Are all vector spaces equally valid for describing the laws of nature?

Question

Is that what is going on in quantum mechanics when we talk about the equivalence of Schrodinger mechanics and Heisenberg matrix mechanics? Since one is a vector space composed of differential equations and the other a vector space composed of matrices. Is that why Dirac came up with bra ket notation...to express quantum mechanics in terms of a general vector space? I'm just curious.

Answer 1

We say that different representations are equivalent because they are "mathematically isomorphic" which is another word for equivalent, resulting in a delicously pointless circularity. But it sounds like you're a little confused.

"Schrodinger mechanics" isn't a term I've ever heard. The solutions to the shroedinger differential equation are functions that can be generalized to vectors living in hilbert space. I sure wouldn't say that the vector space is composed of differential equations- solutions to a differential equation, maybe. The matrices in heisenberg representation live in a hilbert space (i think), but they ARE vector spaces. I don't use this representation, and in my experience it's uses are fairly limited (but I'm sure it still has plenty of uses). With that in mind, I'm fairly sure that the correct statement is that the matrices form hilbert subspaces. Yes, dirac's braket notation is a formulation in an abstract vector space.

Answer 2

Wait until you understand what you're talking about to ask the question. A scientist will choose units, coordinate systems, mathematical notations, integral/differential/tensor/other expressions, and so on according to several factors: which is most practical for the problem at hand; which he is most comfortable and competent with; which his audience is most comfortable with, which shows the desired result with most clarity, and so on.

Answer 3

you totally only asked this question to show off that you know all those terms. But as far as I know, it is a trivial matter describing differntial equations using a matrix so everythin you have to say sounds good to me. It also sounds like you know more about this topic than I do.