What Advanced Mathematics is Essential for Understanding Quantum Mechanics?

[Starship]

What kind of math does QM use (beyond calculus, differential equations and linear algebra)?

 

[Locrian]

Some complex analysis helped me quite a bit.

 

[Starship]

So basically all the topics of multivariate calculus, complex analysis (including special functions), calculus of variations (a generalization of calculus), Fourier analysis and functional analysis. I guess that modern topics in algebra such as groups, rings and fields are also very helpful.

 

[dextercioby]

Less on rings and fields,but a lotta group theory representations...

Daniel.

 

[selfAdjoint]

Less on rings and fields,but a lotta group theory representations...

Daniel.

Unless you get beyond basic QM, when rings (in the form of algebras: rings with a product) and fields get very important. Lie algebras, Von Neumann algebras, Clifford algebras, ...

 

[Starship]

If one goes behind non-relativistic QM, there are the huge fields of modern analysis, abstract algebra, and topology. There are no clear boundaries.

 

[dextercioby]

Of course not.This is theoretical physics,after all.There's never too much mathematics...

Daniel.

P.S.Did someone mention diff.geom.for nice bundle homological & cohomological approaches to quantization (including the famous BRST)...?

 

[robphy]

One might add Probability Theory, Logic, and Symplectic Geometry.

 

[Starship]

I think that symplectic manifolds and nambu dynamics can be seen as a branch of symplectic topology and even differential topology. Thus, one might also add global analysis to this.

 

[NEOclassic]

Don't forget vector-cross-product math. Angular momentum and its orthogonality aspect extend not only from QM atomic orbits to also the perpenicularity of the axis of magnetic rotation of the Milky Way relative to the plane of matter. Cheers, Jim