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Entropy

A Method for Choosing an Initial Time Eigenstate in Classical and Quantum Systems

Gabino Torres-Vega, Mónica Noemí Jiménez-García

Published Jun 17, 2013• 2 citations
Publisher

Abstract

A subject of interest in classical and quantum mechanics is the development of the appropriate treatment of the time variable. In this paper we introduce a method of choosing the initial time eigensurface and how this method can be used to generate time-energy coordinates and, consequently, time-energy representations for classical and quantum systems.

DOI
10.3390/e15062415

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Representation of quantum mechanical wavefunctions by transformation generators. III. Canonical transformations in an extended phase space

C Jung, H. Krüger

For pt.II see ibid., vol.16, no.7, p.1397-408 (1983). It is shown how quantum mechanical wavefunctions can be obtained from a sequence of simple canonical transformations, which map the given system onto a simple reference system. The resulting wavefunctions are at least uniformly valid up to order h(cross). Under some more restrictive conditions for the individual transformation steps the authors even find the exact wavefunctions. The essential point of the paper is to enlarge the conventional coordinate-momentum phase space by taking time and energy as an additional conjugate pair. In this extended space they exploit the possibility of using transformations which intermix energy and time with position coordinates and momenta. Compared with transformations in the conventional position-momentum phase space, they gain the advantage that scattering states and bound states can be treated in a unified way. Therefore this method is appropriate for systems with mixed spectra. In addition it allows for more flexibility in choosing the individual transformation steps. The practicability of the method is demonstrated by several examples.

Journal of Physics A Mathematical and General•Apr 1, 1984•0 citations
A subject of interest in classical and quantum mechanics is the development of the appropriate treatment of the time variable. In this paper we introduce a method of choosing the initial time eigensurface and how this method can be used to generate time-energy coordinates and, consequently, time-energy representations for classical and quantum systems.
We discuss quantum time states formed with a finite number of energy eigenstates with the purpose of obtaining a time coordinate. These time states are eigenstates of the recently introduced discrete time operator. The coordinate and momentum representations of these time eigenstates resemble classical time curves and become classical at high energies. To illustrate this behavior, we consider the simple example of the particle-in-a-box model. We can follow the quantum-classical transition of the system. Among the many existing solutions for the particle in a box, we use a set which leads to time eigenstates for use as a coordinate system.
The usual method for obtaining the eigenstates of an operator is to solve the corresponding eigenvalue equation. This procedure cannot be applied when the operator of interest is not known at all. We develop a method which generates the eigenstates of an operator, and the operator itself, which will be conjugate to a given known operator. This is particularly useful for the case of the time operator in Quantum Mechanics. We illustrate the method by obtaining time eigenstates for the free particle.
An aspect of quantum mechanics that has not been fully understood is the energy shift generated by the time operator. In this study, we introduce the use of the eigensurfaces of dynamical variables and commutators in classical mechanics to study the classical analogue of the quantum translation of energy. We determine that there is a conjugate dynamical system that is conjugate to Hamilton's equations of motion, and then we generate the analogue of the time operator and use it in the translation of points along the energy direction, i.e. the classical analogue of the Pauli theorem. The theory is illustrated with a nonlinear oscillator model.
For pt.II see ibid., vol.16, no.7, p.1397-408 (1983). It is shown how quantum mechanical wavefunctions can be obtained from a sequence of simple canonical transformations, which map the given system onto a simple reference system. The resulting wavefunctions are at least uniformly valid up to order h(cross). Under some more restrictive conditions for the individual transformation steps the authors even find the exact wavefunctions. The essential point of the paper is to enlarge the conventional coordinate-momentum phase space by taking time and energy as an additional conjugate pair. In this extended space they exploit the possibility of using transformations which intermix energy and time with position coordinates and momenta. Compared with transformations in the conventional position-momentum phase space, they gain the advantage that scattering states and bound states can be treated in a unified way. Therefore this method is appropriate for systems with mixed spectra. In addition it allows for more flexibility in choosing the individual transformation steps. The practicability of the method is demonstrated by several examples.