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

Abstract

The transformation of a classical system into its quantum counterpart is usually done through the well known procedure of canonical quantization. However, on non-Cartesian domains, or on bounded Cartesian domains, this procedure can be plagued with theoretical inconsistencies. An alternative approach is {\it affine quantization} (AQ) Fantoni and Klauder (arXiv:2109.13447,Phys. Rev. D {\bf 103}, 076013 (2021)), resulting in different conjugate variables that lead to a more consistent quantization formalism. To highlight these issues, we examine a deceptively simple, but important, problem: that of the harmonic oscillator potential on the semibounded domain: ${\cal D} = (-b,\infty)$. The AQ version of this corresponds 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 which the eigenenergies can be determined exactly (through non-wavefunction dependent methods), confirming the results of Gouba ( arXiv:2005.08696 ,J. High Energy Phys., Gravitation Cosmol. {\bf 7}, 352-365 (2021)). We investigate the limit $b \rightarrow \infty$, confirming that the full harmonic oscillator problem is recovered. The adopted computational methods are in keeping with the underlying theoretical framework of AQ. Specifically, one method is an affine map invariant variational procedure, made possible through a moment problem quantization reformulation. The other method focuses on boundedness (i.e. $L^2$) as an explicit quantization criteria. Both methods lead to converging bounds to the discrete state energies; and thus confirming the accuracy of our results, particularly as applied to a singular potential problem.