Math Required for Quantum Mechanics and General Relativity

[IndustriaL]

Hey, what kind of mathematics are needed to understand the bulk of QM and GR?

 

[werty]

Ordinary calculus probably works, for both subjects. If you only want to understand it and make a few calculations. If you want to do research you need a lot more mathematics.

 

[IndustriaL]

Well, I'm a high school student and I'm very interested in Quantum Mechanics and General Relativity. I even took a class on classical physics here at school, but just the basic stuff.. I have pre-calculus down and just wanted to know what was needed to contribute thanks a lot :D btw.. that was a really quick response

 

[dextercioby]

Not to go deep into details and the formalism and just gather a superficial knowledge of QM:linear algebra,calculus,complex analysis+special functions and ODE+PDE-s.

For GR,well,calculus and linear algebra is more that enough to understand the nongeometrical exposure by Dirac [1].

Daniel.

------------------------------------------------------------------------------
[1] P.A.M.Dirac "General Relativity",1975.

 

[chroot]

The short answer is that QM requires linear algebra, calculus, and operators. GR in full splendor requires all that plus differential geometry and tensor calculus.

- Warren

 

[Kruger]

If you start with the Schrödinger equation you will see what you all need. i think if one can solve this equation he has understood the mathematical concept.

 

[aav]

Hey, what kind of mathematics are needed to understand the bulk of QM and GR?

John Baez has a nice link about "how to learn math and physics". As far as QM is concerned, you will need at least:

Calculus
Multivariable calculus
Linear algebra
Ordinary differential equations
Partial differential equations
Complex analysis

 

[masudr]

Complex analysis is not necessary to use/understand QM.

 

[dextercioby]

And the evaluating of integrals using special functions and the theorem of residues should one pick from a cookbook...?Think about scattering and its integrals (usually Laplace and Fourier's transforms).

Daniel.

 

[masudr]

Hmm... I would still say that complex analysis is not necessary to understand QM, although I will now say that it is necessary to use QM.

 

[quantumdude]

If you start with the Schrödinger equation you will see what you all need. i think if one can solve this equation he has understood the mathematical concept.

The trouble with this is that the Schrodinger equation and the deductions that follow from it do not exhaust the whole of QM. For instance, how can you derive spin from Schrodinger? You can't.

 

[X-43D]

Is it possible to understand GR from the quantum mechanical perspective?

Quantum mechanics doesn't put much emphasis on differential geometry (Spivak style) but instead makes heavy use of algebra.

 

[dextercioby]

You could formulate each theory in the other's favorite framework.The way the way these 2 are still taught is called "traditional".But that still doesn't mean that geometrical quantization,for example,is useless.

Daniel.

 

[pmb_phy]

Not to go deep into details and the formalism and just gather a superficial knowledge of QM:linear algebra,calculus,complex analysis+special functions and ODE+PDE-s.

Do you actually mean Linear Differential Equations when you wrote "ODE"? ODE refers to ordinary differential equations to distinguish it from partial differential equations. ODE refers to non-linear diff eq.s. This was actually a class I took and I too was initially confused by the difference until I saw the text and spoke to the prof.

Pete

 

[Gokul43201]

Well, I'm a high school student and I'm very interested in Quantum Mechanics and General Relativity. I even took a class on classical physics here at school, but just the basic stuff.. I have pre-calculus down and just wanted to know what was needed to contribute thanks a lot :D btw.. that was a really quick response

You have a long way to go, and a lot of classical physics to learn before you can start to appreciate or even understand QM or GR. Thinking about QM without the fundamentals (of classical mechanics -lagrangian and hamiltonian formulations, statistical mechanics and electrodynamics) laid down, is not the best way to go.

 

[pmb_phy]

You have a long way to go, and a lot of classical physics to learn before you can start to appreciate or even understand QM or GR. Thinking about QM without the fundamentals (of classical mechanics -lagrangian and hamiltonian formulations, statistical mechanics and electrodynamics) laid down, is not the best way to go.

Let's not exagerate now. There are many good books with zero math in them which doa good job at describing QM to the layman.

Pete

 

[Gokul43201]

But the OP was suggesting that he wanted to learn QM the mathematical way.

 

[dextercioby]

There's only one way to learn the formalism of QM and that is:realizing this is theoretical physics and mathematics should be central.

Start with topology,the key ingredient of functional analysis.

Daniel.

 

[HackaB]

What topology book do you recommend?

 

[dextercioby]

I don't reccomend a specific book on topology.You'll have to figure out by yourself what kind of mathematics you need to brush on,if you read the first 4 chapters of Bogolubov,Logunov & Todorov "Introduction to Axiomatic Quantum Field Theory",Benjamin/Cummings,1975.

Daniel.

 

[pmb_phy]

There's only one way to learn the formalism of QM and that is:realizing this is theoretical physics and mathematics should be central.

Start with topology,the key ingredient of functional analysis.

Daniel.

Topology is not required to learn QM.

 

[dextercioby]

I don't deny you the right to disagree.After all,everyone is free to do whatever he likes,just as long as they don't make false claims,like "I know Quantum Mechanics"...

Daniel.

 

[HackaB]

I don't reccomend a specific book on topology.You'll have to figure out by yourself what kind of mathematics you need to brush on,if you read the first 4 chapters of Bogolubov,Logunov & Todorov "Introduction to Axiomatic Quantum Field Theory",Benjamin/Cummings,1975.

Daniel.

Okay, I'll take a look at that if they have it at the library. I've seen several books on QM authored by Bogolubov, so hopefully that was one of them.

I don't deny you the right to disagree.After all,everyone is free to do whatever he likes,just as long as they don't make false claims,like "I know Quantum Mechanics"...

Would you say that you know quantum mechanics?

 

[aav]

From someone who knows a little quantum mechanics here's some advice.

For someone just starting out, IMHO starting out with topology to learn the mathematics of QM would probably be... overwhelming. It is certainly possible to approach the subject at different levels of mathematical rigor and sophistication, but why start at the penthouse?

For basement level QM math, I'd recommend something like Cohen-Tannoudji, especially Ch 2 (plus complements) for the mathematical foundations. Then Dennery and Krzywicki (notice they start out with complex analysis). That's a whole lot of math right there, enough to keep you busy for quite a while.

Then afterwards, after you get a "feel" for the math, you can revisit it again from a more rigorous perspective starting with, say, Kelley's General Topology and work your way up to the minutiae of Hilbert spaces...

 

[dextercioby]

Okay, I'll take a look at that if they have it at the library. I've seen several books on QM authored by Bogolubov, so hopefully that was one of them.

Him & Landau are Russia's greatest theorists.

Would you say that you know quantum mechanics?

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

Daniel.

 

[pmb_phy]

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

Basement level?? Clarify please.

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

 

[pmb_phy]

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.?

 

[James Jackson]

Perhaps because QM is a theoretical mathematical construct...

 

[aav]

Basement level?? Clarify please.
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.

 

[pmb_phy]

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]

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...

 

[pmb_phy]

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

 

[X-43D]

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

 

[selfAdjoint]

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

 

[pmb_phy]

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