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Journal of Physics A Mathematical and General

Combined variable-phase supersymmetric quantum mechanics

P K Bera, T. Nandi, B. Talukdar

Published Oct 21, 1993• 4 citations
Publisher

Abstract

Methods of supersymmetric quantum mechanics (SUSYQM) are used to derive relations between phase functions which occur in the context of the variable-phase approach to potential scattering. It is shown that the phase-equivalent potentials obtained by the application of SUSYQM are not off-shell equivalent. A case study is presented in support of this.

DOI
10.1088/0305-4470/26/20/003

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Methods of supersymmetric quantum mechanics (SUSYQM) are used to derive relations between phase functions which occur in the context of the variable-phase approach to potential scattering. It is shown that the phase-equivalent potentials obtained by the application of SUSYQM are not off-shell equivalent. A case study is presented in support of this.
We analyze transition potentials (V(r)→r∼0αr−2)(V(r) \stackrel{r\sim 0}{\rightarrow} {αr^{-2}})(V(r)→r∼0αr−2) in non-relativistic quantum mechanics using the techniques of supersymmetry. For the range −1/4<α<3/4-1/4 < α< 3/4−1/4<α<3/4, the eigenvalue problem becomes ill-defined (since it is not possible to choose a unique eigenfunction based on square integrability and boundary conditions). It is shown that supersymmetric quantum mechanics (SUSYQM) provides a natural prescription for a unique determination of the spectrum. Interestingly, our SUSYQM based approach picks out the same "less singular" wave functions as the conventional approach, and thus provides a simple justification for the usual practice in the literature. Two examples (the Pöschl-Teller II potential and a two anyon system on the plane) have been worked out for illustrative purposes.
We analyse transition potentials, i.e. potentials exhibiting limiting inverse-square behaviour V(r) to r approximately 0 alpha r -2 in non-relativistic quantum mechanics using the techniques of supersymmetry. For the range - 1/4 <or= alpha < 3/4 , the eigenvalue problem becomes ill defined (since it is not possible to choose a unique eigenfunction based on square integrability and boundary conditions). It is shown that supersymmetric quantum mechanics (SUSYQM) provides a natural prescription for a unique determination of the spectrum. Interestingly, our SUSYQM-based approach picks out the same 'less singular' wavefunctions as the conventional approach and thus provides a simple justification for the usual practice in the literature. Two examples (the Poschl-Teller II potential and a two-anyon system on the plane) have been worked out for illustrative purposes.
With a view to increasing the scope of applications of supersymmetric quantum mechanics (SUSY QM), we formulate the same in a complex phase space. Within this framework, the concept of shape invariance is reinvestigated and an insight into the eigenvalue spectra of non-Hermitian Hamiltonians is sought. The results are applied to a variety of potentials. We claim that the shape invariance for these potentials in the complex phase space can be retrieved in terms of the prescriptions already proposed in the conventional SUSY QM, in that the transformation of potential parameters takes the form of a reflection in the parameter space. Interestingly, some of these features turn out to be the generalization of the concept of quasi-parity used recently in the context of SUSY QM of PT-symmetric potentials. Further, a correspondence of the present approach with other complex formulations of SUSY QM and also in two real dimensions is demonstrated.
First it is shown in an abstract mathematical framework of supersymmetric quantum theory that, under some conditions for the perturbation, the wave and the scattering operators on bosonic states have some connections with those on fermionic ones. In particular, a unitary equivalence between the two scattering operators is proved. Then, a class of supersymmetric quantum mechanical models on Rn is discussed and the abstract general results are applied to obtain some more explicit results on the supersymmetric potential scattering.