CANONICAL OPERATORS FOR SIMPLE HARMONIC OSCILLATOR

Angle and action operators (w,j) for the simple harmonic oscillator are treated as resulting from a canonical transformation of coordinate and momentum operators (q,k) generated by a one-sided unitary operator U such that U †U = 1 and UU† commutes with k but not with q. From the discrete spectrum of the number operator n, eigenvectors |η〉 are constructed for every real value of η; the set {|η〉} is complete and orthogonal. Another complete set {|W〉} is obtained, consisting of the Fourier transforms of the kets in the set {|η〉}. The angle operator is w = U†qU = ∫ dW|W〉 W 〈W|. The set {|W〉} is not orthogonal; |W〉 is not an eigenvector of w. If ν is defined as ∫ dW|W> exp (-2πiW)〈W|, then the creation and destruction operators are given by a = νn 1/2, a† = n1/2ν†. ν is a one-sided unitary operator such that νν† = 1, but ν†ν = 1 - |0〉 〈0|, where |0〉 is the ground state of the oscillator; ν and ν† are similar to the operators E- and E+ of Carruthers and Nieto. The Weyl transforms of w and j = 2πh(n + 1/2 1) are the classical angle and action variables of the oscillator. The Weyl transform is formulated in terms of the coherent states of the oscillator. A time operator canonical to the Hamiltonian is defined as t = 2πw/ω (ω/2π = frequency). The observables for the oscillator are also given in the Heisenberg picture and their classical limits are considered.