Nonuniqueness of the factorization scheme in quantum mechanics
01 Aug 1989-International Journal of Theoretical Physics (Kluwer Academic Publishers-Plenum Publishers)-Vol. 28, Iss: 8, pp 911-916
TL;DR: In this paper, the consequences of the nonuniqueness of the factorizability of a quantum mechanical Hamiltonian in one dimension were investigated. But they did not consider the non-uniqueness in terms of the energy spectrum of the harmonic oscillator.
Abstract: We enquire into the consequences of the nonuniqueness of the factorizability of a quantum mechanical Hamiltonian in one dimension. This leads to a hierarchy of potentials, a particular class of which is endowed with the energy spectrum of the harmonic oscillator.
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TL;DR: In this paper, a constructive supersymmetric quantum mechanical method based on a reparametrization of the corresponding conventional superpotential and on the addition of an extra rational contribution expressed in terms of some polynomial g is proposed.
Abstract: New exactly solvable rationally-extended radial oscillator and Scarf I potentials are generated by using a constructive supersymmetric quantum mechanical method based on a reparametrization of the corresponding conventional superpotential and on the addition of an extra rational contribution expressed in terms of some polynomial g. The cases where g is linear or quadratic are considered. In the former, the extended potentials are strictly isospectral to the conventional ones with reparametrized couplings and are shape invariant. In the latter, there appears a variety of extended potentials, some with the same charac- teristics as the previous ones and others with an extra bound state below the conventional potential spectrum. Furthermore, the wavefunctions of the extended potentials are con- structed. In the linear case, they contain ( + 1)th-degree polynomials with = 0,1,2,..., which are shown to beX1-Laguerre orX1-Jacobi exceptional orthogonal polynomials. In the quadratic case, several extensions of these polynomials appear. Among them, two different kinds of ( + 2)th-degree Laguerre-type polynomials and a single one of ( + 2)th-degree Jacobi-type polynomials with = 0,1,2,... are identified. They are candidates for the still unknown X2-Laguerre and X2-Jacobi exceptional orthogonal polynomials, respectively.
150 citations
TL;DR: In this article, the authors develop a systematic approach to construct completely solvable rational potentials, which are then used to solve the singularity problem in supersymmetric quantum systems.
Abstract: We develop a systematic approach to construct novel completely solvable rational potentials. Second-order supersymmetric quantum mechanics dictates the latter to be isospectral to some well-studied quantum systems. \( \mathcal{P}\mathcal{T} \) symmetry may facilitate reconciling our approach to the requirement that the rationally extended potentials be singularity free. Some examples are shown.
134 citations
TL;DR: In this paper, the authors develop a systematic approach to construct completely solvable rational potentials, in which the second-order supersymmetric quantum mechanics dictates the latter to be isospectral to some well-studied quantum systems.
Abstract: We develop a systematic approach to construct novel completely solvable rational potentials. Second-order supersymmetric quantum mechanics dictates the latter to be isospectral to some well-studied quantum systems. $\cal PT$ symmetry may facilitate reconciling our approach to the requirement that the rationally-extended potentials be singularity free. Some examples are shown.
116 citations
TL;DR: In this article, the nonuniqueness of the factorization is exploited to derive new isospectral nonsingular potentials, which may have many possible applications in atomic and molecular physics.
Abstract: Factorization of quantum-mechanical potentials has a long history extending back to the earliest days of the subject. In the present article, the nonuniqueness of the factorization is exploited to derive new isospectral nonsingular potentials. Many one-parameter families of potentials can be generated from known potentials using a factorization that involves superpotentials defined in terms of excited states of a potential. For these cases an operator representation is available. If ladder operators are known for the original potential, then a straightforward procedure exists for defining such operators for its isospectral partners. The generality of the method is illustrated with a number of examples which may have many possible applications in atomic and molecular physics.
24 citations
TL;DR: In this article, the necessary and sufficient conditions for a type A N -fold supercharge to possess intermediate Hamiltonians for the N = 2 case were derived, and it was shown that whenever it has (at least) one intermediate Hamiltonian, it can admit second-order parasupersymmetry and generalized 2-fold superalgebra.
19 citations
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Journal Article•
759 citations
TL;DR: In this article, a one-parameter family of potentials in one dimension was constructed with the energy spectrum coinciding with that of the harmonic oscillator, which is a new derivation of a class of possible potentials previously obtained by Abraham and Moses with the help of the Gelfand-Levitan formalism.
Abstract: A one‐parameter family of potentials in one dimension is constructed with the energy spectrum coinciding with that of the harmonic oscillator. This is a new derivation of a class of potentials previously obtained by Abraham and Moses with the help of the Gelfand–Levitan formalism.
379 citations
TL;DR: In this article, the basic concepts of supersymmetry and its characteristic features: anticommuting variables, supercharges, the cancellation of divergences, the vanishing of the vacuum energy, the degeneracy of energy spectra, and spontaneous breaking of superymmetry are discussed.
Abstract: This review is an elementary introduction to supersymmetry. The example of supersymmetric quantum mechanics is used to discuss the basic concepts of supersymmetry and its characteristic features: anticommuting variables, supercharges, the cancellation of divergences, the vanishing of the vacuum energy, the degeneracy of energy spectra, and the spontaneous breaking of supersymmetry. The form taken by supersymmetry in problems in quantum mechanics and nuclear physics is discussed. The use of a supersymmetric formalism in statistical physics and field theory is also discussed.
256 citations
TL;DR: In this paper, the idea of generating a family of isospectral Hamiltonians from a given Hamiltonian using supersymmetric quantum mechanics is exploited in constructing the 'partner' stability equation for the phi 4 soliton stability equation.
Abstract: The idea of generating a family of isospectral Hamiltonians from a given Hamiltonian using supersymmetric quantum mechanics is exploited in constructing the 'partner' stability equation for the phi 4 soliton stability equation. From the 'partner' stability equation, the soliton profile and the potential that admits the soliton solution are derived.
35 citations
TL;DR: In this paper, a correct procedure for constructing supersymmetry in 3D is presented and the degeneracies are found between states of the same l but different n and Z and the previous results on the Coulomb and the three-dimensional isotropic oscillator problems are reestablished.
Abstract: A correct procedure for constructing supersymmetry in three dimensions is presented. The degeneracies are found between states of the same l but different n and Z and the previous results on the Coulomb and the three-dimensional isotropic oscillator problems are reestablished. The authors also consider the hydrogen-helium problem and find supersymmetry to hold to a good approximation.
20 citations