Generally speaking, there are three forms of quantum mechanics, namely wave mechanics, matrix mechanics and path integral theory. The first two can be considered as "corresponding" to Hamiltonian mechanics, while path integral "corresponding" to Lagrangian mechanics.
As for the mathematical basis of quantum mechanics, it is actually functional analysis. Functional analysis can be divided into linear functional analysis and nonlinear functional analysis according to whether the operator studied is linear or not, and functional analysis in metric space and functional analysis in general topological space according to the topology of basic space. But if you don't study functional analysis, you can go directly to quantum mechanics. Because it is very troublesome to memorize real variable functions before learning functional analysis. I studied quantum mechanics first, then real variable function and functional analysis. Sweating.
The problem version of quantum mechanics has been discussed a lot.
Classical theoretical mechanics is divided into Lagrangian mechanics and Hamiltonian mechanics (there is actually another theory, that is, Hamiltonian-Jacobian theory, whose basic idea is to regard the evolution of phase space points with time as a regular transformation between fixed points and fixed points in phase space, which ... thanks, but it seems a bit deep. How to make it clear in a few words when you meet an interview? After all, I only learned advanced algebra, mathematical physics methods and linear algebra in mathematics. I remember what the teacher told us at the beginning was only the difference between quantum mechanics and classical mechanics, micro and macro, probability and certainty, wave-particle duality and so on. Looking back, it is not clear how much the two are related (physically, not mathematically). Fyl7 (in-sTAtion conTAct ta) analytical mechanics is the key:) bbshitedu (in-station contact ta) should be analytical mechanics. Is the function useful? I don't know. Let's talk about integrity first. Mathematics is to wipe the bottom of physics and make physics more complete and flawless. You will know whether the yzcluster (intra-station contact TA) functional is useful, especially in path integral quantum mechanics and quantum field theory. Have you learned all this?
If you just think that "let's talk about completeness, and mathematics is to wipe the bottom of physics, as if physics is more complete and flawless", then you are wrong. :) bbshitedu (in-station contact ta) That's a variational method, and it seems that there is a variational method first and then a functional. In essence, functional analysis includes three parts: space theory, operator theory and their application as their interconnection with other disciplines, which are organically combined. In fact, most subjects of pure mathematics and applied mathematics are widely related to functional analysis.
However, as far as physicists are concerned, it doesn't matter if you really don't study functional analysis. We can simply understand these disciplines of quantum mechanics from the physical point of view. I don't think it will have much impact. :) kwxia (Station Contact TA) Well ... An accident caused controversy ... So after upgrading from Newtonian mechanics to analytical mechanics, quantum mechanics was naturally introduced to solve microscopic phenomena? The order of bbshitedu (in-station contact TA) is from experimental phenomena to physical logical reasoning to mathematical description, and new phenomena are pointed out through mathematical description. Quantum mechanics is no exception.