Topics in quantum mechanics

Abstract

The present paper deals with three independent subjects. I. We show how for classical canonical transformation we can pass, with the help of Wigner distribution functions, from their representation U in the configurational Hilbert space to a kernel K in phase space. The latter is a much more transparent way of looking at representations of canonical transformations, as the classical limit is reached when ℏ→0 and successive quantum corrections are related with powers of ℏ2n, n=1,2,… . II. We discuss the coherent states solution for a charged particle in a constant magnetic field and show that it is the appropriate one for getting the classical limit of the problem, i.e., motion in a circle around any point in the plane perpendicular to the field and with the square of the radius proportional to the energy of the particle. III. We show that it is possible to have just one equation involving n α’s and β matrices to get relativistic wave equations that can have spins with values up to n/2. We then decompose the α’s and β’s into direct products of ordinary spin matrices and a new type of them that we call sign spin. The problem reduces then to that of the generators of a SU(4) group, entirely similar to the one in the spin-isospin theory of nuclear physics. For a free particle of arbitrary spin the symmetry group is actually the unitary symplectic subgroup of SU(4), i.e., Sp(4). As the latter is isomorphic to O(5), we can characterize our states by the canonical chain O(5)⊃O(4)⊃O(3)⊃O(2), and from it obtain the spin and mass content of our relativistic equation.