Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

AIP conference proceedings

An algebraic function operator expectation value based eigenstate determinations for quantum systems with one degree of freedom

Berfin Kalay, Metin Demіralp

Published Jan 1, 2015• 1 citation
Publisher

Abstract

This proceedings paper aims to show the efficiency of an expectation value identity for a given algebraic function operator which is assumed to be depending pn only position operator. We show that this expectation value formula becomes enabled to determine the eigenstates of the quantum system Hamiltonian as long as it is autonomous and an appropriate basis set in position operator is used. This approach produces a denumerable infinite recursion which may be considered as revisited but at the same time generalized form of the recursions over the natural number powers of the position operator. The content of this short paper is devoted not only to the formulation of the new method but also to show that this novel approach is capable of catching the eigenvalues and eigenfunctions for Hydrogen-like systems, beyond that, it can give a hand to us to reveal the wavefunction structure. So it has also somehow a confirmative nature.

DOI
10.1063/1.4938943

Related papers

Algebraic function operator expectation value based quantum eigenstate determination: A case of twisted or bent Hamiltonian, or, a spatially univariate quantum system on a curved space

N. Abdülbaki Baykara

Recent studies on quantum evolutionary problems in Demiralp’s group have arrived at a stage where the construction of an expectation value formula for a given algebraic function operator depending on only position operator becomes possible. It has also been shown that this formula turns into an algebraic recursion amongst some finite number of consecutive elements in a set of expectation values of an appropriately chosen basis set over the natural number powers of the position operator as long as the function under consideration and the system Hamiltonian are both autonomous. This recursion corresponds to a denumerable infinite number of algebraic equations whose solutions can or can not be obtained analytically. This idea is not completely original. There are many recursive relations amongst the expectation values of the natural number powers of position operator. However, those recursions may not be always efficient to get the system energy values and especially the eigenstate wavefunctions. The present approach is somehow improved and generalized form of those expansions. We focus on this issue for a specific system where the Hamiltonian is defined on the coordinate of a curved space instead of the Cartesian one.

AIP conference proceedings•Jan 1, 2015•0 citations

Extended space expectation values of position related operators for hydrogen-like quantum system evolutions

Berfin Kalay, Metin Demіralp

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.

AIP conference proceedings•Jan 1, 2014•0 citations

Extended space expectation values in quantum dynamical system evolutions

Metin Demіralp

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.

AIP conference proceedings•Jan 1, 2014•2 citations

Phase Space Consideration in Quantum Dynamical Fluctuation Expansion

Esma Meral, Metin Demіralp

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.

CRC Press eBooks•May 4, 2022•0 citations

Real eigenvalues are determined by the recursion of eigenstates

Tong Liu, Youguo Wang

Quantum physics is generally concerned with real eigenvalues due to the unitarity of time evolution. With the introduction of $\mathcal{PT}$ symmetry, a widely accepted consensus is that, even if the Hamiltonian of the system is not Hermitian, the eigenvalues can still be pure real under specific symmetry. Hence, great enthusiasm has been devoted to exploring the eigenvalue problem of non-Hermitian systems. In this work, from a distinct perspective, we demonstrate that real eigenvalues can also emerge under the appropriate recursive condition of eigenstates. Consequently, our findings provide another path to extract the real energy spectrum of non-Hermitian systems, which guarantees the conservation of probability and stimulates future experimental observations.

arXiv:2309.09418•Sep 18, 2023•0 citations

Quantum geometry of expectation values

Chaoming Song

We propose a framework for the quantum geometry of expectation values over arbitrary sets of operators and establish a link between this geometry and the eigenstates of Hamiltonian families generated by these operators. We show that the boundary of expectation value space corresponds to the ground state, which presents a natural bound that generalizes Heisenberg's uncertainty principle. To demonstrate the versatility of our framework, we present several practical applications, including providing a stronger nonlinear quantum bound that violates the Bell inequality and an explicit construction of the density functional. Our approach provides an alternative time-independent quantum formulation that transforms the linear problem in a high-dimensional Hilbert space into a nonlinear algebro-geometric problem in a low dimension, enabling us to gain new insights into quantum systems.

arXiv:2301.05921•Physical Review A•Jun 9, 2023•6 citations
This proceedings paper aims to show the efficiency of an expectation value identity for a given algebraic function operator which is assumed to be depending pn only position operator. We show that this expectation value formula becomes enabled to determine the eigenstates of the quantum system Hamiltonian as long as it is autonomous and an appropriate basis set in position operator is used. This approach produces a denumerable infinite recursion which may be considered as revisited but at the same time generalized form of the recursions over the natural number powers of the position operator. The content of this short paper is devoted not only to the formulation of the new method but also to show that this novel approach is capable of catching the eigenvalues and eigenfunctions for Hydrogen-like systems, beyond that, it can give a hand to us to reveal the wavefunction structure. So it has also somehow a confirmative nature.
Recent studies on quantum evolutionary problems in Demiralp’s group have arrived at a stage where the construction of an expectation value formula for a given algebraic function operator depending on only position operator becomes possible. It has also been shown that this formula turns into an algebraic recursion amongst some finite number of consecutive elements in a set of expectation values of an appropriately chosen basis set over the natural number powers of the position operator as long as the function under consideration and the system Hamiltonian are both autonomous. This recursion corresponds to a denumerable infinite number of algebraic equations whose solutions can or can not be obtained analytically. This idea is not completely original. There are many recursive relations amongst the expectation values of the natural number powers of position operator. However, those recursions may not be always efficient to get the system energy values and especially the eigenstate wavefunctions. The present approach is somehow improved and generalized form of those expansions. We focus on this issue for a specific system where the Hamiltonian is defined on the coordinate of a curved space instead of the Cartesian one.
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.
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.
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.
Quantum physics is generally concerned with real eigenvalues due to the unitarity of time evolution. With the introduction of PT\mathcal{PT}PT symmetry, a widely accepted consensus is that, even if the Hamiltonian of the system is not Hermitian, the eigenvalues can still be pure real under specific symmetry. Hence, great enthusiasm has been devoted to exploring the eigenvalue problem of non-Hermitian systems. In this work, from a distinct perspective, we demonstrate that real eigenvalues can also emerge under the appropriate recursive condition of eigenstates. Consequently, our findings provide another path to extract the real energy spectrum of non-Hermitian systems, which guarantees the conservation of probability and stimulates future experimental observations.
We propose a framework for the quantum geometry of expectation values over arbitrary sets of operators and establish a link between this geometry and the eigenstates of Hamiltonian families generated by these operators. We show that the boundary of expectation value space corresponds to the ground state, which presents a natural bound that generalizes Heisenberg's uncertainty principle. To demonstrate the versatility of our framework, we present several practical applications, including providing a stronger nonlinear quantum bound that violates the Bell inequality and an explicit construction of the density functional. Our approach provides an alternative time-independent quantum formulation that transforms the linear problem in a high-dimensional Hilbert space into a nonlinear algebro-geometric problem in a low dimension, enabling us to gain new insights into quantum systems.