Dynamical Invariants and Quantization of the One-Dimensional Time-Dependent, Damped, and Driven Harmonic Oscillator
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GENERAL AND APPLIED PHYSICS
Dynamical Invariants and Quantization of the One-Dimensional Time-Dependent, Damped, and Driven Harmonic Oscillator M. C. Bertin1
· J. R. B. Peleteiro1 · B. M. Pimentel2 · J. A. Ramirez3
Received: 27 April 2020 © Sociedade Brasileira de F´ısica 2020
Abstract In this paper, it is proposed a quantization procedure for the one-dimensional harmonic oscillator with time-dependent frequency, time-dependent driven force, and time-dependent dissipative term. The method is based on the construction of dynamical invariants previously proposed by the authors, in which fundamental importance is given to the linear invariants of the oscillator. Keywords Dynamical invariants · Dissipative systems · Quantum damped oscillator · Time-dependent systems
1 Introduction Dynamical invariants were first used by Ermakov to show the connection between solutions of some special differential equations, referred to as Steen-Ermakov equations [1]. These equations were first studied by Steen [2] and then rediscovered by other authors [3, 4]. After that, Ray and Reid used the Ermakov approach to construct invariants for a much broader class of differential equations [5–7]. This purely mathematical interest was the start point of significant developments in classical and quantum dynamics. The importance of the dynamical invariants of a system should not be underrated. In classical mechanics, the dynamical constants of motion are the variables that allow complete M. C. Bertin
[email protected] J. R. B. Peleteiro [email protected] B. M. Pimentel [email protected] J. A. Ramirez [email protected] 1
Instituto de F´ısica, Universidade Federal da Bahia, R. Bar˜ao de Jeremoabo, s/n, Ondina, Salvador, Bahia, Brazil
2
Instituto de F´ısica Te´orica, S˜ao Paulo State University, R. Dr. Bento Teobaldo Ferraz, 271, Bloco II, Barra-Funda, S˜ao Paulo, Brazil
3
S˜ao Paulo, Brazil
integration of dynamical systems. In classical field theories, symmetries of Lagrangian systems are related to continuity equations and time invariants through the Noether theorem [8]. In quantum field theory, Casimir invariants of symmetry groups are essential to the understanding of the fundamental particle structure of our universe [9]. In quantum mechanics, a complete characterization of a quantum system is achieved by the knowledge of a complete set of time-invariant observables, which are also generators of a complete symmetry of the system. The process of quantization, therefore, is accomplished by finding an invariant set of stationary eigenvectors which generates, hopefully, a Hilbert space. Symmetries are linked to invariants, and invariants are linked to the very existence of quantum states, on a very fundamental level. In time-dependent systems, dynamical invariants play a major role, since the energy is no longer conserved, and sometimes even defined. Particularly, in quantum mechanics, systems with time-dependent Hamiltonians do not have well-defined energy spectra. Even in the case where a complete basis of eigenvectors exist
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