Canonical Transformations, Quantization, Mutually Unbiased and Other Complete Bases

作     者：Donald J. Kouri Cameron L. Williams Nikhil Pandya 

作者机构：Department of Physics University of Houston Houston TX USA Department of Mathematics University of Houston Houston TX USA 

出 版 物：《Applied Mathematics》 (应用数学（英文）)

年 卷 期：2017年第8卷第7期

页      面：901-919页

学科分类：07[理学] 0701[理学-数学] 070101[理学-基础数学] 

主　　题：Canonical Transformations Quantization Mutually Unbiased Bases Complete Bases 

摘      要：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.