Remarks on semiclassical quantization rule for broken SUSY

Inomata, Akira; Junker, Georg; Супармі, A.

doi:10.1088/0305-4470/26/9/020

Cited by 16 publications

(22 citation statements)

References 5 publications

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“…In the usual semiclassical stationary phase approximation the action functional S x(t)] is expanded about the classical path x cl (t) as S x(t)] ' S cl (x 00 ; x 0 ; ) + S x(t)] + 2 S x(t)]: (17) The classical action S cl S cl (x 00 ; x 0 ; ) = S x cl (t)] is an action evaluated along the classical path from x 0 to x 00 , determined by S x(t)] = 0. Then the propagator is given by the formula of Van Vleck, Pauli and Morette (VMP) 24], K(x 00 ; x 0 ; ) ' xed X xcl(t) r i 2 h @ 2 S cl @x 0 @x 00 exp i h S cl : (18) If there are classical paths more than one, we have to add up the contributions from all these paths. Thus, the summation in (18) is to cover all the classical paths x cl (t) between x 0 and x 00 with a xed time interval .…”

Section: The Quasi-classical Approachmentioning

confidence: 99%

“…Then the propagator is given by the formula of Van Vleck, Pauli and Morette (VMP) 24], K(x 00 ; x 0 ; ) ' xed X xcl(t) r i 2 h @ 2 S cl @x 0 @x 00 exp i h S cl : (18) If there are classical paths more than one, we have to add up the contributions from all these paths. Thus, the summation in (18) is to cover all the classical paths x cl (t) between x 0 and x 00 with a xed time interval . This approximation formula is known to give rise to the exact propagator for the free particle, the harmonic oscillator and more general quadratic systems, but does not directly provide the WKB quantization rule (2).…”

Section: The Quasi-classical Approachmentioning

confidence: 99%

“…In a previous work 16], we have proposed the following formula for a broken SUSY case, xR Z xL q 2m(Ẽ 2 (x)) dx = n + 1 2 h (4) with 2 (x L ) = 2 (x R ) =Ẽ. It has also been shown that this \broken SUSY formula" can reproduce exact spectra for the radial harmonic oscillator and the P oschl-Teller system 17,18].…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Quasiclassical path-integral approach to supersymmetric quantum mechanics

Inomata

Junker

1994

Phys. Rev. A

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From Feynman's path integral, we derive quasi-classical quantization rules in supersymmetric quantum mechanics (SUSY-QM). First, we derive a SUSY counterpart of Gutzwiller's formula, from which we obtain the quantization rule of Comtet, Bandrauk and Campbell when SUSY is good. When SUSY is broken, we arrive at a new quantization formula, which is found as good as and even sometime better than the WKB formula in evaluating energy spectra for certain onedimensional bound state problems. The wave functions in the stationary phase approximation are also derived for SUSY and broken SUSY cases. Insofar as a broken SUSY case is concerned, there are strong indications that the new quasi-classical approximation formula always overestimates the energy eigenvalues while WKB always underestimates.

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Section: The Quasi-classical Approachmentioning

confidence: 99%

Section: The Quasi-classical Approachmentioning

confidence: 99%

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Quasiclassical path-integral approach to supersymmetric quantum mechanics

Inomata

Junker

1994

Phys. Rev. A

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“…It has recently been shown that for the cases of shape invariant three dimensional oscillator as well as for P oschl-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.…”

Section: Introductionmentioning

confidence: 99%

Supersymmetry and quantum mechanics

1995

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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 m uch deeper understanding of why certain potentials are analytically solvable and an array o f p o w erful 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 eld and other potentials in terms of supersymmetric quantum mechanics. Finally, w e 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.

show abstract

“…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.…”

Section: Introductionmentioning

confidence: 99%

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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Remarks on semiclassical quantization rule for broken SUSY

Cited by 16 publications

References 5 publications

Quasiclassical path-integral approach to supersymmetric quantum mechanics

Quasiclassical path-integral approach to supersymmetric quantum mechanics

Supersymmetry and quantum mechanics

Parasupersymmetry in quantum mechanics

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