>>12047170I mean, there are sometimes 'postulates' of quantum mechanics, but I am emphasizing that these are not so hard and fixed-they serve as physical guidelines.
Generally, the postulates of QM are something like each quantum state is associated with a vector [actually a ray] in hilbert space, each observable is associated with a hermitian operator with the state of a particle of measured property value a as being an eigenvector of that operator with eigenscalar a, the measurement of an observable collapses it onto an eigenstate, the existence of a probability function of sorts on a quantum state and observable, and the time-dependent Schrodinger equation or some variations of equivalence. In fact, see Dirac-von Neumann axioms for probably the closest rigorization of this I've seen [though I haven't actually studied it this far]
But these axioms don't serve to actually elucidate what is physically going on. That connection is entirely unmathematical and based on physical insight.
You can't use this to actually 'show' that momentum operator in position space is i?? for instance [or better said, perhaps, this is an arbitrary example of an operator in the formalization-the physical meaning isn't granted by the axioms] Or, for perhaps an even stronger example, of spin. You could imagine QM without spin, but as it happens spin exists and so we introduce it based on physical insight.
Trust me, as a mathematically oriented person, I like seeing more and more detailed descriptions of physics. I've seen classical mechanics formalized in quite an interesting way by arnold in his book, quantum mechanics in the above mentioned way-there was a book on relativity by Reichenbach making an axiomatization of relativity. I got interested in this as 'einstein synchronization' isn't the only method [see Reichenbach synchronization] and all of this is fine but I really don't think much mind is payed attention to how rigorous our axiomatization of physics is.
