We apply supersymmetric quantum mechanics (SUSY QM) to multidimensional Schrödinger equations involving nonseparable potentials, which result in a system of coupled differential equations, where instead of the conventional definition of a scalar superpotential we introduce a superpotential matrix and succeed in applying SUSY QM to the coupled system. Finally, we discuss the shape-invariance condition for the potential matrix of such a system.

TAPAN DAS

Ancient Indian History and Culture

B CHAKRABARTI

University of Calcutta (informally known as Calcutta University or CU) is a collegiate public state university located in Kolkata, West Bengal, India.&nbsp;Within India it is recognized as a &quot;Five-Star University&quot; and accredited &quot;A&quot; Grade by National Assessment and Accreditation Council.

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The University of Calcutta has secured the Fifth position among the Universities of India in the prestigious &quot;Indian University Ranking 2019&quot; list, released by the NIRF of the Ministry of Human Resource Development, Government of India.

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It was established on 24 January 1857, and was one of the first institutions in Asia to be established as a multidisciplinary and Western-style university. Its alumni and faculty include several heads of state, heads of government, social reformers, prominent artists, only academy award winner and Dirac medal winner in India, many Fellows of the Royal Society and five Nobel laureates as of 2019 which is highest in South Asia.

University Of Calcutta

Application of supersymmetry to a coupled system of equations: the concept of a superpotential matrix

Journal of Physics A: Mathematical and General

We calculate ground and several low-lying excited states of various two electron atoms using hyperspherical adiabatic approximation (HAA). Its spectracular accuracy, as compared to exact numerical results of a set of coupled differential equations, demands a better understanding of its mechanism. It is seen that factorizability of the potential matrix into a product of a common function of the global length (hyperradius) and a constant matrix, is responsible for this remarkable success. This is a result of the shape invariance condition in multidimensional supersymmetric quantum mechanics.

International Journal of Modern Physics A

Quality of hyperspherical adiabatic approximation for excited states of atomic systems

Usual prescriptions for the supersymmetric Wentzel-Kramers-Brillouin (SWKB) formulation are applicable either in the case of broken or unbroken supersymmetry. However, they fail for one-dimensional potentials where there is a mixture of broken and unbroken supersymmetry, which corresponds to an oscillating superpotential. Such a situation arises when two or more identical wells interact by tunnelling through a potential barrier, exhibiting quasi-degeneracy. The simplest examples of twofold quasi-degeneracy are the double oscillator well and the double finite square well which are also very important from the point of view of application in real physics problems. The quasi-degenerate states are associated with six supersymmetric turning points in contrast with four classical turning points. This disagreement in the number of classical and supersymmetric turning points is responsible for the failure of the usual SWKB formulation and makes the case more interesting. Here we propose a modified version of the SWKB formulation using the partner potential to calculate the quasi-degenerate states. We apply our reformulated SWKB prescription to the double finite square well, which gives an excellent result.

Reformulation of SWKB quantization condition for quasi-degeneracy in one dimension involving oscillating superpotential

We investigate the reliability of the adiabatic approximation in the hyperspherical harmonics formalism. It has been applied to a number of two-electron systems, both compact and loosely bound, with the interaction between the constituent particles being purely Coulombic. The results are compared with the exact ones obtained by solving the set of coupled differential equations numerically. The accuracy of this approximation for two-electron systems is compared with that for systems with nuclear interactions, and we find that the former is better. An explanation has been provided for better agreement with Coulombic systems, particularly for noncompact ones. © 1997 The American Physical Society.

Physical Review A - Atomic, Molecular, and Optical Physics

Adiabatic approximation in atomic three-body systems

Coupled differential equations resulting from an application of the hyperspherical-harmonic expansion method to a number of two-electron atoms have been solved by an exact numerical method (renormalized Numerov method). Calculations are feasible (for the available computer facility) for the inclusion of up to 20 (K=20) partial waves. Although the convergence trend is discernible, the binding energies for K=20 are far from the converged values. An application of a convergence theorem by Schneider [Phys. Lett. 40B, 439 (1972)] permits extrapolation of the results to large K values. Such converged results agree closely with an accurate multiparameter variational calculation by Pekeris [Phys. Rev. 126, 1471 (1962)]. Even though the hyperspherical-harmonic expansion is notoriously slow for the long-range Coulomb interaction, the present prescription is a highly economical and accurate method for few-electron atoms. We have applied this method to the ground states of H-, He, Li+, and Be2+, as well as to the first excited singlet state of He. © 1994 The American Physical Society.

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ISSN	0305-4470