Semiclassical approach to quantum-mechanical problems with broken supersymmetry

Journal Article · Wed Sep 01 04:00:00 EDT 1993 · Physical Review A; (United States)

DOI:https://doi.org/10.1103/PhysRevA.48.1845· OSTI ID:6217883

Dutt, R ^[1]; Gangopadhyaya, A ^[2]; Khare, A ^[3]; Pagnamenta, A; Sukhatme, U ^[4]

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1. Department of Physics, Visva-Bharati University, Santiniketan 731235, West Bengal (India)
2. Department of Physics, Loyola University Chicago, Chicago, Illinois 60626 (United States)
3. Institute of Physics, Sachivalaya Marg, Bhubaneswar 751005 (India)
4. Physics Department, University of Illinois at Chicago, Chicago, Illinois 60680 (United States)

The semiclassical WKB approximation method is reexamined in the context of nonrelativistic quantum-mechanical bound-state problems with broken supersymmetry (SUSY). This gives rise to an alternative quantization condition (denoted by BSWKB) which is different from the standard WKB formula and also different from the previously studied supersymmetric (SWKB) formula for unbroken SUSY. It is shown that to leading order in [h bar], the BSWKB condition yields exact energy eigenvalues for shape-invariant potentials with broken SUSY (harmonic oscillator, Poeschl-Teller I and II) which are known to be analytically solvable. Further, we show explicitly that the higher-order corrections to these energy eigenvalues, up to sixth order in [h bar], vanish identically. We also consider a number of examples of potentials with broken supersymmetry that are not analytically solvable. In particular, for the broken SUSY superpotential [ital W]=[ital Ax][sup 2[ital d]] [[ital A][gt]0, [ital d]=(integer)], we evaluate contributions up to the sixth order and show that these results are in excellent agreement with numerical solutions of the Schroedinger equation. While the numerical BSWKB results in lowest order are not always better than the corresponding WKB results, they are still a considerable improvement because they guarantee equality of the corresponding energy eigenvalues for the supersymmetric partner potentials [ital V][sub +] and [ital V][sub [minus]]. This is of special importance in those situations where these partner potentials are not related by parity.

• DOE Contract Number: FG02-84ER40173
• OSTI ID: 6217883
• Journal Information: Physical Review A; (United States), Journal Name: Physical Review A; (United States) Vol. 48:3; ISSN 1050-2947; ISSN PLRAAN
• Country of Publication: United States
• Language: English

Related Subjects

661100* -- Classical & Quantum Mechanics-- (1992-)
71 CLASSICAL AND QUANTUM MECHANICS
GENERAL PHYSICS
BOUNDARY CONDITIONS
EIGENVALUES
POTENTIALS
QUANTIZATION
SUPERSYMMETRY
SYMMETRY
SYMMETRY BREAKING
WKB APPROXIMATION