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CRC Press eBooks

Phase Space Consideration in Quantum Dynamical Fluctuation Expansion

Esma Meral, Metin Demіralp

Published May 4, 2022• 0 citations
Publisher

Abstract

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.

DOI
10.1201/9780429077166-97

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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.
The time variant power series expansion for the expectation value of a given quantum dynamical operator is well-known and well-investigated issue in quantum dynamics. However, depending on the operator and Hamiltonian singularities this expansion either may not exist or may not converge for all time instances except the beginning of the evolution. This work focuses on this issue and seeks certain cures for the negativities. We work in the extended space obtained by adding all images of the initial wave function under the system Hamiltonian’s positive integer powers. This requires the introduction of certain appropriately defined weight operators. The resulting better convergence in the temporal power series urges us to call the new defined entities “extended space expectation values” even though they are constructed over certain weight oparetors and are somehow pseudo expectation 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.
This work considers the quantum optimal control of one dimensional harmonic oscillator under linear control agents. The system’s potential energy function contains a perturbation term which is bounded everywhere in the space variable’s domain. We use the most preferred cost functional to construct the necessary equations. Equations are converted to a boundary value problem for a set of ordinary differential equations containing the expectation values of certain operators and the terms corresponding to the transitions between the states described by the wave and costate functions. Resulting equations involve certain undesired expectation values and transition terms. We use a recently developed scheme called fluctuation expansion to approximate these terms at the sharply localized wave and costate function limits. This enables us to construct n infinite number of ordinary differential equations and accompanying boundary conditions whose both halves are given at the beginning and end of the control. These equations are truncated and then the resulting boundary value problem is solved iteratively.
In this study, expectation value dynamics of the quantum harmonic oscillator under the influence of the external dipol effects is discussed in some details. The dipole function of the system has been taken as an odd cubic spatial polynomial and the external field amplitude has been assumed to be known. Instead of the wave function of the system, expectation values of the position and momentum operators are considered as unknowns and certain matrix ordinary differential equations are obtained by using the fluctuation free matrix representation concept. The resulting equations are in the category of Quantum Mechanical Matrix Ordinary Differential Equations (QMMODE) and can be solved by a recently developed scheme called Characteristic Evolutions Method (CEM). We mention also how to implement this approach. The purpose of this work can be considered as a preparation to quantum optimal control problem solutions via expectation value dynamics.