Time-dependent Schrödinger equation with non-central potentials

Abstract. We elaborate further on the metric representation that is obtained by transferring the timedependence from a Hermitian Hamiltonian to the metric operator in a related non-Hermitian system. We provide further insight into the procedure on how to employ the time-dependent Dyson relation and the quasi-Hermiticity relation to solve time-dependent Hermitian Hamiltonian systems. By solving both equations separately we argue here that it is in general easier to solve the former. We solve the mutually related time-dependent Schrödinger equation for a Hermitian and non-Hermitian spin 1/2, 1 and 3/2 model with time-independent and time-dependent metric, respectively. In all models the overdetermined coupled system of equations for the Dyson map can be decoupled algebraic manipulations and reduces to simple linear differential equations and an equation that can be converted into the non-linear Ermakov-Pinney equation.

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“…[18][19][20][21][22][23], and also some quantization schemes [24,25]. The general solution to this equation is known to be…”

Section: Solutions Of the Time-dependent Dyson Relationmentioning

confidence: 99%

Metric versus observable operator representation, higher spin models

Fring

Frith

2018

Eur. Phys. J. Plus

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“…The finally obtained wave functions are expressed in terms of these eigenfunctions and time-dependent phase factors, given in (49), that correspond to global phases. The difference of this research from the previous one, performed by Ferkous et al [30], is that we used the NU method in order to derive exact quantum solutions of the system whereas Ferkous et al did not. Quantum wave functions that we derived in this work were represented in terms of spherical harmonics ( , ), while those of Ferkous et al were represented in terms of the Jacobi polynomials.…”

Section: Resultsmentioning

confidence: 94%

“…According to the Lewis-Riesenfeld method which is available for deriving quantum solutions, it is necessary to construct an invariant operator of the system. By solving (6), we have the following invariant operator [30]:…”

Section: Hamiltonian and Invariant Operatormentioning

confidence: 99%

Quantization of a 3D Nonstationary Harmonic plus an Inverse Harmonic Potential System

Medjber

Bekkar

Choi

2016

Advances in Mathematical Physics

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The Schrödinger solutions for a three-dimensional central potential system whose Hamiltonian is composed of a time-dependent harmonic plus an inverse harmonic potential are investigated. Because of the time-dependence of parameters, we cannot solve the Schrödinger solutions relying only on the conventional method of separation of variables. To overcome this difficulty, special mathematical methods, which are the invariant operator method, the unitary transformation method, and the Nikiforov-Uvarov method, are used when we derive solutions of the Schrödinger equation for the system. In particular, the Nikiforov-Uvarov method with an appropriate coordinate transformation enabled us to reduce the eigenvalue equation of the invariant operator, which is a second-order differential equation, to a hypergeometric-type equation that is convenient to treat. Through this procedure, we derived exact Schrödinger solutions (wave functions) of the system. It is confirmed that the wave functions are represented in terms of time-dependent radial functions, spherical harmonics, and general time-varying global phases. Such wave functions are useful for studying various quantum properties of the system. As an example, the uncertainty relations for position and momentum are derived by taking advantage of the wave functions.

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“…Quantum problem of molecular interactions such as the Coulomb-type and harmonic interactions can be described in terms of central [36][37][38] or noncentral [7,[39][40][41] time-dependent Hamiltonians. Attractive or repulsive forces between various molecules including non-bonded atoms are responsible for a specific formation of molecular structure and its change.…”

Section: Hamiltonian and Invariantmentioning

confidence: 99%

“…To attain accurate results when we study molecular systems, it is necessary to introduce an exact Hamiltonian that yields actual time dependence of molecule behaviors. If we consider the convention that time-varying factors have usually been neglected on most studies of dynamical systems, the recent tendency [3,[36][37][38][39][40][41][42][43][44][45][46][47][48] for considering time dependence of physical parameters in this field may open up a new trend in the analysis of molecular interactions.…”

Section: Introductionmentioning

confidence: 99%

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

2017

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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.

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Time-dependent Schrödinger equation with non-central potentials

Cited by 10 publications

References 42 publications

Metric versus observable operator representation, higher spin models

Metric versus observable operator representation, higher spin models

Quantization of a 3D Nonstationary Harmonic plus an Inverse Harmonic Potential System

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

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