Exact and approximated solutions for the harmonic and anharmonic repulsive oscillators: Matrix method
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THE EUROPEAN PHYSICAL JOURNAL D
Regular Article
Exact and approximated solutions for the harmonic and anharmonic repulsive oscillators: Matrix method Braulio M. Villegas-Mart´ıneza , H´ector Manuel Moya-Cessa, and Francisco Soto-Eguibar ´ Instituto Nacional de Astrof´ısica, Optica y Electr´ onica, INAOE, Calle Luis Enrique Erro 1, Santa Mar´ıa Tonantzintla, San Andr´es Cholula, Puebla 72840, Mexico Received 2 March 2020 / Received in final form 18 May 2020 Published online 1 July 2020 c EDP Sciences / Societ`
a Italiana di Fisica / Springer-Verlag GmbH Germany, part of Springer Nature, 2020 Abstract. We study the repulsive harmonic oscillator and an extension of this system, with an additional linear anharmonicity term. The system is solved in exact and perturbative form, the latter by using the socalled normalized perturbative matrix method. The perturbative solution up to second order is compared with the exact solution when the system is initially in coherent and a Schr¨ odinger-cat states.
1 Introduction In 1926, Schr¨odinger supplied the theoretical basis for analyzing the time evolution of a quantum system through his famous equation that now bears his name [1–4]. Soon after its advent, an intrinsic interest emerged to get exactly solvable models by using the Schr¨ odinger equation; however, only a very small number of problems fulfil this specific requirement [2–6]. Since the vast majority of physical systems are rather complicated to be treated exactly [7], one may try to find approximated solutions with the aid of other methods, like perturbation theories [8,9]. Among the available perturbative recipes, one which stands out is the Normalized Matrix Perturbation Method (NMPM) [10–13]. Such as its name suggests, this scheme is devoted to seek normalized perturbative solutions of the time-dependent Schr¨ odinger equation, ordered in power series of block-diagonal tridiagonal matrices. These solutions possess time dependent factors that enable to determine the temporal evolution of the corrections [10–13]. The extension of this approach has gradually grown due to its capability to deal with mixed states, where the Lindblad master equation is treated perturbatively [12]; additionally, the NMPM is capable of generating a dual Dyson series in matrix form, which it works for both weak and strong perturbations [13]. Up to now, the NMPM has been tested on systems where one can find simultaneously a well defined unitary evolution operator for the unperturbed Hamiltonian and its eigenstates. The simple harmonic oscillator is an instance model that exhibits this nature; indeed, this singular example maintains the first statement true whereas the second one may fail if, and only if, the oscillator frequency switches from real to imaginary, i.e. ω → iω [14–18]. If this happens, the harmonic oscillator turns upside-down and gives rise to a repulsive a
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oscillator. Time dependent repulsive oscillators have been studied in the framework of invariants for time dependent Hamiltonians by Pedros
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