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

The transformation of a classical system into its quantum counterpart isusually done through the well known procedure of canonical quantization.However, on non-Cartesian domains, or on bounded Cartesian domains, thisprocedure can be plagued with theoretical inconsistencies. An alternativeapproach is {\it affine quantization} (AQ) Fantoni and Klauder(arXiv:2109.13447,Phys. Rev. D {\bf 103}, 076013 (2021)), resulting indifferent conjugate variables that lead to a more consistent quantizationformalism. To highlight these issues, we examine a deceptively simple, butimportant, problem: that of the harmonic oscillator potential on thesemibounded domain: ${\cal D} = (-b,\infty)$. The AQ version of thiscorresponds to the (rescaled) system, ${\cal H} = \frac{1}{2}\Big(-\partial_x^2+ \frac{3}{4(x+b)^2} + x^2\Big)$. We solve this system numerically for $b > 0$.The case $b = 0$ corresponds to an {\it exactly solvable} potential, for whichthe eigenenergies can be determined exactly (through non-wavefunction dependentmethods), confirming the results of Gouba ( arXiv:2005.08696 ,J. High EnergyPhys., Gravitation Cosmol. {\bf 7}, 352-365 (2021)). We investigate the limit$b \rightarrow \infty$, confirming that the full harmonic oscillator problem isrecovered. The adopted computational methods are in keeping with the underlyingtheoretical framework of AQ. Specifically, one method is an affine mapinvariant variational procedure, made possible through a moment problemquantization reformulation. The other method focuses on boundedness (i.e.$L^2$) as an explicit quantization criteria. Both methods lead to convergingbounds to the discrete state energies; and thus confirming the accuracy of ourresults, particularly as applied to a singular potential problem.

Affine Quantization of the Harmonic Oscillator on the Semi-bounded domain $(-b,\infty)$ for $b: 0 \rightarrow \infty$