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AIP conference proceedings

Extended space expectation values in quantum dynamical system evolutions

Metin Demіralp

Published Jan 1, 2014• 2 citations
Publisher

Abstract

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.

DOI
10.1063/1.4897873

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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.
We have recently shown that the operator expectation values for the quantum dynamical systems of singular Hamiltonians, cannot be expanded into temporal Maclaurin series. Depending on the nature of the system under consideration and the relevant operator, either a finite number of terms can exist while the others become infinite or the series can convergence only at the beginning of the evolution even if the coefficients of the series all exist. This however happens when the initial wave packet has some features leading us to nonintegrabilities in certain expectation values. The reason why this happens underlies beneath the fact that the certain images of the initial wave packet under certain natural number powers of the Hamiltonian become nonintegrable. This negativity can be bypassed by using certain weight functions in the expectation value definition and corresponds to working on an extended space constructed from the Hilbert space in which the initial and any time wave function lay. We have used a two parameter weight function in our most recent publication. Here, in this work, we extend the weight function to multiparameter structures where the number of parameters is more than two and may climb even up to infinity.
The expectation value definitions over an extended space from the considered Hilbert space of the system under consideration is given in another paper of the second author in this symposium. There, in that paper, the conceptuality rather than specification is emphasized on. This work uses that conceptuality to investigate the time evolutions of the position related operators' expectation values not in its standard meaning but rather in a new version of the definition over not the original Hilbert space but in the space obtained by extensions via introducing the images of the given initial wave packet under the positive integer powers of the system Hamiltonian. These images may not be residing in the same space of the initial wave packet when certain singularities appear in the structure of the system Hamiltonian. This may break down the existence of the integrals in the definitions of the expectation values. The cure is the use of basis functions in the abovementioned extended space and the sandwiching of the target operator whose expectation value is under questioning by an appropriately chosen operator guaranteeing the existence of the relevant integrals. Work specifically focuses on the hydrogen-like quantum systems whose Hamiltonians contain a polar singularity at the origin.
This work is a new extension to our a very recent work whose paper will appear in the proceedings of a very recent international conference. What we have done in the previous work is the use of a weight operator to suppress the singularities causing nonexistence of some of temporal Maclaurin expansion coefficients. The weight operator has been constructed in such a way that certain number of expectation values of position operator’s first positive integer powers with and without the chosen weight operator match. Therein this match has not been considered for the momentum operator’s corresponding power expectation values and a finite linear combination of the spatial variable’s first reciprocal powers has been used in the construction of the weight operator. Here, that approach is extended to the case where matches for both position and momentum operators are considered and the weight operator involves finite linear combinations of the spatial variable’s both positive integer powers and their reciprocals.
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 usual definition of the time evolution operator e−iHt/ℏ=∑n=0∞1n!(−iℏHt)n, where H is the Hamiltonian of the system, as given in almost every book on quantum mechanics, causes problems in some situations. The operators that appear in quantum mechanics are either bounded or unbounded. Unbounded operators are not defined for all the vectors (wave functions) of the Hilbert space of the system; when applied to some states, they give a non-normalizable state. Therefore, if H is an unbounded operator, the definition in terms of the power series expansion does not make sense because it may diverge or result in a non-normalizable wave function. In this article, we explain why this is so and suggest, as an alternative, another definition used by mathematicians.