Deformed Shape Invariant Superpotentials in Quantum Mechanics and Expansions in Powers of ℏ
Abstract
:1. Introduction
2. Deformed Shape Invariance in Deformed Supersymmetric Quantum Mechanics
- (i)
- As for conventional SE, they should be square integrable on the (finite or infinite) interval of definition of the potentials —i.e.,
- (ii)
3. Deformed Shape Invariance for Superpotentials with no Explicit Dependence on ℏ
3.1. Example of the Pöschl-Teller Potential
3.2. Example of the Radial Harmonic Oscillator Potential
3.3. Lists of Results
4. Deformed Shape Invariance for Superpotentials with an Explicit Dependence on ℏ
5. Conclusions
Funding
Acknowledgments
Conflicts of Interest
Appendix A. Going from Previously Used Conventions to the Present Ones
References
- Cooper, F.; Khare, A.; Sukhatme, U. Supersymmetry and quantum mechanics. Phys. Rep. 1995, 251, 267–385. [Google Scholar] [CrossRef] [Green Version]
- Junker, G. Supersymmetric Methods in Quantum and Statistical Physics; Springer: Berlin, Germany, 1996. [Google Scholar]
- Bagchi, B. Supersymmetry in Quantum and Classical Physics; Chapman and Hall/CRC: Boca Raton, FL, USA, 2000. [Google Scholar]
- Gangopadhyaya, A.; Mallow, J.; Rasinariu, C. Supersymmetric Quantum Mechanics: An Introduction; World Scientific: Singapore, 2010. [Google Scholar]
- Gendenshtein, L. Derivation of exact spectra of the Schrödinger equation by means of supersymmetry. JETP Lett. 1983, 38, 356–359. [Google Scholar]
- Darboux, G. Leçons sur la Théorie Générale des Surfaces, 2nd ed.; Gauthier-Villars: Paris, France, 1912. [Google Scholar]
- Schrödinger, E. A method of determining quantum-mechanical eigenvalues and eigenfunctions. Proc. R. Ir. Acad. 1940, A46, 9–16. [Google Scholar]
- Schrödinger, E. Further studies on solving eigenvalue problems by factorization. Proc. R. Ir. Acad. 1941, A46, 183–206. [Google Scholar]
- Schrödinger, E. The factorization of the hypergeometric equation. Proc. R. Ir. Acad. 1941, A47, 53–54. [Google Scholar]
- Infeld, L.; Hull, T.E. The factorization method. Rev. Mod. Phys. 1951, 23, 21–68. [Google Scholar] [CrossRef]
- Gómez-Ullate, D.; Grandati, Y.; Milson, R. Extended Krein-Adler theorem for the translationally shape invariant potentials. J. Math. Phys. 2014, 55, 043510. [Google Scholar] [CrossRef] [Green Version]
- Quesne, C. Exceptional orthogonal polynomials, exactly solvable potentials and supersymmetry. J. Phys. A 2008, 41, 392001. [Google Scholar] [CrossRef]
- Bagchi, B.; Quesne, C.; Roychoudhury, R. Isospectrality of conventional and new extended potentials, second-order supersymmetry and role of PT symmetry. Pramana J. Phys. 2009, 73, 337–347. [Google Scholar] [CrossRef] [Green Version]
- Quesne, C. Solvable rational potentials and exceptional orthogonal polynomials in supersymmetric quantum mechanics. SIGMA 2009, 5, 084. [Google Scholar] [CrossRef] [Green Version]
- Odake, S.; Sasaki, R. Infinitely many shape invariant potentials and new orthogonal polynomials. Phys. Lett. B 2009, 679, 414–417. [Google Scholar] [CrossRef] [Green Version]
- Gómez-Ullate, D.; Kamran, R.; Milson, R. An extended class of orthogonal polynomials defined by a Sturm-Liouville problem. J. Math. Anal. Appl. 2009, 359, 352–367. [Google Scholar] [CrossRef] [Green Version]
- Kempf, A. Uncertainty relation in quantum mechanics with quantum group symmetry. J. Math. Phys. 1994, 35, 4483–4496. [Google Scholar] [CrossRef] [Green Version]
- Hinrichsen, H.; Kempf, A. Maximal localization in the presence of minimal uncertainties in positions and in momenta. J. Math. Phys. 1996, 37, 2121–2137. [Google Scholar] [CrossRef] [Green Version]
- Kempf, A. Non-pointlike particles in harmonic oscillators. J. Phys. A 1997, 30, 2093–2102. [Google Scholar] [CrossRef]
- Witten, E. Reflections on the fate of spacetime. Phys. Today 1996, 49, 24–30. [Google Scholar] [CrossRef]
- Bastard, G. Wave Mechanics Applied to Semiconductor Heterostructures; Editions de Physique: Les Ulis, France, 1988. [Google Scholar]
- Weisbuch, C.; Vinter, B. Quantum Semiconductor Heterostructures; Academic: New York, NY, USA, 1997. [Google Scholar]
- Serra, L.; Lipparini, E. Spin response of unpolarized quantum dots. Europhys. Lett. 1997, 40, 667–672. [Google Scholar] [CrossRef]
- Harrison, P.; Valavanis, A. Quantum Wells, Wires and Dots: Theoretical and Computational Physics of Semiconductor Nanostructures; Wiley: Chichester, UK, 2016. [Google Scholar]
- Barranco, M.; Pi, M.; Gatica, S.M.; Hernández, E.S.; Navarro, J. Structure and energetics of mixed 4He-3He drops. Phys. Rev. B 1997, 56, 8997–9003. [Google Scholar] [CrossRef] [Green Version]
- Geller, M.R.; Kohn, W. Quantum mechanics of electrons in crystals with graded composition. Phys. Rev. Lett. 1993, 70, 3103–3106. [Google Scholar] [CrossRef] [Green Version]
- Arias de Saavedra, F.; Boronat, J.; Polls, A.; Fabrocini, A. Effective mass of one 4He atom in liquid 3He. Phys. Rev. B 1994, 50. [Google Scholar] [CrossRef] [Green Version]
- Puente, A.; Serra, L.; Casas, M. Dipole excitation of Na clusters with a non-local energy density functional. Z. Phys. D 1994, 31, 283–286. [Google Scholar] [CrossRef]
- Ring, P.; Schuck, P. The Nuclear Many Body Problem; Springer: New York, NY, USA, 1980. [Google Scholar]
- Bonatsos, D.; Georgoudis, P.E.; Lenis, D.; Minkov, N.; Quesne, C. Bohr Hamiltonian with a deformation-dependent mass term for the Davidson potential. Phys. Rev. C 2011, 83, 044321. [Google Scholar] [CrossRef] [Green Version]
- Willatzen, M.; Lassen, B. The Ben Daniel-Duke model in general nanowire structures. J. Phys. Condens. Matter 2007, 19, 136217. [Google Scholar] [CrossRef]
- Chamel, N. Effective mass of free neutrons in neutron star crust. Nucl. Phys. A 2006, 773, 263–278. [Google Scholar] [CrossRef] [Green Version]
- Infeld, L. On a new treatment of some eigenvalue problems. Phys. Rev. 1941, 59, 737–747. [Google Scholar] [CrossRef]
- Stevenson, A.F. Note on the “Kepler problem” in a spherical space, and the factorization method of solving eigenvalue problems. Phys. Rev. 1941, 59, 842. [Google Scholar] [CrossRef]
- Infeld, L.; Schild, A. A note on the Kepler problem in a space of constant negative curvature. Phys. Rev. 1945, 67, 121. [Google Scholar] [CrossRef]
- Kalnins, E.G.; Miller, W., Jr.; Pogosyan, G.S. Superintegrability and associated polynomial solutions: Euclidean space and the sphere in two dimensions. J. Math. Phys. 1996, 37, 6439–6467. [Google Scholar] [CrossRef] [Green Version]
- Kalnins, E.G.; Miller, W., Jr.; Pogosyan, G.S. Superintegrability on the two-dimensional hyperboloid. J. Math. Phys. 1997, 38, 5416–5433. [Google Scholar] [CrossRef] [Green Version]
- Quesne, C.; Tkachuk, V.M. Deformed algebras, position-dependent effective masses and curved spaces: An exactly solvable Coulomb problem. J. Phys. A 2004, 37, 4267–4281. [Google Scholar] [CrossRef]
- Bagchi, B.; Banerjee, A.; Quesne, C.; Tkachuk, V.M. Deformed shape invariance and exactly solvable Hamiltonians with position-dependent effective mass. J. Phys. A 2005, 38, 2929–2945. [Google Scholar] [CrossRef] [Green Version]
- Quesne, C. Point canonical transformations versus deformed shape invariance for position-dependent mass Schödinger equations. SIGMA 2009, 5, 046. [Google Scholar]
- Quesne, C. Quantum oscillator and Kepler-Coulomb problems in curved spaces: Deformed shape invariance, point canonical transformations, and rational extensions. J. Math. Phys. 2016, 57, 102101. [Google Scholar] [CrossRef] [Green Version]
- Gangopadhyaya, A.; Mallow, J.V. Generating shape invariant potentials. Int. J. Mod. Phys. A 2008, 23, 4949–4978. [Google Scholar] [CrossRef]
- Bougie, J.; Gangopadhyaya, A.; Mallow, J.V. Generation of a complete set of additive shape-invariant potentials from an Euler equation. Phys. Rev. Lett. 2010, 105, 210402. [Google Scholar] [CrossRef] [Green Version]
- Bougie, J.; Gangopadhyaya, A.; Mallow, J.; Rasinariu, C. Supersymmetric quantum mechanics and solvable models. Symmetry 2012, 4, 452–473. [Google Scholar] [CrossRef] [Green Version]
- Bougie, J.; Gangopadhyaya, A.; Mallow, J.V.; Rasinariu, C. Generation of a novel exactly solvable potential. Phys. Lett. A 2015, 379, 2180–2183. [Google Scholar] [CrossRef] [Green Version]
- Mallow, J.V.; Gangopadhyaya, A.; Bougie, J.; Rasinariu, C. Inter-relations between additive shape invariant superpotentials. Phys. Lett. A 2020, 384, 126129. [Google Scholar] [CrossRef] [Green Version]
- Mustafa, O.; Mazharimousavi, S.H. Ordering ambiguity revisited via position dependent mass pseudo-momentum operators. Int. J. Theor. Phys. 2007, 46, 1786–1796. [Google Scholar] [CrossRef] [Green Version]
- Quesne, C. Revisiting (quasi-)exactly solvable rational extensions of the Morse potential. Int. J. Mod. Phys. A 2012, 27, 1250073. [Google Scholar] [CrossRef] [Green Version]
- Quesne, C. Novel enlarged shape invariance property and exactly solvable rational extensions of the Rosen-Morse II and Eckart potentials. SIGMA 2012, 8, 080. [Google Scholar] [CrossRef] [Green Version]
- Grandati, Y. New rational extensions of solvable potentials with finite bound state spectrum. Phys. Lett. A 2012, 376, 2866–2872. [Google Scholar] [CrossRef] [Green Version]
- Grandati, Y.; Quesne, C. Confluent chains of DBT: Enlarged shape invariance and new orthogonal polynomials. SIGMA 2015, 11, 061. [Google Scholar] [CrossRef] [Green Version]
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Quesne, C. Deformed Shape Invariant Superpotentials in Quantum Mechanics and Expansions in Powers of ℏ. Symmetry 2020, 12, 1853. https://doi.org/10.3390/sym12111853
Quesne C. Deformed Shape Invariant Superpotentials in Quantum Mechanics and Expansions in Powers of ℏ. Symmetry. 2020; 12(11):1853. https://doi.org/10.3390/sym12111853
Chicago/Turabian StyleQuesne, Christiane. 2020. "Deformed Shape Invariant Superpotentials in Quantum Mechanics and Expansions in Powers of ℏ" Symmetry 12, no. 11: 1853. https://doi.org/10.3390/sym12111853