Derivation of the clocked schroedinger equation by using directly the schroedinger equation and applications

Abstract

The clocked Schroedinger equation was proposed by Sokolovski using the Feynman path integrals with constraint (Phys. Rev. A 52, R2, 1995). Sokolovski pointed out that the clocked Schroedinger equation (clocked SE) cannot be derived directly from the Schroedinger equation (SE). In this thesis, we show that the clocked Schroedinger equation can be derived by starting from the ordinary Schroedinger equation. We derive the clocked SE by two methods: (i) substitution of the wave function into the ordinary SE and (ii) reduction of a composite system, composed of the observed system and the apparatus system or the pointer, as defined by von Neumann. For applications, we provide the ket state in the Schroedinger picture from the entangled state and the differential equation for solving the probability amplitude of a quantum system as a function of time from the timeindependent SE. We use the Jaynes-Cummings model for an example.