How to Introduce Time Operator.pdf

How to Introduce Time Operator Zhi-Yong Wang*, Cai-Dong Xiong School of Physical Electronics, University of Electronic Science and Technology of China, Chengdu 610054 Abstract Time operator can be introduced by three different approaches: by pertaining it to dynamical variables; by quantizing the classical expression of time; taken as the restriction of energy shift generator to the Hilbert space of a physical system. PACS: 03.65.Ca; 03.65.Ta; 03.65.Xp Keywords: time operator; self-adjointness; energy shift; time-energy uncertainty 1. Introduction Traditionally, time enters quantum mechanics as a parameter rather than a dynamical operator. As a consequence, the investigations on tunneling time, arrival time and traversal time, etc., still remain controversial today [1-19]. On the one hand, one imposes self-adjointness as a requirement for any observable; on the other hand, according to Paulis argument [20-23], there is no self-adjoint time operator canonically conjugating to a Hamiltonian if the Hamiltonian spectrum is bounded from below. A way out of this dilemma set by Paulis objection is based on the use of positive operator valued measures (POVMs) [19, 22-26]: quantum observables are generally positive operator valued measures, e.g., quantum observables are extended to maximally symmetric but not necessarily self-adjoint operators, in such a way one preserves the requirement that time operator be conjugate to the Hamiltonian but abandons the self-adjointness of time operator. In this paper, general time operators are constructed by three different approaches. In the following, the natural units of measurement ( 1c= = ) is applied. * E-mail address: zywang@uestc.edu.cn 1 2. Mandelstam–Tamm version of non-self-adjoint time operator The time-energy uncertainty relation has been a controversial issue and has many versions. However, the Mandelstam–Tamm version [27] of the time-energy uncertainty is the most widely accepted nowad