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

Sutee Boonchui, Wichit Sritrakool, Virulh Sa-Yakanit

Published Jan 1, 2005• 0 citations
Publisher

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.

DOI
10.58837/chula.the.2005.1039

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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.
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.
An alternative method is proposed for deriving the time dependent Schroedinger equation from the pictures of wave and matrix mechanics. The derivation is of a mixed classical quantum character, since time is treated as a classical variable, thus avoiding any controversy over its meaning in quantum mechanics. The derivation method proposed in this paper requires no ad hoc assumption and avoids going through a second-order differential equation that can be reduced to the well known time-dependent Schroedinger equation only postulating a complex wavefunction with an exponential time dependence, as did by Schroedinger in its original paper of 1926.
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.
Nowadays the task of molecular modeling plays the important role in many branches of science and engineering. But the existing models do not take quantum effects into account because of known calculating issues. The main difficulty in modeling of the evolution of multi particle quantum states by solving the Schroedinger equation consists iii exponential growth of required time and memory with the corresponding increase of entangled particles. The approach to this problem considered in works [3 4] consists in applying Feynman path integrals instead of solving Schroedinger equation. In terms of the amplitude quanta method based on this approach the wave function of a particle is calculated as a sum of the large amount of a.q. that probably will allow the linear growth of required memory with the corresponding increase of entangled particles. This approach also allows to the easy transition from the classical description of a particle to the quantum and back. The given work is devoted to the development of the quantum state evolution model based on the a. q. method in one and two-dimensional cases.
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.