system
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human
What does Math in Quantum Physics "look like"? Is it still much like applied math, or does it start to look more like doing proofs?

I've heard that a lot of the really theory-heavy areas of math show up in quantum mechanics a lot; things like linear algebra, group theory and complex analysis. Moreover, it seems like there are some folks in the pure math camp that are also doing stuff related to QM.

Does quantum physics mostly rely on the computational results coming out of these fields, or do you start working at a higher level of abstraction and do a lot of proving?
gpt
The two main ways of "doing" quantum mechanics are through the Schroedinger equation and through Heisenberg matrix mechanics. The path integral formulation is generally used in quantum field theory.

The Schroedinger equation is a second order differential equation, which is solved according to the boundary conditions giving a wavefunction describing a probability amplitude. The expectation values of various things (like the average position of a particle) is found by integrating the product of the wavefunction, the thing, and the adjoint of the wavefunction.

In matrix quantum mechanics, the vectors representing the state of a system are eigenvectors of the Hamiltonian matrix, with each eigenvalue being a discrete energy the system can take. Bra-ket notation is generally used here.