Dynamics of localized states in N = 4 SUSY QM

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ynamics of Localized States in N = 4 SUSY QM1 V. P. Berezovoj and M. I. Konchatnij A.I. Akhiezer Institute of Theoretical Physics, National Science Center “Kharkov Institute of Physics and Technology”, 1, Akademicheskaya St., Kharkov, 61108 Ukraine email: [email protected] Abstract—A consistent approach is offered for investigating the temporal dynamics of localized states. It is based on exactly solvable quantum mechanical models with multiwell potentials and the associate propaga tors. The Hamiltonian states with multiwell potentials form an adequate basis for expanding wave packets (WP) of various types and degrees of localization. Special features of WP tunneling have been studied with due regard for all Hamiltonian states with symmetric and asymmetric potentials. DOI: 10.1134/S1063779612050048 1

In most cases, theoretical analysis of tunneling transitions in doublewell potentials is carried out in the twomode approximation [1]. The significant characteristics in this approach are the energy differ ence of the ground state and the first excited state ( Δ = E1 − E 0 ), and also, their wave functions. The use of this approximation provides explanation for the general properties of tunneling processes; how ever, it fails to give interpretation of many fine effects [2–4]. Besides, the analysis of the processes with multiwell potentials is hampered by the fact that the models employed operate, as a rule, on phenomeno logical or piecewise potentials (e.g., with cross linked rectangular wells and barriers or parabolas), which are far from real potentials. It should be noted that there exist exactly solvable models with multi well potentials [5–7], which can be applied to describe the tunneling processes. The WP time evo lution is described in terms of the propagators, which take into account the contribution of the whole Hamiltonian spectrum. In papers [8–10] an approach has been proposed to construct new prop agators on the basis of the propagators already known in the context of supersymmetric quantum mechanics. It is of interest to apply the mentioned procedure for obtaining propagators in the exactly solvable quantum mechanical models with multi well potentials. The goal of the present work has been to study the peculiarities of localized state dynamics in multiwell potentials in the consistent approach. Exactly solv able models obtained in the N = 4 SUSY QM [7] have been used as potentials, both symmetric and asym metric, and the WP dynamics is described by means of the propagators corresponding to these models and calculated in the approach [8–10]. 1 The article is published in the original.

1. N = 4 SUSY QM AND MULTIWELL POTENTIALS The procedure of constructing Hamiltonians with multiwell potentials in the N = 4 SUSY QM, consid ered in detail in [7]. It can be realized through adding additional levels with the energy ε below the ground state energy E 0 of the basic Hamiltonian H 0 . In this case the multiwell structure of the potentials derived ( E0 – ε ) will be most indicative fo