QM mathematics

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

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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 alot more mathematics.

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 alot :D btw.. that was a really quick response

dextercioby
Homework Helper
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.

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

chroot
Staff Emeritus
Gold Member
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

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
IndustriaL said:
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

Complex analysis is not necessary to use/understand QM.

dextercioby
Homework Helper
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.

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.

Tom Mattson
Staff Emeritus
Gold Member
Kruger said:
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.

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
Homework Helper
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.

dextercioby said:
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
Staff Emeritus
Gold Member
IndustriaL said:
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 alot :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.

Gokul43201 said:
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
Staff Emeritus
Gold Member
But the OP was suggesting that he wanted to learn QM the mathematical way.

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

Daniel.

What topology book do you recommend?

dextercioby
Homework Helper
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.

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

Daniel.
Topology is not required to learn QM.

dextercioby
Homework Helper
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.

dextercioby said:
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.

dextercioby said:
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
Homework Helper
HackaB said:
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.

HackaB said:
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.