Semiclassical approach to quantum-mechanical problems with broken supersymmetry

Dutt, Ranabir; Gangopadhyaya, Asim; Khare, Avinash; Pagnamenta, A.; Sukhatme, U.

doi:10.1103/physreva.48.1845

Cited by 24 publications

(45 citation statements)

References 27 publications

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“…We now show that the entire spectrum of H 1 can be very easily obtained algebraically by using the shape invariance condition (96 Hamiltonian H n will have the same spectrum as H 1 except that the first n−1 levels of H 1 will be absent in H n . On repeatedly using the shape invariance condition (96), it is then clear that…”

Section: General Formulas For Bound State Spectrum Wave Functions Anmentioning

confidence: 99%

“…For the simplest possibility of m = 1, one has a number of solutions to the shape invariance condition (96). In Table 4.1, we give expressions for the various shape invariant potentials V 1 (x), superpotentials W (x), parameters a 1 and a 2 and the corresponding energy eigenvalues E (1) n [54,55].…”

Section: Solutions Involving Translationmentioning

confidence: 99%

“…It has recently been shown that for the cases of shape invariant three dimensional oscillator as well as for Pöschl-Teller I and II potentials with broken SUSY, this lowest order BSWKB calculation gives the exact spectrum [95,97]. Recently, Dutt et al [96] have also developed a systematic higher order BSWKB expansion and using it have shown that in all the three (shape invariant) cases, the higher order corrections to O(h 6 ) are zero. Further, the energy eigenvalue spectrum has also been obtained in the case of several non-SIP and it turns out that in many cases BSWKB does as well as (if not better than) the usual WKB approximation [96].…”

Section: Introductionmentioning

confidence: 99%

“…Let us now discuss the interesting question of the classification of various solutions to the shape invariance condition (96). This is clearly an important problem because once such a classification is available, then one discovers new SIPs which are solvable by purely algebraic methods.…”

Section: Strategies For Categorizing Shape Invariant Potentialsmentioning

confidence: 99%

“…Recently, Dutt et al [96] have also developed a systematic higher order BSWKB expansion and using it have shown that in all the three (shape invariant) cases, the higher order corrections to O(h 6 ) are zero. Further, the energy eigenvalue spectrum has also been obtained in the case of several non-SIP and it turns out that in many cases BSWKB does as well as (if not better than) the usual WKB approximation [96].…”

Section: Introductionmentioning

confidence: 99%

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Parasupersymmetry in quantum mechanics

Khare

1993

Journal of Mathematical Physics

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In the past ten years, the ideas of supersymmetry have been profitably applied to many nonrelativistic quantum mechanical problems. In particular, there is now a much deeper understanding of why certain potentials are analytically solvable and an array of powerful new approximation methods for handling potentials which are not exactly solvable. In this report, we review the theoretical formulation of supersymmetric quantum mechanics and discuss many applications. Exactly solvable potentials can be understood in terms of a few basic ideas which include supersymmetric partner potentials, shape invariance and operator transformations. Familiar solvable potentials all 1 have the property of shape invariance. We describe new exactly solvable shape invariant potentials which include the recently discovered self-similar potentials as a special case. The connection between inverse scattering, isospectral potentials and supersymmetric quantum mechanics is discussed and multi-soliton solutions of the KdV equation are constructed. Approximation methods are also discussed within the framework of supersymmetric quantum mechanics and in particular it is shown that a supersymmetry inspired WKB approximation is exact for a class of shape invariant potentials. Supersymmetry ideas give particularly nice results for the tunneling rate in a double well potential and for improving large N expansions. We also discuss the problem of a charged Dirac particle in an external magnetic field and other potentials in terms of supersymmetric quantum mechanics. Finally, we discuss structures more general than supersymmetric quantum mechanics such as parasupersymmetric quantum mechanics in which there is a symmetry between a boson and a para-fermion of order p.

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Section: General Formulas For Bound State Spectrum Wave Functions Anmentioning

confidence: 99%

Section: Solutions Involving Translationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Strategies For Categorizing Shape Invariant Potentialsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Parasupersymmetry in quantum mechanics

Khare

1993

Journal of Mathematical Physics

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Uncommonly accurate energies for the general quartic oscillator

Okun

Burke

2020

Int J of Quantum Chemistry

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Recent advances in the asymptotic analysis of energy levels of potentials produce relative errors in eigenvalue sums of order 10−34, but few non‐trivial potentials have been solved numerically to such accuracy. We solve the general quartic potential (arbitrary linear combination of x2 and x4) beyond this level of accuracy using a basis of several hundred oscillator states. We list the lowest 20 eigenvalues for 9 such potentials. We confirm the known asymptotic expansion for the levels of the pure quartic oscillator, and extract the next two terms in the asymptotic expansion. We give analytic formulas for expansion in up to three even basis states. We confirm the virial theorem for the various energy components to similar accuracy. The sextic oscillator levels are also given. These benchmark results should be useful for extreme tests of approximations in several areas of chemical physics and beyond.

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Semiclassical approach to quantum-mechanical problems with broken supersymmetry

Cited by 24 publications

References 27 publications

Parasupersymmetry in quantum mechanics

Parasupersymmetry in quantum mechanics

Uncommonly accurate energies for the general quartic oscillator

Symbolic Computations

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