don't understand this. Boas's book is meant to introduce to you almost all the math you need to understand QM. You need SKILLS know how to use the math! That's why you need repeated drill exercises.

Only after you understand the math can you understand the "theory of QM". How do you think you'd expect to understand how to solve the quantum harmonic potential if you don't know what Hermite polynomials are, or how would you solve a spherical potential if you don't know how to find solutions that give you the Bessel function and the spherical harmonics? These are how the "math corresponds to the theory".

Zz.

I'm not saying I don't want to practice. Of course I understand practice is an essential part of mastering a scientific discipline. But how much practice suffices?

Annecdote: I was surprised how clear entanglement became to me after reaching chapter 8 of the Theoretical Minimum. So I guess understanding the math has some strange effect on understanding the matter haha! QM is (I suspect) a beautiful theoretical framework.

 

[ZapperZ]

I'm not saying I don't want to practice. Of course I understand practice is an essential part of mastering a scientific discipline.

Actually, no. I re-read what you typed, and until this last post, you appear to not consider this at all, and I had to explicitly state this.

But how much practice suffices?

Until you think you can solve that type of a problem. Since you intend to do your own self-study, you have to do your own judgement. Can you, after you work through the chapter on Bessel function, be able to solve the radial part of the Hydrogen Schrodinger Equation? That is your measuring stick.

Zz.

 

[entropy1]

Actually, no. I re-read what you typed, and until this last post, you appear to not consider this at all, and I had to explicitly state this.

Sorry for the misunderstanding. Indeed, I hadn't made it clear enough.

 

[A. Neumaier]

But how much practice suffices?

There is a compromise between first practicing all the math or first trying to go for the theory. You can read any theory book of your choice, and whenever you encounter a concept or calculation that you can't make sense of you look it up in the math book and practice that part. This should work well while keeping you motivated. It will also show you which exercises you need to do!

 

[atyy]

The Bessel functions or spherical harmonics or whatever are not that important for the conceptual structure of quantum mechanics. You can look these special functions up when you need them. Even the important numerical calculations can have mathematical errors, see the note added in proof on p17-18 of http://www.fisica.unam.mx/grupos/altasenergias/kinoshita.pdf.

The key is entirely in finite dimensional vector spaces, which are the easy sort of linear algebra. Infinite dimensional vector spaces are also used in QM, but while there are mathematical subtleties, there are no physical ones. This is why Sakurai starts quantum mechanics using spin 1/2, because the overall structure of QM is very simple.

 

[eloheim]

However, there are a lot more excercises in the book than solutions, so you have to have access to the teacher's manual to study most of them at home. So Boas isn't that consistent with respect to her philosophy. I don't mean to attack you or her, I'm just making this observation. There very many excercises in the book and I think I'm going to limit my practice of them.

From what I've heard here about Boas (the text) I'm interested to give it a look.

One thing though I'm 100% on board with you about is the exercise solutions thing when you're engaging in independent study. I can recall discovering several perfect-looking textbooks that I had to immediately give up on because there was no way to check the exercises! I'm about to go investigate Boas more but does anyone know if this is going to be a dealbreaker??

 

[A. Neumaier]

The key is entirely in finite dimensional vector spaces

Only regarding the interpretation issues. For understanding the physics, infinite dimensions are essential. Already the harmonic oscillator needs infinite dimensions. The uncertainty relations needs canonical commutation relations and hence infinite dimensions. Understanding the quantum mechanics of atoms and hence the periodic systems needs infinite dimensions. Quantum optics needs infinite dimensions since it is interaction with harmonic oscillators.

Almost everything done in quantum mechanics is done in infinite dimensions. Only quantum information theory can do without it - but restricting quantum mechanics to the latter robs it of almost all really useful applications.

 

[atyy]

Only regarding the interpretation issues. For understanding the physics, infinite dimensions are essential. Already the harmonic oscillator needs infinite dimensions. The uncertainty relations needs canonical commutation relations and hence infinite dimensions. Understanding the quantum mechanics of atoms and hence the periodic systems needs infinite dimensions. Quantum optics needs infinite dimensions since it is interaction with harmonic oscillators.

Almost everything done in quantum mechanics is done in infinite dimensions. Only quantum information theory can do without it - but restricting quantum mechanics to the latter robs it of almost all really useful applications.

But you don't need to know the infinite dimensional stuff rigourously to do the applications. So for example, a text at the level of Dirac's is good enough.

One can use one's intuition for finite dimensional vector spaces to get by on the infinite dimensional ones. That'll be enough to get lots of useful things like atomic spectra and Rutherford scattering.

One will get some things wrong, but that will provide material for self-amusement https://www.physicsforums.com/threads/quantum-challenge-mathematical-paradoxes.868292/

 

[A. Neumaier]

you don't need to know the infinite dimensional stuff rigourously

You don't need to know it rigorously but you need the practice of doing the related computations correctly and understanding how to use the corresponding properties and arguments.

For example, one needs to be able to solve ordinary differential equations, especially linear ones, multivariate Fourier transforms, power series, a lot of complex analysis, the Gamma function, multivariate Gaussian and many other integrals, generating functions, the Laplace equation and associated special functions, separation of variables, Greens functions, Hilbert spaces, the spectral theorem, tensors, symplectic forms, etc. Not everything from the start but everything in the right place.

 

[bhobba]

Is there a level at which one could say you know 'enough' to 'understand' QM? And if not, does that mean I will never understand it? And if that is the case, what do I learn from studying QM?

Nobody understands QM in the sense they know all about it - you continually learn all the time.

But after Susskind I recommend the following in this order:
https://www.amazon.com/dp/0674843924/?tag=pfamazon01-20
https://www.amazon.com/dp/0071765638/?tag=pfamazon01-20
https://www.amazon.com/dp/0805387145/?tag=pfamazon01-20

After that there are a number of routes you can take (and it will take a while because you will need to learn the math as you go - I was fortunate in that I already had a degree in math that included Hilbert spacers etc):

If you want to see QM developed at an advanced level axiomatically then get Ballentine:
https://www.amazon.com/dp/9814578584/?tag=pfamazon01-20

If you want to delve into interpretational issues then get Schlosshauer
https://www.amazon.com/dp/3540357734/?tag=pfamazon01-20

If you want to go into QFT get:
https://www.amazon.com/dp/019969933X/?tag=pfamazon01-20

Be warned, especially without the mathematical background it no easy task. But if you take your time and persevere its doable.

Thanks
Bill

 

[entropy1]

Be warned, especially without the mathematical background it no easy task. But if you take your time and persevere its doable.

I certainly am very motivated. Sometimes it is just too much fun and I have to pause because of the excitement haha! However, I also might have problems focussing due to medications. Do you know if that is a dealbreaker for studying this kind of material?

 

[bhobba]

I certainly am very motivated. Sometimes it is just too much fun and I have to pause because of the excitement haha! However, I also might have problems focussing due to medications. Do you know if that is a dealbreaker for studying this kind of material?

Nope.

I too am on medications that affect focusing (specifically Avanza for depression). It simply takes longer - that's all.

Thanks
Bill

 

[entropy1]

One more question: what do we actually learn from studying QM except for the formalism? Does the formalism give any more insight in nature, and if so, which? Does the 'grasp' one gets on the math give any satisfaction, and if so, why?

 

[Demystifier]

One more question: what do we actually learn from studying QM except for the formalism? Does the formalism give any more insight in nature, and if so, which? Does the 'grasp' one gets on the math give any satisfaction, and if so, why?

We learn how to fool ourselves that we understand something which we really don't.

 

[LarryS]

The adagium of most quantumphysics-afficionado's is: "Shut up and calculate" - 'learn the formalism'. So I started with Leonard Susskind's 'Theoretical minimum' textbooks.

So now I know a little (very little) about the formalism, I started to wonder to which extent I have to go to educate myself in order to understand what I need to know. Is what you learn ever enough? And if not, why start with quantummechanics at all? Is it at all satisfying to study QM? Or is it that you learn more precisely what you don't know?

So my question is: after the 'Theoretical minumum' series, what is essential to know about QM? I have planned "Mathematical Methods in the Physical Sciences" by Mary Boas, follow by "An Introduction to Quantummechanics" by David Griffiths. This is quite a lifelong planning for me it seems to me. So, do I know anything more than I did when I've read all this? Is it worth it to read all this?

Can anyone elaborate on this? Much appreciated.

Susskind has done a TON of physics video courses in conjunction with Stanford University. If you have watched those only listed under theoreticalminimum.com you have probably not watched them all. There maybe additional courses under cosmolearning.com and also just under youtube.com. I have found all these courses extremely helpful in understanding modern physics. Also, IMHO, you will need to watch the series on Classical Field Theory if you want to understand QFT.

 

[entropy1]

I would also like to throw in a piece of advice: If you are new on the subject of QM, it might pay to start re-reading "The Theoretical Minimum: QM" when you've reached chapter 8. If you go back to the beginning, much of what is said there makes even more sense with the knowledge gained from the first read. My two cents.

 

[Giulio Prisco]

Is there a level at which one could say you know 'enough' to 'understand' QM? And if not, does that mean I will never understand it?

No and no. The fact that scientists are still arguing over interpretations of QM means that there are still gaps in our understanding of QM. In this sense, nobody knows enough to understand QM. But you can leave interpretations aside and come to "understand" QM well enough to use it for practical specific applications (they say "shut up and calculate"). Also, there is always the possibility of new theoretical developments in quantum interpretations (never say never).

 

[A. Neumaier]

One more question: what do we actually learn from studying QM except for the formalism? Does the formalism give any more insight in nature, and if so, which?

It gives a nearly endless number of insights when studied in enough breadth and depth.
It shows that all our concepts are uncertain when applied to Nature. It explains the meaning of chemical reactions and why some chemical elements bond much more than others. It explains the periodic system of elements. It allows to predict the possible results of chemical experiments and allows one to design new chemical substances. It explains the color of metals, gold, and other substances. It explains why the sun is burning for millions of years without changing much. It explains why a chair is hard enough to sit on although it consists of a myriad of pointlike particles that together make up less than 1% of the space occupied. It explains why water freezes at zero degree Celsius and boils at 100 degrees. It explains things like superconductivity or superfluidity. It allows you to understand why transistors work and how to design their properties. It predicts the results of high energy collision experiments that make headlines in newspapers. It allows to take part (a bit) in the thoughts of numerous Nobel prize winners in physics and chemistry. It allows you to read the Scientific American with a deeper understanding.

Does the 'grasp' one gets on the math give any satisfaction, and if so, why?

For many it does. For many others it is boring like math was for all their life. Why? Because it is interesting for the first group, and the interests of the second group lie elsewhere. You need to find out for yourself to which group you belong.

 

[entropy1]

For many it does. For many others it is boring like math was for all their life. Why? because it is interesting for the first group, and the interests of the second group lie elsewhere. You need to find out for yourself to which group you belong.

I have to say I am not a big hero at math; to the contrary: I'm really not good at it, so that is frustrating to me at times when studying QM. But I have to say that what I've studied from The Theoretical Minumum (it's really the basics, I know ) gives me a lot greater understanding of for instance entanglement. I had an idea in my head, and then, "puff!", it suddenly makes a lot of sense! (Or at least I think it does... )

 

[Giulio Prisco]

entropy1 - Think of math (at least of the thin math used by Susskind in the TM books) as very precise words. Math is really formalized common sense. The core concepts of QM can be explained with rather simple math. In comparison, fluid dynamics is a nightmare.

 

[Giulio Prisco]

By the way, the TM course has only two books so far but many excellent video lectures covering most of modern physics, including advanced quantum mechanics.

 

[entropy1]

So I guess I am not going to find out how it is possible that two entangled particles exhibit a (measurement-)correlation except for the fact that the formalism describes how the pure quantumstate of the pair leads to the correlation?

 

[Giulio Prisco]

So I guess I am not going to find out how it is possible that two entangled particles exhibit a (measurement-)correlation except for the fact that the formalism describes how the pure quantumstate of the pair leads to the correlation?

But you just explained it! ;-)

OK, you want to find an intuitive model of entanglement that you can visualize in your mind like you visualize a rock falling to the ground. I'm afraid that no, you aren't going to find it. Evolution has programmed us to throw rocks, not entangled particles.

 

[Giulio Prisco]

@entropy1 - Actually you can find find intuitive models of entanglement that you can visualize in your mind like you visualize a rock falling to the ground, just don't take them too seriously and don't push them too far. For example:

Two entangled particle are really "one thing," not two things. So picture 3D space as a 2D plane. Picture a circle in an orthogonal plane, with the center in the first plane. The two entangled particles are the intersections of the circle (one thing) and the plane. Now color half of the circle white and the other half black. Rotate the circle in its plane around its center (as a model of "what really happens"). The two intersections (particles) will always be correlated, if one is white the other is black.

 

[entropy1]

@entropy1 - Actually you can find find intuitive models of entanglement that you can visualize in your mind like you visualize a rock falling to the ground, just don't take them too seriously and don't push them too far. For example:

Two entangled particle are really "one thing," not two things. So picture 3D space as a 2D plane. Picture a circle in an orthogonal plane, with the center in the first plane. The two entangled particles are the intersections of the circle (one thing) and the plane. Now color half of the circle white and the other half black. Rotate the circle in its plane around its center (as a model of "what really happens"). The two intersections (particles) will always be correlated, if one is white the other is black.

Yes, however, doesn't that impose non-locality as a fact?

 

[Demystifier]

Yes, however, doesn't that impose non-locality as a fact?

So?

 

[Giulio Prisco]

Yes, however, doesn't that impose non-locality as a fact?

Yes it does.

 

[entropy1]

Yes it does.

Well, then it's easy.

 

[ogg]

A couple of random points.
In his video lectures, Susskind 'explains' entanglement by asking students something along the following (sorry, its been a few years): If I have two coins: a penny and a dime and place one in my pocket and the other in a friend's. If she then travels 10 light-years from me and at an agreed upon time looks into her pocket, how long will it take her to determine what coin is in my pocket? (Assuming no change of clothes, etc.) If your answer is anything LESS than 10 years, then you seemingly have a violation of locality, since any signal I send (at agreed upon time) will take >= 10 yrs to reach her.
Second point: QFT and more specifically the Standard Model (of Particle Physics) is not the same as QM, but QM is used as an umbrella term describing both QM and QFT. It is QFT which is the more "fundamental" basis for our understanding of the physical world.
Third: While QM/QFT involves relatively "simple" math, Yang-Mills Theory (QFT) has yet to be proved to be mathematically consistent (see Wikipedia entries, especially Constructive quantum field theory; as well as Millennium Prize Problems; Yang-Mills Theory; QFT; etc.)
Fourth: Relativistic QFT involves (surprize, surprize!) relativistic physics. General Relativity is NOT simple math. (although the need to go much beyond the much simpler Special Relativity is moot).
Fifth: As a starting "rule of thumb" you need to practice something for ~10,000 hours to become "skilled". This is the type of time commitment you should plan on IF your goal is to "understand" QFT or QM. People like me who have NOT put in the sweat and time might be able to follow along in the solution of a non-trivial problem, but can't claim that given a random physical system (real world) that we could correctly predict the outcome of a specified experiment a priori. I am satisfied with understanding QM in very broad strokes and in only the smallest most simplified systems. Your mileage may vary.

 

[Demystifier]

Well, then it's easy.

All conceptual puzzles in physics are easy when you think of them in the right way.

 

[RockyMarciano]

There is no dependence of one theory on the other, neither regarding the math nor regarding measurement issues.

(Sorry for the late reply, I couldn't answer earlier.)
The dependence I referred above is just paraphrasing the words of the great physicist Lev Landau in the first pages(2-3) of his Quantum mechanics(nonrelativistic) volume in theoretical physics. He wrote:"...we first examine the special nature of the interrelation between quantum mechanics and classical mechanics. A more general theory can usually be formulated in a logically complete manner, independently of a less general theory which forms a limiting case of it.[...] It is in principle impossible, however, to formulate the basic concepts of quantum mechanics without using classical mechanics [...] Hence it is clear that, for a system composed only of quantum objects, it would be entirely impossible to construct any logically independent mechanics.[...] Thus quantum mechanics occupies a very unusual place among physical theories: it contains classical mechanics as a limiting case, yet at the same time it requires this limiting case for its own formulation."

 

[A. Neumaier]

It is in principle impossible, however, to formulate the basic concepts of quantum mechanics without using classical mechanics

This was considered true when Landau wrote his book, but it is no longer true since we know better how macroscopic (i.e., classical) properties derive from microscopic (i.e., quantum) ones.

 

[entropy1]

I was wondering if Ballentine would be a nice follow up for Susskind's TM?

 

[Truecrimson]

How well can you handle Griffiths right now? (Not that one has to get past Griffiths first but Ballentine will be harder than that.)