Canonical Quantization, Quasi-Hermiticity, Observables and the Construction of Complete Basis Sets

TL;DR

This paper develops a framework for constructing complete basis sets in quantum mechanics using supersymmetric transformations, leading to new operator representations that are quasi-Hermitian and support various basis types including mutually unbiased and orthogonal bases.

Contribution

It introduces a method to generate generalized position and momentum operators via canonical transformations, resulting in new basis sets and operator representations in quantum anharmonic oscillators.

Findings

Operators are quasi-Hermitian in the original position basis and Hermitian in the W-representation.

Constructed basis sets include mutually unbiased, orthogonal, and biorthogonal bases.

New W-harmonic oscillators and coherent states are derived with specific spectral properties.

Abstract

We consider the problem of designing a variety of "system guided" basis sets for quantum mechanical anharmonic oscillators. Using ideas based on supersymmetric quantum mechanics, we design canonical transformations of the usual position and momentum to generate generalized "Cartesian-like positions, W and momenta, p_W" with unit Poisson brackets. These are quantized following Dirac, leading to an infinite family of potential "operator observables". The fundamental issue is that all but one of the operators are not Hermitian (formally self-adjoint) in the original position representation. 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, p_W, 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. The MUB include W-generalized Fourier transform kernels whose eigenvectors are the W-harmonic oscillator eigenstates, with the spectrum (p/m i, p/m i). The W, p_W satisfy the uncertainty product relation: Delta-W * Delta-p_W gte 1/2.