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

Physica Scripta

Generalized Fredholm operator equation and its solution based on intermediate coordinate-momentum representation

Hong-Yi Fan, Qin Guo

Published Oct 16, 2007• 6 citations
Publisher

Abstract

Based on the intermediate coordinate-momentum representations, we construct the appropriate generalized Fredholm operator equation and then find its solution. The application of this equation for deriving Hermite-polynomial operator formulae is demonstrated. The connection of the generalized Wigner operator to the form of the Gaussian bivariate distribution of random variables in statistics is shown, which is useful for studying the tomogram of quantum states.

DOI
10.1088/0031-8949/76/5/022

Related papers

Operator Fredholm Integration Equation in Intermediate Coordinate-Momentum Representation and Its Analogue to Thermo Conduction Equation

Fan Hong-Yi, Xubing Tang

Using the technique of integration within an ordered product (IWOP) of operators we construct intermediate coordinate-momentum representation, with which we build a type of operator Fredholm integration equation that is an operator generalization of the solution of thermo conduction equation. Then we seach for the solution of operator Fredholm integration equations, which provides us with a new approach for deriving some operator identities

Communications in Theoretical Physics•May 1, 2008•0 citations

Wigner function and tomogram of the Hermite polynomial state

Meng Xiang-Guo, Wang Ji-Suo, Li Yan-Ling

Using the coherent state representation of Wigner operator and the technique of integration within an ordered product (IWOP) of operators, this paper derives the Wigner function for the Hermite polynomial state (HPS). The tomogram of the HPS is calculated with the aid of newly introduced intermediate coordinate-momentum representation in quantum optics.

Chinese Physics•Aug 1, 2007•8 citations

QUANTUM PHASE SPACE THEORY BASED ON INTERMEDIATE COORDINATE-MOMENTUM REPRESENTATION

Hong-Yi Fan, Qin Guo

Through the Radon transformation of the normally ordered Wigner operator we introduce two mutually conjugate intermediate coordinate-momentum representations. Based on them we construct the appropriate quantum phase space theory which includes the new Wigner operator adapting to this space. We find that the new Wigner function's marginal distribution for a state |Ψ〉 is just the quantum tomogram of the state |Ψ〉.

Modern Physics Letters B•Nov 20, 2007•2 citations

Solution to Operator Fredholm Equation in Bipartite Entangled State Representation and Its Applications

Fan Hong‐Yi, Ji‐Suo Wang

Based on the bipartite entangled state representation and using the technique of integration within an ordered product (IWOP) of operators we construct the corresponding operator Fredholm equations and then derive their solutions. As its application we deduce some new bosonic operator identities and new relations about the two-variable Hermite polynomials.

Communications in Theoretical Physics•Apr 1, 2007•4 citations

Coordinate-Momentum Intermediate Representation and Marginal Distributions of Quantum Mechanical Bivariate Normal Distribution

Fan Hong-Yi, Lou Sen‐Yue

We introduce bivariate normal distribution operator for state vector |ψ〉 and find that its marginal distribution leads to one-dimensional normal distribution corresponding to the measurement probability | λ,ν 〈x|·ψ〉| 2 , where |x〉 λ,ν is the coordinate-momentum intermediate representation. As a by-product, the one-dimensional normal distribution in statistics can be explained as a Radon transform of two-dimensional Gaussian function.

Communications in Theoretical Physics•Mar 1, 2008•1 citation

New relationship between quantum state’s tomogram and its wave function

Ji‐Suo Wang, Xiang‐Gao Meng, Hong-Yi Fan

By searching for Q- and P-ordering form of the coordinate-momentum intermediate representation |x⟩λ,νλ,ν⟨x| (formed by the eigenvector of λQ+νP with eigenvalue being x, Q,P=iħ), and noting that |x⟩λ,νλ,ν⟨x|=δ(x-λQ-νP) is just the Radon transformation of the Wigner operator, we find new relationship between a quantum state ψ’s tomogram and its wave function in coordinate (or momentum) representation, i.e. the tomogram can be split into two parts, one is ψq’s Gaussian integration transformation with the parameter ν and another is ψ∗p’s Gaussian integration transformation with the parameter λ. In this way, the quantum tomogram of number state is conveniently deduced. We also derive Radon transform of δ(p-P)δ(q-Q) and δ(q-Q)δ(p-P), which can be either viewed as P- and Q-ordered operator correspondence of the classical function δ(x-λq-νp).

Journal of Modern Optics•Feb 13, 2017•9 citations
Based on the intermediate coordinate-momentum representations, we construct the appropriate generalized Fredholm operator equation and then find its solution. The application of this equation for deriving Hermite-polynomial operator formulae is demonstrated. The connection of the generalized Wigner operator to the form of the Gaussian bivariate distribution of random variables in statistics is shown, which is useful for studying the tomogram of quantum states.
By searching for Q- and P-ordering form of the coordinate-momentum intermediate representation |x⟩λ,νλ,ν⟨x| (formed by the eigenvector of λQ+νP with eigenvalue being x, Q,P=iħ), and noting that |x⟩λ,νλ,ν⟨x|=δ(x-λQ-νP) is just the Radon transformation of the Wigner operator, we find new relationship between a quantum state ψ’s tomogram and its wave function in coordinate (or momentum) representation, i.e. the tomogram can be split into two parts, one is ψq’s Gaussian integration transformation with the parameter ν and another is ψ∗p’s Gaussian integration transformation with the parameter λ. In this way, the quantum tomogram of number state is conveniently deduced. We also derive Radon transform of δ(p-P)δ(q-Q) and δ(q-Q)δ(p-P), which can be either viewed as P- and Q-ordered operator correspondence of the classical function δ(x-λq-νp).