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Physical Review A

Clocked Schrödinger equation in the meaning of the measurement system

Sutee Boonchui, Virulh Sa‐Yakanit, Wichit Sritrakool

Published Jan 18, 2006• 2 citations
Publisher

Abstract

The clocked Schr\"odinger equation was proposed by Sokolovski using the Feynman path integral with constraint (Phys. Rev. A 52, R2, 1995). Sokolovski pointed out that the clocked Schr\"odinger equation cannot be derived directly from the Schr\"odinger equation. In this paper, we show that the clocked Schr\"odinger equation can be derived by starting from the normal Schr\"odinger equation for a composite system, composed of the observed system and the measuring device, as defined by von Neumann. Details of the derivation and the physical meaning of the clocked Schr\"odinger equation are given.

DOI
10.1103/physreva.73.012108

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The clocked Schr"odinger equation was proposed by Sokolovski using the Feynman path integral with constraint (Phys. Rev. A 52, R2, 1995). Sokolovski pointed out that the clocked Schr"odinger equation cannot be derived directly from the Schr"odinger equation. In this paper, we show that the clocked Schr"odinger equation can be derived by starting from the normal Schr"odinger equation for a composite system, composed of the observed system and the measuring device, as defined by von Neumann. Details of the derivation and the physical meaning of the clocked Schr"odinger equation are given.
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.
A derivation of the time-dependent Schrödinger equation from the time-independent one is considered. Instead of time, the coordinate of an additional degree of freedom, the clock, is introduced into the original time-independent Schrödinger equation. It is shown that the standard time-dependent Schrödinger equation can be obtained for the semiclassical clock only. For elucidation of the physical meaning of the equation obtained in this way, various types of clocks are discussed. In addition, the corresponding equation for the density matrix and formulas for the mean values of operators are derived.
A derivation of the time-dependent Schrödinger equation from the time-independent one is considered. Instead of time, the coordinate of an additional degree of freedom, the clock, is introduced into the original time-independent Schrödinger equation. It is shown that the standard time-dependent Schrödinger equation can be obtained for the semiclassical clock only. For elucidation of the physical meaning of the equation obtained in this way, various types of clocks are discussed. In addition, the corresponding equation for the density matrix and formulas for the mean values of operators are derived.
Abstract The equation of motion in the standard formulation of non-relativistic quantum mechanics, the Schrödinger equation, is based on the Hamiltonian. In contrast, in Feynman’s path-integral formulation of quantum mechanics, the equation of motion is the propagation equation, which is based on the Lagrangian. That these two different equations of motion are equivalent was shown by Feynman, who provided a derivation of the Schrödinger equation from the propagation equation. Surprisingly, however, while in classical mechanics there exists a simple relationship between the Hamiltonian and the Lagrangian, there is nothing in Feynman’s derivation that gives any indication of this relationship. Here I show that the equations of motion in the Hamiltonian and Lagrangian formulations of quantum mechanics are, in fact, simply related to each other.
It is shown how the time-dependent Schrödinger equation may be simply derived from the dynamical postulate of Feynman's path integral formulation of quantum mechanics and the Hamilton–Jacobi equation of classical mechanics. Schrödinger's own published derivations of quantum wave equations, the first of which was also based on the Hamilton–Jacobi equation, are also reviewed. The derivation of the time-dependent equation is based on an a priori assumption equivalent to Feynman's dynamical postulate. de Broglie's concepts of ‘matter waves’ and their phase and group velocities are also critically discussed.