Canonical Transformations, Quantization, Mutually Unbiased and Other Complete Bases

Donald J Kouri,Cameron L Williams,Nikhil Pandya

doi:10.4236/am.2017.87071

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.

Highlights

IntroductionThe harmonic oscillator (with its symmetric quadratic dependence on the position and momentum operators) provides a complete set of eigenstates that are eigenfunctions of the Fourier transform, and the ground state minimizes the product of the position and momentum uncertainties (variances)
The requirement that quantum mechanical operators for observables be self-adjoint is sufficient to ensure that the above properties hold, but one can ask “is it necessary”? Certainly, an examination of the textbooks and most of the literature dealing with quantization of classical mechanics would suggest that it is [1]-[7]
The Jacobian of the canonical transformation automatically supplies the required measure for self-adjointness

Summary

Introduction

The harmonic oscillator (with its symmetric quadratic dependence on the position and momentum operators) provides a complete set of eigenstates that are eigenfunctions of the Fourier transform, and the ground state minimizes the product of the position and momentum uncertainties (variances). In light of the role played by the position and momentum coordinates in obtaining complete bases of various types, it is natural to inquire as to the role that is played by canonical transformations of the type x → W , px → pW , such that the Poisson bracket satisfies {W , pW } = 1 This will naturally involve how one proceeds from classical to quantum mechanics (we shall use the Dirac canonical quantization procedure). While the paper focusses on quantum mechanics, we point out the relevance of some of the results for certain classical processes

Classical Dynamical Considerations

Dirac Quantization of the Canonically Conjugate Classical Variables

Non-unitary Transformations and the New Momentum Operators

Results and Discussion

Conclusion