As the first step in an approach to the solution of Hilbert's sixth problem, a general scheme of mechanics, called `supmech', is developed integrating noncommutative symplectic geometry and noncommutative probability theory in an algebraic framework; it has quantum mechanics (QM) and classical mechanics as special subdisciplines and facilitates an autonomous development of QM and satisfactory treatments of quantum-classical correspondence and quantum measurements (including a straightforward \emph{derivation} of the von Neumann reduction rule). The scheme associates, with every `experimentally accessible' system, a symplectic superalgebra and operates essentially as noncommutative Hamiltonian mechanics incorporating the extra condition that the sets of observables and pure states be mutually separating. The latter condition serves to smoothly connect the algebraically defined quantum systems to ilbert space-based ones; the rigged Hilbert space - based Dirac bra-ket formalism naturally appears. The formalism has a natural place for commutative superselection rules. Noncommutative analogues of objects like the momentum map and the Poincar
eˊ-Cartan form are introduced and some related symplectic geometry developed.
With some simple examples behind us, we are now ready to list the rules of quantum mechanics: that is, the rules that we need. Unfortunately, there are more than the three laws of classical mechanics (Newton's laws) or the four laws of thermodynamics. But the rules we need can be written on one page. The remainder of the chapter illustrates the meaning of some of these rules, using simple wave functions that we have already derived. The best way to understand these rules is by practicing their application. For this reason, more end-of-chapter problems are present here than in any other chapter.
This chapter outlines the formal rules of quantum mechanics. A formal mathematical object that possesses these properties was introduced by Dirac and called the Dirac delta function. It is discussed further in Appendix C. The association of wave functions with vectors is actually more than a mere analogy. It is possible to formulate quantum mechanics abstractly in the language of generalized vectors and to regard wave functions as simply one particular representation of the vector algebra. The chapter discusses that the wave functions that arise from the standard physical problems to be considered do form a complete set. In quantum mechanics one frequently deals with infinite-dimensional space and the question of completeness is mathematically delicate. In classical mechanics there is a well-known association between the existence of geometrical symmetries in a mechanical system and the constancy in time of certain dynamical quantities.