# QM mathematics

aav said:
For basement level QM math, I'd recommend something like Cohen-Tannoudji, especially Ch 2 (plus complements) for the mathematical foundations.

I took quantum mechanics in both undergrad and graduate school. I no class and in no text did I ever read anything which referred to topology. E.g. see

http://www.geocities.com/physics_world/qm/state_space.htm

What is the benefit of using topology in QM?

Pete

dextercioby said:
Him & Landau are Russia's greatest theorists.

Nope.It's not modesty,but i'm learning QM the right way.Using as much mathematics as possible.

Daniel.
Why do you consider using as much math as possible "the right way.?

Perhaps because QM is a theoretical mathematical construct...

aav
pmb_phy said:
I took quantum mechanics in both undergrad and graduate school. I no class and in no text did I ever read anything which referred to topology. E.g. see
http://www.geocities.com/physics_world/qm/state_space.htm
What is the benefit of using topology in QM?
Pete
Topology as a mathematical prerequisite for functional analysis, when you start discussing stuff like Lebesque integration, measure theory, L2 spaces, the Riesz-Fischer theorem, generalized functions, etc etc which are required in a rigorous formulation of the math of QM.

aav said:
Topology as a mathematical prerequisite for functional analysis, when you start discussing stuff like Lebesque integration, measure theory, L2 spaces, the Riesz-Fischer theorem, generalized functions, etc etc which are required in a rigorous formulation of the math of QM.
And yet I know functional analysis and never studied topology. What you're saying is similar to saying that real analysis is a prereq for calculus. While true, one never needs to study real analysis to understand most if not all of calculus. I took real analysis because my second major was math and was required but it was a very difficult course and only served to give me more confidence in calculus.

Pete

aav
pmb_phy said:
What you're saying is similar to saying that real analysis is a prereq for calculus. While true, one never needs to study real analysis to understand most if not all of calculus.
Pete
Yes, I agree, from a practical point of view real analysis is not needed for calculus, but the suggestion Daniel gave (as I understand it) was to study QM from a axiomatic point of view. I simply recommended doing the "usual" math first...

But alas, Art is long and life is short...

aav said:
Yes, I agree, from a practical point of view real analysis is not needed for calculus, but the suggestion Daniel gave (as I understand it) was to study QM from a axiomatic point of view. I simply recommended doing the "usual" math first...

But alas, Art is long and life is short...
True. Far too short. I was once so naive as to think I could learn all branches of math that has ever been applied to anykind of physics. One problem with that. By the time you learn the last you have forgotten a lot of the first.

I see no reason to learn all the math one wishes to in order to be more and more of an expert in a particular branch of physics. But its just crazy talk to learn th subject like this. It delays the understanding for no real reason and it also will depend on why one is studying it. If one whishes to be a mathematical physicist then that would seem the next move after grad quantum. But if one is an experimentalist then I can't fathom a reason for most applications (save things like quantum computing etc.)

Pete

I think it doesn't much more than multivaraite calculus, linear algebra and differential geometry/topology to understand GR and QM.

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X-43D said:
I think it doesn't much more than multivaraite calculus, linear algebra and differential geometry/topology to understand GR and QM.

Well if you want to go beyond the obvious with spin, you need a little bit about groups and representations

Well if you want to go beyond the obvious with spin, you need a little bit about groups and representations
Please give an example where group theory is absolutely necessary. Thanks.

Pete

dextercioby
Homework Helper
The first postulate of QM...?:uhh:

Daniel.

dextercioby said:
The first postulate of QM...?:uhh:

Daniel.
The first postulate of QM is as given in Cohen-Tannoudji et al is defined as
At a fixed time t0, the state of a physical system is defined by specifying a ket |$\psi(t_0)$> belonging to the state space.
There is nothing in there which demands one have learned group theory. One need only understand what a state space is. Whether it is a group one is not required to know.

So what you're telling me is that you were not able to comphrehend this until you learned group theory? Unless you didn't catch the "absolutely" in my question?

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dextercioby
Homework Helper
Nope.In the "lightweight" version i've been learnt in school,the first postulate reads

lecture notes on QM said:
"The state of a quantum system at a certain moment of time is described by sequence at most countable

$$\left\{|\psi_{k}\rangle,p_{k}\right\}$$

,in which $|\psi_{k}\rangle$ are normalized vectors from a separable Hilbert space called "the space of states associated to the system" and $p_{k}$ are real nonnegative numbers satisfying the normalization condition

$$\sum_{k} p_{k} =1$$

and are called weights associated to the vectors $|\psi_{k}\rangle$.
This is the elementary version taught in my school.It doesn't account for supraselection rules.

[Sidenote]There are more rigurous formulations using
*rigged Hilbert spaces.
*unit rays and Wigner's theorem.
*Bargmann's theorem and projective representations of symmetry groups.
*coherent subspaces accounting for supraselection rules.[/Sidenote]

Daniel.

dextercioby said:
Nope.
Nope what? "Nope" that is not what Cohen-Tannoudji et al states? I don't know why you're trying to respond to a question I directed to selfAdjoint but you haven't even addressed the question as if yet. The question was
Please give an example where group theory is absolutely necessary.
In the "lightweight" version i've been learnt in school,the first postulate reads..
For what reason do you refer to it as "lightweight"? Is it because you consider that which is more mathematically oriented is defined as "heavy weight"? If so then that's pretty off-topic here.

So what part of
This is the elementary version taught in my school.It doesn't account for supraselection rules.

There are more rigurous formulations using
*rigged Hilbert spaces.
*unit rays and Wigner's theorem.
*Bargmann's theorem and projective representations of symmetry groups.
*coherent subspaces accounting for supraselection rules.
does what is impossible to do without group theory? You're not sticking to the question posed to selfadjoint. Seems more to me that you're trying to impress someone with math.

dextercioby
Homework Helper
pmb_phy said:
Nope what? "Nope" that is not what Cohen-Tannoudji et al states?
Yes.In the form stated,it's higly incomplete.A pure state is a very particular case,not really encountered in experiments.Thank god my QM teacher used C-Tann. only for exercises/applications and not theory

I know what the question was.Can one teach angular momentum in QM without group theory...?I guess not.Can one teach the 6 postulates (especially the I-st and the VI-th) without group theory?Maybe,but that would be missing the essence of the formalism.

pmb_phy said:
For what reason do you refer to it as "lightweight"? Is it because you consider that which is more mathematically oriented is defined as "heavy weight"? If so then that's pretty off-topic here.
Yes,"lightweight",judging by the amounts of mathematics necessary to know in order to fully understand it.Maybe it's offtopic.

pmb_phy said:
does what is impossible to do without group theory? You're not sticking to the question posed to selfadjoint. Seems more to me that you're trying to impress someone with math.
I'm not trying to impress anyone with math.I have a serious problem with people trying to minimize the role of mathematics in theoretical physics,that's all,not that you were doing it...

Daniel.

Dude! The person asking this question wanted a simple answer. He appears to be a highschool student. Do you want to scare the heck out of him so that he feels that he'll never be able to do QM or GR? Distinguish between the terms "need" and "its cool to.."

Sorry but you seem to be irritating me again. Catch ya next month if you can chill out. I have no time for anxiety/stress. Too dangerous for me at the moment.

dextercioby
Homework Helper
Nope,just make everyone realize that "to do" science requires many things/abilities.To do theoretical physics requires MATHEMATICS.

The end.

Daniel.

reilly
There are levels of the game,and myriads of styles of using math in physics. But, there is, in my opinion, one central truth of quantum mechanics that should always be remembered: QM is weird because Nature is weird. Waves, particles, spin, discrete spectra, and on and on, that's Nature. Physics first, math second. Thus, as is always ultimately the case in physics, math serves as a powerful and logical tool for describing Nature. How much rigor, how much measure theory, how much cohomology? Depends. But little or none until the student has mastered Dirac's QM book -- mastery judged by an appropriate exam, of course.

The basic historical physics of the early days of QM are very important. Freshman physics texts deal with these issues, as well as with QM iteself. Do a Google, check out used book stores -- look for "The Cosmic Code" by Heinz Pagels. Has a terrific discussion of QM for the layman,including Bell's Thrm. Get the ideas, the physics in hand before jumping into fullblown QM. Scientific American has had lots of good stuff over the years.

To do QM at the graduate level, you'll need graduate level E&M and mechanics, some stat mech is good, undergraduate atomic and nuclear physics, boundary value problems/mathematical physics, linear algebra and group theory, and, preferably, complex variable theory. To do QM justice, you need a practical mastery of much of the prerequisite material, which takes time, like the time to do an undergraduate degree -- at least for most of us.

Enrico Fermi used to ask during oral exams, "How far can a bird fly?" Theoretical physics? You bet. Needs much math? Nope.

Do you really need to understand compact sets, measure theory, weak vs. strong topologies, semigroups and so forth to understand QM? Probabably not. To do string theory and field theory, probably yes, and more.

Remember that both Bohr and Einstein went remarkably far with great insight and simple algebra.

Regards,
Reilly Atkinson

There is a conflict between quantum mechanics and general relativity. Not only do the 2 theories make use of different mathematics but the physical approach is also very different.

For general relativity all one needs to know is multivariable calculus, linear algebra and differential geometry. For QM and QFT one needs PDEs, real analysis, measure theory, functional analysis + operator theory etc...

Kane O'Donnell
There is nothing in there which demands one have learned group theory. One need only understand what a state space is. Whether it is a group one is not required to know.
Well, sorry to jump in late, but a state space is a Hilbert space, which is a complete inner product space. An inner product space is a vector space, which is, amongst other things, defined in terms of it's group structure. So at this rudimentary level, yes, you should have *seen* some group theory. It's not necessary to solve the Schrodinger equation or do a lot of the undergrad applications that you see, but that barely constitutes "using" quantum mechanics, let alone understanding. Much of the real importance QM is in the representation and manifestation of symmetries, something much more abstract but *directly* and inescapably connected to group theory. Then again, at undergrad level maybe that doesn't count. What do people think?

You don't have to be an expert, of course, but an *awful* lot of the properties that students (and everyone else) use in proving properties come from linear algebra, they aren't just some magical element that QM invents.

HOWEVER, there is a reason people don't jump straight into QM the way Daniel suggests - it's just too bloody hard for most students to be learning a bunch of applications (in optics and surface physics, for example) whilst also studying QM to a high degree of rigour. Most students are introduced slowly, just like they're introduced slowly to calculus by first studying the 'dodgy' version of limits and the Newtonian tangent definition of the derivatve, and then studying real analysis, then complex analysis as a generalisation, then topology to generalise further. The thing is, topology is *better* studied at an advanced level because you can be more precise without alienating the students. You just have to work up to it!

So what you're telling me is that you were not able to comphrehend this until you learned group theory? Unless you didn't catch the "absolutely" in my question?
Only a Sith deals in absolutes.

As a final comment I should say that when we say "you need to know this branch of mathematics", there are two ways in which you have to "know" it. Neither of them means knowing everything. The first is that you need to have experience with doing calculations in the subject so that you can use the machinery. The second is that you need to have seen the important theorems and definitions so that you understand the properties of the construct. Both take a lot of time and at undergrad level it's not an easy path just to take a textbook and try and learn from it. This is *especially* true of GR, where the physical content of the theory (ie, "Oi, Which Of These Lines Are Straight, Then, Eh?" ) is easily lost in indices and theorems about the curvature tensor.

Kane

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Hilbert space is only preliminary. To go deep into the math structure of QM, you at least need these: spectral theory; distribution theory; operator semigroups; Lie groups and Lie algebras; and most importantly, C*-/von Neumann/Weyl algebras...... these are of the level of modern math at 1960s