Information entropy of conditionally exactly solvable potentials

Abstract

We evaluate Shannon entropy for the position and momentum eigenstates of some conditionally exactly solvable potentials which are isospectral to harmonic oscillator and whose solutions are given in terms of exceptional orthogonal polynomials. The Bialynicki-Birula-Mycielski (BBM) inequality has also been tested for a number of states.

Quantum Hamilton–Jacobi route to exceptional Laguerre polynomials and the corresponding rational potentials

“…Over the years these new polynomials appeared in connection with nonlinear oscillators [20]- [21] and superintegrability [22]. They can also be found in the context of quantum information theory [23], discrete quantum mechanics [24] and Schrödinger equation with position dependent mass and other studies [25]- [27].…”

Darboux partners of pseudoscalar Dirac potentials associated with exceptional orthogonal polynomials

“…for Pauli matrices σ 1 and σ 3 . Expression (10) represents a pseudoscalar potential for the Dirac equation (9). In conclusion, we have converted a solution ψ of our initial Schrödinger equation to a solution of the Dirac equation (9) for the pseudoscalar potential (10).…”

“…Along with the study of the mathematical properties of these exceptional polynomial families [6], the latter has found applications in a number of physical problems, e.g. quantum superintegrability [7], Dirac operators minimally or non minimally coupled to external fields and Fokker Planck equations [8], entropy measures in quantum information theory [9], solutions of Schrödinger equation with some conditionally exactly solvable potentials [10], solutions for position dependent mass systems [12], discrete quantum mechanics [17], quantum Hamilton-Jacobi formalism [11], N -fold Supersymmetry and its dynamical breaking in the context of position dependent mass [13].…”

Time dependent potentials associated with exceptional orthogonal polynomials

“…The multi-indexed version X m 1 , m 2 , ..., m k of X m EOPs are constructed by using multi-step Darboux transformation, 21 Crum-Adler mechanism, 22 higher-order SUSYQM, 23 and multi-step Darboux-Backlund transformation. 11 Recently, EOPs have been studied in diverse scenarios such as quantum Hamilton-Jacobi formalism, 24 position dependent mass systems, 25 N-fold supersymmetry, 26,27 Fokker-Planck equations, 28 conditionally exactly solvable potentials, 29,30 and quantum mechanical scattering. 31 All above mentioned cases are related to time independent Hamiltonians.…”