Free motion quantum theory of time in a box

Abstract

The concept of time in the quantum domain has been controversial since the inception of quantum mechanics. In the usual single Hilbert space formulation of quantum mechanics time stands as a mere parameter, but time undoubtedly acquires dynamical significance in questions involving the occurrence of an event, for example, when does a nucleon decays or when does a particle arrives at a given spatial point. Furthermore a proper interpretation of the time-energy uncertainty principle requires more than a parametric treatment of time. Each of the above mentioned examples requires a selfadjoint time operator conjugate to the Hamiltonian in keeping with the rules of quantum mechanics. With the Hamiltonian generally possessing a pure point spectrum and is bounded below, this requirement has been tagged impossible to realize as it goes against the commonly held interpretation of the canonical quantization rule (CQR): That the spectrum of canonically conjugate operators span the entire real line. This difficulty has led to so much confusion in dealing with the problems mentioned above. In this paper we demonstrate that we can construct a self-adjoint time operator, which is at the same time canonically conjugate to the Hamiltonian in a dense subspace of the Hilbert space. The object of our investigation is a spinless free particle trapped in a box.