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arXiv:2603.29266

Analyzing Uniform WKB for Deformed QM Or How Not to Quantize the SW Curve

Dharmesh Jain

hep-thquant-phSubmitted Mar 31, 2026• Announced Apr 1, 2026
arXivHTMLPDF

Abstract

We uncover an inconsistency in the uniform WKB quantization of deformed quantum mechanics.
Comments
5 pages, 1 Mathematica notebook

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WKB theory provides a plausible link between classical mechanics and quantum mechanics in its semi-classical limit. Connecting the WKB wave function across the turning points in a quantum well, leads to the WKB quantization condition. In this work, I focus on some improvements and recent developments related to the WKB quantization condition. First I discuss how the combination of super-symmetric quantum mechanics and WKB, gives the SWKB quantization condition which is exact for a large class of potentials called shape invariant potentials. Next I turn to the fact that there is always a probability of refection when the potential is not constant and the phase of the wave function should account for this refection. WKB theory ignores refection except at turning points. I explain the work of Friedrich and Trost who showed that by including the correct "refection phase" at a turning point, the WKB quantization condition can be made to give exact bound state energies. Next I discuss the work of Cao and collaborators which takes refection of the wave function into account everywhere. We show that Cao's method provides a way to compute the F-T refection phase. Finally I discuss a paper of Fabre and Guery-Odelin who used the exponential potential to study the accuracy of WKB. In their results the accuracy deteriorates as the energy increases, which is inconsistent with Bohr's correspondence principle. Using the Friedrich and Trost method, we resolved this problem.
Different approaches are compared to formulation of quantum mechanics of a particle on the curved spaces. At first, the canonical, quasi-classical and path integration formalisms are considered for quantization of geodesic motion on the Rimannian configurational spaces. A unique rule of ordering of operators in the canonical formalism and a unique definition of the path integral are established and, thus, a part of ambiguities in the quantum counterpart of geodesic motion is removed. A geometric interpretation is proposed for non-invariance of the quantum mechanics on coordinate transformations. An approach alternative to the quantization of geodesic motion is surveyed, which starts with the quantum theory of a neutral scalar field. Consequences of this alternative approach and the three formalisms of quantization are compared. In particular, the field theoretical approach generates a deformation of the canonical commutation relations between coordinates and momenta of a prticle. A possible cosmological consequence of the deformation is presented in short. Key words: quantum mechanics, Riemannian space, geodesic motion, deformation.