English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

We consider the problem of designing a variety of "system guided" basis setsfor quantum mechanical anharmonic oscillators. Using ideas based onsupersymmetric quantum mechanics, we design canonical transformations of theusual position and momentum to generate generalized "Cartesian-like positions,W and momenta, p_W" with unit Poisson brackets. These are quantized followingDirac, leading to an infinite family of potential "operator observables". Thefundamental issue is that all but one of the operators are not Hermitian(formally self-adjoint) in the original position representation. We show thatthe resulting operators are "quasi-Hermitian" relative to the x-representationand that all are Hermitian in the W-representation. Depending on how one treatsthe Jacobian of the canonical transformation in the expression for theclassical momentum, p_W, quantization yields a) continuous mutually unbiasedbases (MUB) b) orthogonal bases (with Dirac delta normalization) c)biorthogonal bases (with Dirac delta normalization) d) new W-harmonicoscillators yielding standard orthonormal bases (as functions of W) andassociated coherent states. The MUB include W-generalized Fourier transformkernels whose eigenvectors are the W-harmonic oscillator eigenstates, with thespectrum (p/m i, p/m i). The W, p_W satisfy the uncertainty product relation:Delta-W * Delta-p_W gte 1/2.

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