Effective Equations for Expectation Values

T. S. Shankara

Muzaffer Ayvaz, Metin Demіralp

In this study, expectation value dynamics of the quantum harmonic oscillator under the influence of the external dipol effects is discussed in some details. The dipole function of the system has been taken as an odd cubic spatial polynomial and the external field amplitude has been assumed to be known. Instead of the wave function of the system, expectation values of the position and momentum operators are considered as unknowns and certain matrix ordinary differential equations are obtained by using the fluctuation free matrix representation concept. The resulting equations are in the category of Quantum Mechanical Matrix Ordinary Differential Equations (QMMODE) and can be solved by a recently developed scheme called Characteristic Evolutions Method (CEM). We mention also how to implement this approach. The purpose of this work can be considered as a preparation to quantum optimal control problem solutions via expectation value dynamics.

Galina A. Nemova

C. A. Onate, I. B. Okon, M. C. Onyeaju +1 more

The solution of one-dimensional Schrodinger equation for a newly proposed potential called modified shifted Deng-Fan momentum-dependent potential is obtained via supersymmetric approach. The expectation values of momentum and position were calculated using Hellmann Feynman Theorem. The effects of momentum-dependent parameter on the solutions of the system as well as the expectation values were studied. Finally, the special cases of the interacting potential were obtained.

Muzaffer Ayvaz, Metin Demіralp

This study focuses on the construction of the optimal control equations for one dimensional quantum harmonic oscillator under the influence of external dipol effects and the solution of these equations by using Fluctuationlessness Theorem and a recently developed scheme called Characteristic Evolutions Method. The dipole function of the system has been taken as odd cubic spatial polynomial. Optimal control equations of the system under consideration are constructed by using expectation values of the position and the momentum operators instead of the wave and costate evolutions. It is shown that, the resulting equations are systems of ordinary differential equations and there are infinitely many ODEs. The solution strategy is based on the approximation of the expectation values for the operator products in the sense of Fluctuationlessness Theorem.