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

[entropy1]

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.

 

[A. Neumaier]

Is it worth it to read all this?

This depends on your ultimate goals. Why do you learn at all? What do you want to understand/do/achieve, and at which level? In science, learning never ends, as long as one is motivated to learn.

 

[entropy1]

This depends on your ultimate goals. Why do you learn at all? What do you want to understand/do/achieve, and at which level? In science, learning never ends, as long as one is motivated to learn.

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?

 

[A. Neumaier]

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?

Your questions don't have an answer without specifying the context - in this case your values, desires, and goals.
Do you understand life? yourself? your girl friend? Will you ever understand?

You learn some quantum mechanics by studying quantum mechanics, and you learn something about how it relates to other subjects. It is a huge subject - after almost 30 years of learning I still don't know enough to understand it at the deepest level. But I understand it well enough to explain most things about quantum mechanics that interest me to others (lay people and students) in an intelligible way.

 

[entropy1]

Could you illuminate me on whether reading 'The theoretical minimum' can be self-contained as a text? I feel that now I understand pure and mixed states a bit better (I have to re-peruse it though) that I know pretty much enough. So, what do I lack then?

I feel I would be happy if I understood the uncertainty principle, entanglement, the double slit experiment and the delayed quantum eraser. (and their relation)

 

[A. Neumaier]

I feel I would be happy if I understood the uncertainty principle, entanglement, the double slit experiment and the delayed quantum eraser.

There are levels of understanding. In some sense you understand something if you can answer to your satisfaction the questions you have about it. You understand better if you can answer them to the satisfaction of someone else. On a deeper level, you understand something if you can make sense of what others write about it, can discriminate between whether they write nonsense, or something meaningful. This includes noticing when they were sloppy (e.g., an indexing mistake). Reaching this level takes significantly longer. At this stage you should also be able to solve exercises related to the subject. Answering questions of others about the subject is another level that takes more practice, though of a different kind. You reached a really deep level of understanding if you can assess the subject for yourself and arrange the material in a personally motivated and sound way, ready for others to understand. This may take days or years of thinking about the subject, depending on what it is.

You'll probably always lack a lot unless you studied quantum mechanics for a whole life. So don't assess your growth by what you lack but by what you gained. A simple check would be how much of the wikipedia pages on the subject you can read with understanding before you get lost...

 

[jtbell]

Thirty-four years after finishing a Ph.D. in physics, I'm still learning new things about QM on this forum.

 

[entropy1]

There are levels of understanding. In some sense you understand something if you can answer to your satisfaction the questions you have about it. You understand better if you can answer them to the satisfaction of someone else. On a deeper level, you understand something if you can make sense of what others write about it, can discriminate between whether they write nonsense, or something meaningful. This includes noticing when they were sloppy (e.g., an indexing mistake). Reaching this level takes significantly longer. At this stage you should also be able to solve exercises related to the subject. Answering questions of others about the subject is another level that takes more practice, though of a different kind. You reached a really deep level of understanding if you can assess the subject for yourself and arrange the material in a personally motivated and sound way, ready for others to understand. This may take days or years of thinking about the subject, depending on what it is.

Thank you for the elaborated reply! So, how 'deep' can I go by self-study?? (at home)

 

[Demystifier]

So, how 'deep' can I go by self-study?? (at home)

It depends only on you (and on the literature you have access to).

 

[atyy]

But a very important point is that in a sense, the first few lectures are enough. There are a couple of mathematical details, but they are not that important. Even quantum field theory doesn't go beyond elementary quantum mechanics. For example, section 2.1 of http://www.theory.caltech.edu/people/preskill/ph229/notes/chap2.pdf is in an important sense all of quantum mechanics, and all of quantum field theory, and all of string theory.

The Theoretical Minimum is a decent text and will teach you all of quantum mechanics. It is a little idiosyncratic, but no more so than Landau and Lifshitz or Weinberg, which are both great texts. However, it is a bit chatty, so it may not be clear that all of quantum mechanics is very simple.

 

[A. Neumaier]

Thank you for the elaborated reply! So, how 'deep' can I go by self-study?? (at home)

Everything you need (with the exceptions mentioned below) is free on the web, usually in many variants. So if you don't get (physically or mentally) one account try another one. Until you are at research level you can safely ignore all articles and books that are behind a paywall.

Except for your time, concentration and determination - that must be contributed by yourself. You may be interested in reading Chapter C4: How to learn theoretical physics of my theoretical physics FAQ where I responded to others who self-study. Some of the other stuff in the FAQ might also be interesting for you, either now or at some later stage.

 

[entropy1]

Thank you all for the sources and the replies! What I'm going to say may sound odd: I learned from the theoretical minimum how states work (more or less ). A state represents properties (what is 'known') of a physical element like a particle, if you want to call it that. However, a state requires a new concept to be representable: a state space. The state space works perfectly well as a mathematical representation, but it has (I was told) no physical significance! Nonetheless states are mathematically very well defined (it seems to me). So, is the principle of defining a new space for each new concept you want to describe the guiding principle in QM? Then, all concepts would share a common similarity, and the core of QM could be seen as (abstract!) vectors, functions and spaces in a mathematical setting, every new space built on the previous one. Does that make any sense? I hope the question is clear.

 

[A. Neumaier]

is the principle of defining a new space for each new concept you want to describe the guiding principle in QM?

There are spaces for each kind of objects that one may possibly want to study in a geometric way. This is one of the guiding principles in mathematics. And much of understanding quantum mechanics is understanding its mathematical language.

 

[A. Neumaier]

The state space works perfectly well as a mathematical representation, but it has (I was told) no physical significance!

That's not quite true. It is like saying space has no physical significance since one cannot measure it, though it is the arena in which all motions occur.

Similarly, the state space has no direct physical significance but it is the arena where all the dynamics happens, and hence is eminently physical. For example, with an appropriate state space you can shrink a complex molecule in 3 dimensions to a single point in high dimension - and this is everywhere made use of.

 

[StevieTNZ]

You could try these books:
https://www.amazon.com/dp/9814327549/?tag=pfamazon01-20
and
https://www.amazon.com/dp/1584887036/?tag=pfamazon01-20

Browse through the contents page to see what it has, and whether that interests you.

 

[Truecrimson]

The state space works perfectly well as a mathematical representation, but it has (I was told) no physical significance! Nonetheless states are mathematically very well defined (it seems to me). So, is the principle of defining a new space for each new concept you want to describe the guiding principle in QM?

Classical probability theory also has states and state spaces even though we don't usually call them that. (But https://www.amazon.com/dp/0521804426/?tag=pfamazon01-20 did. I recommend the first chapter if you want to see the similarity.) Only the shapes are different. They are simplices (generalization of a triangle) in the classical theory. The shape of quantum state space is spherical for a qubit but more complicated in higher dimensions. This innovation by quantum mechanics is the source of all kinds of counter-intuitive features like entanglement and Bell violation.

You might enjoy how Scott Aaronson "tweaks" classical probability theory a little to get quantum mechanics: http://www.scottaaronson.com/democritus/lec9.html

 

[A. Neumaier]

You might enjoy how Scott Aaronson "tweaks" classical probability theory a little to get quantum mechanics: http://www.scottaaronson.com/democritus/lec9.html

But later you have to unlearn the weird negative probability stuff presented there and replace it by mathematically more respectable notions.

 

[Truecrimson]

I see Aaronson either talks about negative amplitudes or negative "probabilities" always in scare quotes and he does not forget to say that probabilities are always non-negative. So I don't see a problem with it. Unless you mean something else by "mathematically more respectable notions."

 

[A. Neumaier]

Aaronson either talks about negative amplitudes or negative "probabilities" always in scare quotes

Actually he is just confusing the reader with mentioning negative probabilities at all. Crossing out the corresponding parts of sentences and headings doesn't change anything and would be less confusing. Apart from this (and a similar sloppiness where he talks without need about p-norms with real nonnegative p and complex entries), it is indeed ok.
Thus he is just too wordy and emphasizes in his headings some irrelevant nonsense.

 

[atyy]

Thank you all for the sources and the replies! What I'm going to say may sound odd: I learned from the theoretical minimum how states work (more or less ). A state represents properties (what is 'known') of a physical element like a particle, if you want to call it that. However, a state requires a new concept to be representable: a state space. The state space works perfectly well as a mathematical representation, but it has (I was told) no physical significance! Nonetheless states are mathematically very well defined (it seems to me). So, is the principle of defining a new space for each new concept you want to describe the guiding principle in QM? Then, all concepts would share a common similarity, and the core of QM could be seen as (abstract!) vectors, functions and spaces in a mathematical setting, every new space built on the previous one. Does that make any sense? I hope the question is clear.

It is important to understand why we consider such a notion to be suspect. The reason is that in quantum mechanics, we need a "classical" observer to say when a measurement is performed. We don't have any easy way to make the observer fully quantum by incorporating him into a larger Hilbert space. If we do so, we seem to need yet another classical observer to observer the larger Hilbert space.

Since we cannot easily have a wave function of the universe and nothing else, we consider quantum mechanics to be an operational theory. The observer makes a subjective division of the world into a classical measuring apparatus, and a part described by a Hilbert space. Only the measurement outcomes are real (their distribution is described by expectation of observables, including correlation functions). The Hilbert space is not necessarily real, and just a fiction help us calculate the distribution of measurement outcomes.

If we wish to have a wave function of the universe and nothing else, one approach is the Many-Worlds Interpretation. However, it is unclear whether such an interpretation is truly coherent.

 

[A. Neumaier]

in quantum mechanics, we need a "classical" observer to say when a measurement is performed.

This is also needed in classical mechanics since the Hamiltonian formalism doesn't tell it.

we cannot easily have a wave function of the universe and nothing else

Nothing forbids to have this as easily in quantum mechanics as one can have a state of the universe in classical mechanics. The only difficulty in both cases is specifying exactly which state the universe is in. One doesn't need many worlds for it, neither in classical nor in quantum mechanics. One world is enough, and it features already all we know and ever will know.

The observer makes a subjective division of the world into a classical measuring apparatus

In classical mechanics, the observer also makes a subjective division of the world into a classical measuring apparatus and the system to be measured. And there is the problem of how to define the measurement result (a property of the measurement device) and how to relate it to the measured system (which is coupled through the dynamics of the universe, hence there is no simple bijection between what one reads off the device and a property of the measured system).

Thus one has in quantum mechanics no additional difficulties compared to the classical situation. But in quantum mechanics people turn it into a big philosophical problem while in classical mechanics everyone always adhered to shut-up-and-calculate.

So, how 'deep' can I go by self-study?? (at home)

You'll go deep only if you concentrate mainly on the formal side and mostly ignore the details of the many incompatible interpretations.

Quantum mechanics thrives because of the predictions from the formal structure, not from endless discussions about the interpretation.

 

[RockyMarciano]

In classical mechanics, the observer also makes a subjective division of the world into a classical measuring apparatus and the system to be measured. And there is the problem of how to define the measurement result (a property of the measurement device) and how to relate it to the measured system (which is coupled through the dynamics of the universe, hence there is no simple bijection between what one reads off the device and a property of the measured system).

Thus one has in quantum mechanics no additional difficulties compared to the classical situation. But in quantum mechanics people turn it into a big philosophical problem while in classical mechanics everyone always adhered to shut-up-and-calculate.

While what you say about classical mechanics in the first paragraph is true I would say there are clearly additional difficulties in quantum mechanics that lie in the fact that quantum mechanics has a dependence on the classical theory that is obvious in the fact that the observables are ultimately classical and all measurements are classical in that sense. So there would be no difficulties if QM had not this foundational dependence on classical physics(wich would also make easy to consider QM as a fundamental theory, a hard thing as long as this dependency remains). Of course you can always refer to the mathematical formalism disconnected with the physical measurements, but then it is just a mathematical theory, not a physical theory, it couldn't make any prediction.

 

[A. Neumaier]

if QM had not this foundational dependence on classical physics

Only some interpretations have it; it is not really needed - just over and over repeated for historical reasons.

In practice, classical = macroscopic limit of quantum mechanics, describes by statistical mechanics, so the classical is an intrinsic limiting part of the quantum. Observables are macroscopic, hence classical in this sense.

 

[RockyMarciano]

Only some interpretations have it; it is not really needed - just over and over repeated for historical reasons.

In practice, classical = macroscopic limit of quantum mechanics, describes by statistical mechanics, so the classical is an intrinsic limiting part of the quantum. Observables are macroscopic, hence classical in this sense.

Um, do you know any interpretation without measurements(if only for predictions to be possible in the first place)? If so that interpretation is pure math, not a physical theory.
The identity you use between classical and macroscopic limit of QM followed by declaring all observables macroscopic(therefore classical, and the observables are the connection between the formalism and the physical measurements) is another way to see this dependence on the classical theory but with the limiting part understood the other way around.
This follows from simple logic, no theory that has an explicit dependence on other can be considered its generalization. Or expressed in different words: a theory cannot depend on its special case.
Again if you are only referring to the mathematical formalism without reference to measurements and predictions what I'm saying doesn't apply, but then you are not dealing with a physical theory.

Edit:I see you edited the last sentence to avoid the logical issue. So are observables macroscopic?

 

[A. Neumaier]

So are observables macroscopic?

Of course, else a human cannot observe them.

Classical measurements also need the same specification of being macroscopic - so there is again no difference to the quantum case.
The math is in both cases free of measurement issues.

 

[RockyMarciano]

Of course, else a human cannot observe them.Classical measurements also need the same specification of being macroscopic - so there is again no difference to the quantum case.

Right, and hence classical, as you wrote.

The math is in both cases free of measurement issues.

Sure, that's what I said. Math is by definition free of measurement issues.

 

[A. Neumaier]

Sure, that's what I said. Math is by definition free of measurement issues.

yes, and the measurement issues are the same in classical and in quantum mechanics, since one can only observe macroscopic objects.

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

 

[eloheim]

But a very important point is that in a sense, the first few lectures are enough. There are a couple of mathematical details, but they are not that important.

One of the biggest problems for me as someone who intermittently wades deeper into the maths of various areas of physics is getting back up to speed with the mathematics each time, not to mention learning new stuff relevant to whatever physics I'm looking into.

"Mathematical Methods in the Physical Sciences" by Mary Boas.

Why'd you pick this one to study next? Is it general preparation for the maths used in physics? Or if you (or anyone else!) know a good mathematical minimum type book out there I'd love to hear it. Hopefully something that could be run through front to back, but also would work as a good refresher for just a certain concept or subject when consumed in smaller portions...

Thanks

 

[entropy1]

Why'd you pick this one to study next? Is it general preparation for the maths used in physics?

I picked Griffiths as my first book, but it showed to be too advanced for me to start with, so I took the advice to buy Boas. However, (don't remember how) I chose to start with Susskind. Now I'm actually not so sure if Boas is a good choice, for I discovered after buying it that it is mainly very practical and not theoretical and it is mostly about practicing math, which however can be handy in a way. I'm not sure if I should read Boas. Griffiths seems a good follow up to the theoretical minimum. I should mention I have a university degree on computer science, so (long ago!) I aquainted myself a bit with a wide range of math. (I am unfit for work, so I'm not an engineer).

 

[ZapperZ]

I picked Griffiths as my first book, but it showed to be too advanced for me to start with, so I took the advice to buy Boas. However, (don't remember how) I chose to start with Susskind. Now I'm actually not so sure if Boas is a good choice, for I discovered after buying it that it is mainly very practical and not theoretical and it is mostly about practicing math, which however can be handy in a way. I'm not sure if I should read Boas. Griffiths seems a good follow up to the theoretical minimum.

That's a very odd thing to say. That's like saying you want to build a house with your own two hands, but you don't want to learn the skills of using the tools.

Boas's text is meant for students who need the math, but simply do not have the time or the patience to learn the math in depth and under each separate topics. Read her "Intro" and "To the Students". It is meant to get someone up to speed to USE the math. Isn't that what you want to book for, to use the math to be able to understand QM (and Griffith's text?)? You didn't buy it to actually learn all the math from scratch, did you?

Zz.

 

[entropy1]

That's a very odd thing to say. That's like saying you want to build a house with your own two hands, but you don't want to learn the skills of using the tools.

Boas's text is meant for students who need the math, but simply do not have the time or the patience to learn the math in depth and under each separate topics. Read her "Intro" and "To the Students". It is meant to get someone up to speed to USE the math. Isn't that what you want to book for, to use the math to be able to understand QM (and Griffith's text?)? You didn't buy it to actually learn all the math from scratch, did you?

Zz.

I agree that practicing math, in particular in the field of QM, for someone who isn't aquainted with it yet (like me), can be useful. However, I am mostly interested in the theory of QM. Boas is very much focussed on excercising, and I am not sure the book will actually give me insight in how the math corresponds to the theory. I hope and expect is does, but finding out after reading the whole book that you haven't learned what you was looking for, would be disappointing to me. So if you say that Boas will give me insight in QM, I welcome that, but the question remains: are all the excercises really necessary to get insight in the theory of QM? I don't want to seem lazy, but due to my illness I have fatique and limited focus.

 

[ZapperZ]

I agree that practicing math, in particular in the field of QM, for someone who isn't aquainted with it yet (like me), can be useful. However, I am mostly interested in the theory of QM. Boas is very much focussed on excercising, and I am not sure the book will actually give me insight in how the math corresponds to the theory. I hope and expect is does, but finding out after reading the whole book that you haven't learned what you was looking for, would be disappointing to me. So if you say that Boas will give me insight in QM, I welcome that, but the question remains: are all the excercises really necessary to get insight in the theory of QM?

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.

 

[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.

So you say the math skills are essential to understand the matter, am I right? I should mention that as a result from my education I am distantly aquainted with many math principles, so I am afraid I waste my energy on repeating stuff. I guess I have to drop that argument since I have to master the math skills, right?

 

[ZapperZ]

So you say the math skills are essential to understand the matter, am I right? I should mention that as a result from my education I am distantly aquainted with many math principles, so I am afraid I waste my energy on repeating stuff. I guess I have to drop that argument since I have to master the math skills, right?

Again, read the Preface!

One point about your study of this material cannot be emphasized too strongly: To use mathematics effectively in applications, you need not just knowledge, but skill. Skill can be obtained only through practice. You can obtain a certain superficial knowledge of mathematics by listening to lectures, but you cannot obtain skill that way. How many students have I heard say "It looks so easy when you do it," or "I understand it but I can't do the problems!" Such statements show lack of practice and consequent lack of skill. The only way to develop the skill necessary to use this material in your later courses is to practice by solving many problems.

"distantly acquainted" doesn't cut it, and what Boas would consider as a superficial knowledge.

Zz.

 

[entropy1]

Again, read the Preface!
"distantly acquainted" doesn't cut it, and what Boas would consider as a superficial knowledge.

Zz.

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.

 

[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.