TITLE:
Canonical Transformations, Quantization, Mutually Unbiased and Other Complete Bases
AUTHORS:
Donald J. Kouri, Cameron L. Williams, Nikhil Pandya
KEYWORDS:
Canonical Transformations, Quantization, Mutually Unbiased Bases, Complete Bases
JOURNAL NAME:
Applied Mathematics,
Vol.8 No.7,
July
12,
2017
ABSTRACT: Using ideas based on supersymmetric quantum mechanics, we design canonical
transformations of the usual position and momentum to create generalized
“Cartesian-like” positions, W, and momenta, Pw , with unit Poisson
brackets. These are quantized by the usual replacement of the classical , x Px by quantum operators, leading to an infinite family of potential “operator observables”.
However, all but one of the resulting operators are not Hermitian
(formally self-adjoint) in the original position representation. Using either the
chain rule or Dirac quantization, we show that the resulting operators are
“quasi-Hermitian” relative to the x-representation and that all are Hermitian
in the W-representation. Depending on how one treats the Jacobian of the
canonical transformation in the expression for the classical momentum, Pw ,
quantization yields a) continuous mutually unbiased bases (MUB), b) orthogonal
bases (with Dirac delta normalization), c) biorthogonal bases (with Dirac
delta normalization), d) new W-harmonic oscillators yielding standard
orthonormal bases (as functions of W) and associated coherent states and
Wigner distributions. The MUB lead to W-generalized Fourier transform
kernels whose eigenvectors are the W-harmonic oscillator eigenstates, with
the spectrum (±1,±i) , as well as “W-linear chirps”. As expected, W,Pw satisfy
the uncertainty product relation: ΔWΔPw ≥1/2 , h=1.