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Approximation methods for the solution of quantum evolution equations at the zero fluctuation limit

Buket Benek, Metin Demіralp

Published May 27, 2008• 0 citations
OpenAlex

Abstract

This paper is focused on the solution of a univariate quantum dynamical system that can be treated through the dynamics of expectation values of the position and momentum operators. This goal is realised by using the fluctuationlessness theorem. The concerned equations are matrix equations and derived using the wave function of the Schrodinger equation. The quality of the obtained approximated solution is tested by comparing it with the real solution for different n values.

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This paper is focused on the solution of a univariate quantum dynamical system that can be treated through the dynamics of expectation values of the position and momentum operators. This goal is realised by using the fluctuationlessness theorem. The concerned equations are matrix equations and derived using the wave function of the Schrodinger equation. The quality of the obtained approximated solution is tested by comparing it with the real solution for different n values.
The variation of the expectation matrix of position and momentum operator in time can serve us to investigate the evolution of a quantum system in time. This brings the utilization of the ODEs instead of Schrödinger's equation at the expense of incapability for the calculation of the wave function. As long as we deal with the observables which can be expressed in terms of position and momentum operators this may be quite practical to know about the quantum dynamics of the system under consideration. Expectation matrix of an operator becomes a function of the expectation matrices of the position and momentum operator when the fluctuations diminish to zero. At this limit, the time‐variant ODEs for the expectation matrices of the position and momentum operator can be handled by using the matrix algebraic tools even in the case of nonlinearities in the potential function. This work presents certain details about these points.
A new method called “Fluctuation Expansion” has been recently proposed for solving quantum dynamical problems . This method is succesfully applied to the dynamics of expectation values such that not the wave function itself but the expectation values of the position and momentum operator are taken into consideration and then the expectation value of certain entities depending on position and momentum operators are expressed via fluctuation expansion. This brings an infinite number of new unknown, fundamental and associate fluctuation terms. The derivatives of expectation values of the position and momentum operators and fluctuation operators are expressed in terms of these terms again. The resulting infinite set of differential equations and accompanying initial conditions are truncated at certain fluctuation order and then tried to be solved. Here we take a one dimensional time independent Hamiltonian and consider the zeroth and first order truncations. Equations obtained in this way are investigated in the sense of phase space which is defined by the expextation values of the position and momentum operator and the fluctuations entering the truncated equations. We find certain subspaces of the phase space such that there, certain expressions remain constant . This helps us to reduce the dimension of the problem.
The goal of this work is, the utilization of a method recently developed by the author, in quantum dynamical problems. Main idea is based on the relation between quantum and classical dynamics such that quantum dynamical problems become their classical dynamical counterparts when they are formulated via expectation values of certain operators and the probability density tends to be sharply localized. Expectation values can be expanded into a series in ascending appearence multiplicity of the complement of the projection operator which projects the space spanned by the wave function. This expansion contains fluctuation terms as unknowns besides the expectation values. We construct an infinite set of differential equations. The finite truncations of this set enables us to approximate the quantum dynamics of the system under consideration.
We study stochastic evolution equations describing the dynamics of open quantum systems. First, using resolvent approximations, we obtain a suf- ficient condition for regularity of solutions to linear stochastic Schrodinger equa- tions driven by cylindrical Brownian motions applying to many physical systems. Then, we establish well-posedness and norm conservation property of a wide class of open quantum systems described in position representation. Moreover, we prove Ehrenfest-type theorems that describe the evolution of the mean value of quantum observables in open systems. Finally, we give a new criterion for the existence and uniqueness of weak solutions to non-linear stochastic Schrequations. We apply our results to physical systems such as fluctuating ion traps and quantum measurement processes of position.
Our point of departure are the unitary dynamics of closed quantum systems as generated from the Schrödinger equation. We focus on a class of quantum models that typically exhibit roughly exponential relaxation of some observable within this framework. Furthermore, we focus on pure state evolutions. An entropy in accord with Jaynes principle is defined on the basis of the quantum expectation value of the above observable. It is demonstrated that the resulting deterministic entropy dynamics are in a sense in accord with a transient fluctuation theorem. Moreover, we demonstrate that the dynamics of the expectation value are describable in terms of an Ornstein-Uhlenbeck process. These findings are demonstrated numerically and supported by analytical considerations based on quantum typicality.
In this paper we are discussing the question how a continuous quantum system can be simulated by mean field fluctuations of a finite number of qubits. On the kinematical side this leads to a convergence result which states that appropriately chosen fluctuation operators converge in a certain weak sense (i.e. we are comparing expectation values) to canonical position and momentum Q, P of one-degree of freedom, continuous quantum system. This result is substantially stronger than existing methods which rely either on central limit theorem arguments (and are therefore restricted to the Gaussian world) or are valid only if the states of the ensembles are close to the "fully polarized" state. Dynamically this relationship keeps perfectly intact (at least for small times) as long as the continuous system evolves according to a quadratic Hamiltonian. In other words we can approximate the corresponding (Heisenberg picture) time evolution of the canonical operators Q, P up to arbitrary accuracy by the appropriately chosen time evolution of fluctuation operators of the finite systems.