How much theory of functions does one need to study Quantum Mechanics?

[raopeng]

For example, how far should one go in real analysis or functional analysis to learn the necessary mathematical formalism in Quantum Mechanics? I have some grounding in differential equations, algebra, complex analysis, and rudiments of real analysis included in a classical analysis course. Would that suffice to start to self-study Quantum Mechanics? Thanks for the replies

 

[bhobba]

That's enough if you have some linear algebra although some knowledge of Hilbert spaces would be useful but not essential.

Thanks
Bill

 

[dextercioby]

Depends on how <deep> you wish to go. Normally, for self-study you can do nicely with the book by Leslie Ballentine whose level of mathematics is moderate. Indeed, linear algebra is the key mathematical ingredient, for some things you learn in finite dimensions transport to infinite dimensions.

 

[WannabeNewton]

I'm curious, dexter and bhobba, as to whether having a much more extensive knowledge of the mathematics behind QM made it harder for you to read the more physics-centered books (i.e. books centered on actual experiments, physical applications, physical phenomena etc.) in the sense that the potential hand waviness in such books becomes unbearable to sit through?

 

[dextercioby]

Speaking for myself, it did, since for the physical side, I've decided to take it from the notes I've received in class, a decade ago. So for me it was ok, since the level of the notes was satisfactory at the time, I only later turned into a more mathematically minded reader of Quantum Mechanics.

And hand-waviness is acceptable, since most people learning do not pursue mathematics on their own and will not end up as mathematical physicists.

 

[WannabeNewton]

Interesting. So would you say it is better to start out with the less mathematically inclined but more physics inclined texts before going straight into the QM books that focus mainly on the mathematical physics / mathematics?

 

[AlephZero]

So would you say it is better to start out with the less mathematically inclined but more physics inclined texts before going straight into the QM books that focus mainly on the mathematical physics / mathematics?

I would be inclined to answer "yes" for any branch of physics. Science is based on experiment and observation. It isn't a sub-branch of mathematics.

 

[Lavabug]

You could try a good rigorous book written by a master experimentalist... Cohen-Tannoudji. The only requirements to this is some basic linear algebra and (obviously) all of calculus, because everything else necessary is introduced in the text from the ground up (hence the ton of appendices and complements in the book). The info is all over the place, but it is pretty much self-contained.

The explanation of the Stern-Gerlach experiment in that book is the best I've ever seen. No hand waving to be found.

 

[ZombieFeynman]

I would be inclined to answer "yes" for any branch of physics. Science is based on experiment and observation. It isn't a sub-branch of mathematics.

I don't think this could have been said better.

 

[bhobba]

I'm curious, dexter and bhobba, as to whether having a much more extensive knowledge of the mathematics behind QM made it harder for you to read the more physics-centered books (i.e. books centered on actual experiments, physical applications, physical phenomena etc.) in the sense that the potential hand waviness in such books becomes unbearable to sit through?

It did. I did a degree in math and when I learned QM learned it from Dirac's and Von Neumans books - Von Neumann was fine - its math is rigorous - as you would expect from a mathematician of his stature. But Dirac - whoa - maddening. It sent me off on a tangent investigating Rigged Hilbert spaces and distribution theory. Eventually I emerged with a much deeper understanding but it was a detour I could have done without.

As Dextercioby mentions Ballentine is an excellent book - its math at a reasonable level and he explains the issues well - wish it was my first exposure rather than Dirac's.

Start with something like QM demystified to get your sea legs so to speak:
https://www.amazon.com/dp/0071455469/?tag=pfamazon01-20

Its level of math rigor leaves a bit to be desired - so if you have a background in math just gloss over it - it will be explained a lot better in Ballentine:
https://www.amazon.com/dp/9810241054/?tag=pfamazon01-20

Thanks
Bill

 

[bhobba]

Science is based on experiment and observation. It isn't a sub-branch of mathematics.

That's true. But as far as we can tell today physics is written in the language of math. And that language has rules that needs to be respected or you can run into problems. Its fine IMHO for a first brush with QM to be a bit less than mathematically rigorous - providing that is pointed out. This is similar to when you learn calculus - you don't do it at first brush rigorously with epsilon delta proofs. But later you really need to move onto a book like Ballentine that doesn't gloss over such issues - just like with calculus you eventually need to do your epsilonics.

Just as an aside the place I did my degree is one of the few places that allows you to do a Bachelor Mathematics - not a Bachelor Science majoring in math but an actual Bachelor Mathematics - its what I did. Anyway when I did it they forced you to do analysis and epsilonics second semester first year after first semester calculus (this was in Australia BTW - when you start university you already have the equivalent of US Calculus 1 - first semester was US Calculus 2). I personally liked it understanding its importance but nearly everyone else hated it. Pressure was applied and it was removed - its now a third year elective but I was left shaking my head - you can't really say your a mathematician without your epsilonics IMHO. And because of that they removed third year subjects I did on Hilbert Spaces and applications. They were really unpopular as well and are now only available at graduate level - again a pity.

Thanks
Bill

 

[WannabeNewton]

Cool. I'm just curious because one thing I've noticed is that by learning topology and differential geometry to whatever extent and then doing Wald's General Relativity, which I think you would agree is much more focused on mathematical physics than physics, actually seemed to make more physics oriented GR texts much more readable i.e. reading a more mathematically rigorous book on the topic beforehand made the more physics-like texts on GR considerably easier to work through. I was just wondering if the same held for people with similar experiences in QM and if it was worth the detour / extra time spent.

 

[bhobba]

Cool. I'm just curious because one thing I've noticed is that by learning topology and differential geometry to whatever extent and then doing Wald's General Relativity, which I think you would agree is much more focused on mathematical physics than physics, actually seemed to make more physics oriented GR texts much more readable i.e. reading a more mathematically rigorous book on the topic beforehand made the more physics-like texts on GR considerably easier to work through. I was just wondering if the same held for people with similar experiences in QM and if it was worth the detour / extra time spent.

Well actually that's not what I suggest. Similar to learning calculus I would suggest a more elementary book like Griffith first than moving onto a mathematically better one like Ballentine.

Wald is my favorite GR textbook but even there would start with something mathematically less sophisticated first like Shultz, then Sean Carroll's book, then Wald. Each step is mathematically more sophisticated but at the less mathematically advanced level you are building valuable intuition. All you need to understand is if you, like me, think about the math of what you are reading don't get too worried if you spot issues - it will be fixed later. After going through the sequence go back and do it again - with your more mature mathematical knowledge you will get more from it second reading.

Thanks
Bill

 

[WannabeNewton]

Well actually that's not what I suggest. Similar to learning calculus I would suggest a more elementary book like Griffith first than moving onto a mathematically better one like Ballentine.

I'm a die hard fan of Griffiths' electromagnetism text but I could not stand his QM text. There really is a limit to how much a person can hand-wave >.>

Wald is my favorite GR textbook but even there would start with something mathematically less sophisticated first like Shultz, then Sean Carroll's book, then Wald. Each step is mathematically more sophisticated but at the less mathematically advanced level you are building valuable intuition. All you need to understand is if you, like me, think about the math of what you are reading don't get too worried if you spot issues - it will be fixed later. After going through the sequence go back and do it again - with your more mature mathematical knowledge you will get more from it second reading.

Unfortunately I have OCD when it comes to mathematics that is "tossed aside" and "hand-waved" so to speak and I find it hard to continue on until I have seen proofs and justifications of "god-given" mathematical facts that some (or rather many) physics books tend to throw at you. This is why I was wondering if it would serve better, in a general context, to go through more mathematically oriented textbooks on whatever physics subject and then go on (or go back, depending on how you look at it) to the more physics-oriented ones which provide the "physical intuition". I don't speak for everyone of course as not everyone will have the same pet peeves as me or you or others.

 

[bhobba]

Unfortunately I have OCD when it comes to mathematics that is "tossed aside" and "hand-waved" so to speak and I find it hard to continue on until I have seen proofs and justifications of "god-given" mathematical facts that some (or rather many) physics books tend to throw at you. This is why I was wondering if it would serve better, in a general context, to go through more mathematically oriented textbooks on whatever physics subject and then go on (or go back, depending on how you look at it) to the more physics-oriented ones which provide the "physical intuition". I don't speak for everyone of course as not everyone will have the same pet peeves as me or you or others.

Same here mate and I know only too well where you are coming from. You most certainly can proceed directly to Ballentine if you wish. But having been through that with Dirac and consulting advanced tomes on how to make that approach mathematically acceptable then returning to it I simply can not recommend it. Bite your tongue and laugh your head off at its mathematical naivety, but work your way through it. You will develop valuable intuition. That way you can concentrate on the math in the more advanced texts with a bit of intuition about the physics already cemented. Then return to 'hand-wavy' books and you will see how the problems they have can be circumvented. Then go through the more advanced book again - at the end everything will be fine.

Simply think back to when you learned calculus - would you have learned it via epsilon delta proofs from the start? I know I wouldn't have.

Oh - one thing I want to add is QM is bad enough - but QFT is even worse. With that they do things like square Dirac delta functions and its well known that's a mathematical no no (Zee does it for example) - at least no one has been known to figure out how to rigorously do that one. Again persevere - you see how to get around that one with renormalisation and EFT where there is no need to do it in the first place.

Thanks
Bill

 

[Arsenic&Lace]

A researcher I work with in particle physics described that an experience with mathematical rigor in analysis made field theory exceptionally challenging for him, since he was so concerned with rigor that he spent much more time worrying about it than sort of blustering along as physicists are won't to do.

Eventually he realized that for physicists this hack and slash mentality works perfectly fine; indeed, I think that excessive rigor can be quite damaging. Much of mathematics is not really new but rather an exercise in semantics and syntax (if I'm using those linguistic terms properly). It's the process of formulating something with incredible precision; that's really the majority of the battle, actually precisely formulating something. Once this is done, at least some of the time, making the arguments can be quite rapid. The process of formulation produces nothing new; calculus was working perfectly fine before the invention of the epsilon delta formulation, and it can be used and adapted quite fine without this as well.

Honestly I would strongly suggest avoidance of excessive mathematical rigor in physics; my understanding of modern theoretical work is that it can be more of a hindrance than anything else for the production of new theory. Perhaps if you are concerned that path integral QFT is not the same as the series stuff you'll need to use lots of rigor, but this is assuming you're a high level theorist anyway, which most of us never will be.