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

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.

Time Eigenstates for Potential Functions without Extremal Points

“…Any hypersurface that crosses the energy shells can be used as a zero time surface, but the surfaces introduced in [73] and in this paper are easy to use for any potential function with the additional advantage that the same process can be used for quantum systems.…”

“…At the end of this paper, there is a small, incomplete, list of references on this subject . In a previous paper, we have proposed the use of coordinate eigenstates located at the extremal points of the potential function as a zero time eigenstate for the generation of a time coordinate system in classical and in quantum systems [73]. However, that proposal does not work for a potential function without extremal points, as is the case of the linear potential.…”

“…The case of vanishing force was treated in [73]. In that work, there was given a justification for the use of coordinate eigenstates, placed at the zero force places, as zero-time eigenstates for the generation of a time coordinate in classical phase space and in quantum systems.…”