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Journal of Physics Communications

Quantum features of molecular interactions associated with time-dependent non-central potentials

Jeong Ryeol Choi, Salah Menouar, Salim Medjber, Hacene Bekkar

Published Dec 13, 2017• 6 citations
Publisher

Abstract

The research for quantum features of molecular Coulomb interactions subjected to a time-dependent non-central potential has many applications. The wave functions of this system can be obtained by using an invariant operator, which is necessary for investigating a time-dependent Hamiltonian system. Regarding time dependence of the system, one can confirm that the formula of this operator is in general somewhat complicated. Hence, in order to solve its eigenvalue equation, special mathematical techniques beyond separation of variables method, such as the unitary transformation method, the Nikiforov–Uvarov method, and the asymptotic iteration method, should be employed. The double ring-shaped generalized non-central potential of which evolution explicitly depends on time is introduced as a particular case. The complete quantum solutions of the system can be identified from the eigenstates of the invariant operator. These solutions are useful for analyzing dynamical properties of the system.

DOI
10.1088/2399-6528/aa83f6

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The research for quantum features of molecular Coulomb interactions subjected to a time-dependent non-central potential has many applications. The wave functions of this system can be obtained by using an invariant operator, which is necessary for investigating a time-dependent Hamiltonian system. Regarding time dependence of the system, one can confirm that the formula of this operator is in general somewhat complicated. Hence, in order to solve its eigenvalue equation, special mathematical techniques beyond separation of variables method, such as the unitary transformation method, the Nikiforov–Uvarov method, and the asymptotic iteration method, should be employed. The double ring-shaped generalized non-central potential of which evolution explicitly depends on time is introduced as a particular case. The complete quantum solutions of the system can be identified from the eigenstates of the invariant operator. These solutions are useful for analyzing dynamical properties of the system.
Quantum features of a dynamical system subjected to time-dependent non-central potentials are investigated. The entire potential of the system is composed of the inverse quadratic potential and the Coulomb potential. An invariant operator that enables us to treat the time-dependent Hamiltonian system in view of quantum mechanics is introduced in order to derive Schrödinger solutions (wave functions) of the system. To simplify the problem, the invariant operator is transformed to a simple form by unitary transformation. Quantum solutions in the transformed system are easily obtained because the transformed invariant operator is a time-independent simple one. The Nikiforov-Uvarov method is used for solving eigenvalue equation of the transformed invariant operator. The double ring-shaped generalized non-central time-dependent potential is considered as a particular case for further study. From inverse transformation of quantum solutions obtained in the transformed system, the complete quantum solutions in the original system are identified. The quantum properties of the system are addressed on the basis of the wave functions.
Quantum solutions of a time-dependent Hamiltonian for the motion of a time-varying mass subjected to time-dependent singular potentials in three dimensions are investigated. A time-dependent inverse quadratic potential and a Coulomb-like potential are considered as the components of the singular potential of the system. Because the Hamiltonian is a function of time, special techniques for deriving quantum solutions of the system are necessary. A quadratic invariant operator is introduced, and its eigenstates are derived using the Nikiforov-Uvarov method together with a unitary transformation method. The Nikiforov-Uvarov method enables us to solve the eigenvalue equations of the invariant operator, which are second-order linear diffierential equations, by reducing the original equation to a hypergeometric type. According to the invariant operator theory, the wave functions of the system are represented in terms of the eigenstates obtained in such a way. The difference of the wave functions from the eigenstates of the invariant operator is that the wave functions have time-dependent phases while the eigenstates do not. By determining the phases of the wave functions via the help of the Schr¨odinger equation, we identify the full wave functions of the system and address their physical implications.
The one-dimensional Schrödinger equation associated with a time-dependent Coulomb potential is studied. The invariant operator method (Lewis and Riesenfeld) and unitary transformation approach are employed to derive quantum solutions of the system. We obtain an ordinary second-order differential equation whose analytical exact solution has been unknown. It is confirmed that the form of this equation is similar to the radial Schrödinger equation for the hydrogen atom in a (arbitrary) strong magnetic field. The qualitative properties for the eigenstates spectrum are described separately for the different values of the parameter ω 0 appearing in the x 2 term, x being the position, i.e., ω 0 > 0, ω 0 < 0 and ω 0 = 0. For the ω 0 = 0 case, the eigenvalue equation of invariant operator reduces to a solvable form and, consequently, we have provided exact eigenstates of the time-dependent Hamiltonian system.
Explicitly time-dependent Hamiltonians are ubiquitous in applications of quantum theory. It is therefore necessary to solve the time-dependent Schrödinger equation directly. The system’s dynamics is associated with a unitary time-evolution operator (a propagator), formally given as an infinite Dyson series. Time-dependent observables are invariant under unitary time-dependent transformations, where it is sometimes useful to transform the time-evolution from the states into the corresponding operators. This is carried out in part (in full) by transforming to the interaction (Heisenberg) picture. The corresponding equations of motion for the time-dependent operators are introduced. For quadratic potential energy functions, the time evolution of quantum expectation values coincides with the corresponding classical dynamics. This is demonstrated and analyzed in detail for Gaussian wave packets and a coherent state. Finally, we derive exact and approximate expressions for time-dependent transition probabilities and transition rates between quantum states. The validity of time-dependent perturbation theory is analyzed by comparison to exact dynamics.
Quantum states of a particle subjected to time‐dependent singular potentials in one‐dimension are investigated using invariant operator method and the Nikiforov‐Uvarov method. We consider the case that the system is governed by two singular potentials which are the Coulomb potential and the inverse quadratic potential. An invariant operator that is a function of time has been constructed via a fundamental mechanics. This invariant operator is transformed to a simple one using a unitary operator, which is a time‐independent invariant operator. By solving the Schrödinger equation in the transformed system, analytical forms of exact eigenvalues and eigenfunctions of the invariant operator are evaluated in a simple elegant manner with the help of the Nikiforov‐Uvarov method. Eventually, the full wave functions in the original system (untransformed system) are obtained through an inverse unitary transformation from the wave functions in the transformed system. Quantum characteristics of the system associated with the wave functions are addressed in detail.
The general solutions of the Schrödinger equation for a non-central potential are obtained by using the Nikiforov–Uvarov method. The Schrödinger equation with general non-central potential is separated into radial and angular parts, and energy eigenvalues and eigenfunctions are derived analytically. By making special selections, the non-central potential is reduced to Coulomb and Hartmann ring-shaped potentials, and the obtained results are compared with the solutions of Coulomb and Hartmann potentials given in the literature.