Physical Review A

Common eigenkets of three-particle compatible observables

Hong-Yi Fan, Ying Zhang

Published May 1, 1998• 26 citations

Abstract

We give common eigenkets of three compatible observables {${P}_{1}{+P}_{2}{+P}_{3},$ $({\ensuremath{\mu}}_{2}{Q}_{2}+{\ensuremath{\mu}}_{3}{Q}_{3})/({\ensuremath{\mu}}_{2}+{\ensuremath{\mu}}_{3})\ensuremath{-}{Q}_{1},$ ${Q}_{3}\ensuremath{-}{Q}_{2}$}, which are composed of three particles' coordinate ${Q}_{i}$ and momentum ${P}_{i},$ where ${\ensuremath{\mu}}_{i}{=m}_{i}{/(m}_{1}{+m}_{2}{+m}_{3}).$ This set of operators are so-called Jacobi coordinates and momenta. By compatible we mean such observables can be simultaneously determined. Using the technique of integration within an ordered product of operators, we prove that the common eigenkets are complete and orthonormal, and hereby qualified for making up a representation. Applying this representation to solving some new three-body problems is also shown.

DOI: 10.1103/physreva.57.3225

Related papers

Common eigenstates of two particles’ center-of-mass coordinates and mass-weighted relative momentum

Fan Hong-Yi, Ye Xiong

Physical Review AApr 1, 1995124 citations

COMMON EIGENVECTORS OF N PARTICLES' COMPATIBLE OBSERVABLES AND ITS SQUEEZING OPERATOR

Shi-Min Xu, Xing-Lei Xu, Hongqi Li

Modern Physics Letters BJul 10, 20087 citations

Eigenvectors of two particles’ relative position and total momentum

Fan Hong‐Yi, John R. Klauder

Physical Review AFeb 1, 1994384 citations

Solving some two-body dynamical problems in 〈ζ|-〈η| representation

Fan Hong‐Yi, Chen Bozhan

Physical Review AMay 1, 199657 citations

Common eigenstates of many-particle compatible observables

Nian-Quan Jiang, Bai-Qi Jin, Yi Zhang +1 more

We introduce common eigenstates of many-particle compatible observables in n -mode Fock space. These states make a complete and orthogonal set and can be realized by the use of beam splitter devices. The entanglement properties of these states are exhibited, their applications in solving many-body problems and in quantum communication are shown.

Europhysics Letters (EPL)Sep 22, 200832 citations

A New Kind of Bipartite Entangled State Representation and its Application

Shimin Xu, Xing-Lei Xu, Hongqi Li +1 more

We construct four linear composite operators for a two-particle system and give common eigenvectors of those operators. The technique of integration within an ordered product of operators is employed to prove that those common eigenvectors are complete and orthonormal. Therefore, a new intermediate coordinate-momentum representation for a two-particle system is proposed and applied to some two-body dynamic problems.

Communications in Theoretical PhysicsFeb 1, 20091 citation

Physical Review A

Common eigenkets of three-particle compatible observables

Hong-Yi Fan, Ying Zhang

Published May 1, 1998• 26 citations

Publisher

Abstract

We give common eigenkets of three compatible observables {

{P}_{1}{+P}_{2}{+P}_{3},

({\ensuremath{\mu}}_{2}{Q}_{2}+{\ensuremath{\mu}}_{3}{Q}_{3})/({\ensuremath{\mu}}_{2}+{\ensuremath{\mu}}_{3})\ensuremath{-}{Q}_{1},

{Q}_{3}\ensuremath{-}{Q}_{2}

}, which are composed of three particles' coordinate

{Q}_{i}

and momentum

{P}_{i},

where

{\ensuremath{\mu}}_{i}{=m}_{i}{/(m}_{1}{+m}_{2}{+m}_{3}).

This set of operators are so-called Jacobi coordinates and momenta. By compatible we mean such observables can be simultaneously determined. Using the technique of integration within an ordered product of operators, we prove that the common eigenkets are complete and orthonormal, and hereby qualified for making up a representation. Applying this representation to solving some new three-body problems is also shown.

DOI: 10.1103/physreva.57.3225

Related papers

Common eigenstates of two particles’ center-of-mass coordinates and mass-weighted relative momentum

Fan Hong-Yi, Ye Xiong

We give the explicit form of the common eigenstate \ensuremath{\Vert}\ensuremath{\xi}〉 of the center-of-mass coordinate ${\mathrm{\ensuremath{\mu}}}_{1}$ ${\mathit{Q}}_{1}$ +${\mathrm{\ensuremath{\mu}}}_{2}$ ${\mathit{Q}}_{2}$ and the mass-weighted relative momentum ${\mathrm{\ensuremath{\mu}}}_{2}$ ${\mathit{P}}_{1}$ -${\mathrm{\ensuremath{\mu}}}_{1}$ ${\mathit{P}}_{2}$ of two particles, which is more complicated than the common eigenstate of the other pair of commutative operators ${\mathit{Q}}_{2}$ - ${\mathit{Q}}_{1}$ and ${\mathit{P}}_{1}$ + ${\mathit{P}}_{2}$ . The orthonormal and completeness relation of \ensuremath{\Vert}\ensuremath{\xi}〉 are investigated easily by virtue of the technique of integration within an ordered product of operators. The normally ordered squeezing operator for ${\mathrm{\ensuremath{\mu}}}_{1}$ ${\mathit{Q}}_{1}$ +${\mathrm{\ensuremath{\mu}}}_{2}$ ${\mathit{Q}}_{2}$ and ${\mathrm{\ensuremath{\mu}}}_{2}$ ${\mathit{P}}_{1}$ -${\mathrm{\ensuremath{\mu}}}_{1}$ ${\mathit{P}}_{2}$ is also derived by using the \ensuremath{\Vert}\ensuremath{\xi}〉 representation.We give the explicit form of the common eigenstate \ensuremath{\Vert}\ensuremath{\xi}〉 of the center-of-mass coordinate ${\mathrm{\ensuremath{\mu}}}_{1}$${\mathit{Q}}_{1}$+${\mathrm{\ensuremath{\mu}}}_{2}$${\mathit{Q}}_{2}$ and the mass-weighted relative momentum ${\mathrm{\ensuremath{\mu}}}_{2}$${\mathit{P}}_{1}$-${\mathrm{\ensuremath{\mu}}}_{1}$${\mathit{P}}_{2}$ of two particles, which is more complicated than the common eigenstate of the other pair of commutative operators ${\mathit{Q}}_{2}$-${\mathit{Q}}_{1}$ and ${\mathit{P}}_{1}$+${\mathit{P}}_{2}$. The orthonormal and completeness relation of \ensuremath{\Vert}\ensuremath{\xi}〉 are investigated easily by virtue of the technique of integration within an ordered product of operators. The normally ordered squeezing operator for ${\mathrm{\ensuremath{\mu}}}_{1}$${\mathit{Q}}_{1}$+${\mathrm{\ensuremath{\mu}}}_{2}$${\mathit{Q}}_{2}$ and ${\mathrm{\ensuremath{\mu}}}_{2}$${\mathit{P}}_{1}$-${\mathrm{\ensuremath{\mu}}}_{1}$${\mathit{P}}_{2}$ is also derived by using the \ensuremath{\Vert}\ensuremath{\xi}〉 representation.

Physical Review AApr 1, 1995124 citations

COMMON EIGENVECTORS OF N PARTICLES' COMPATIBLE OBSERVABLES AND ITS SQUEEZING OPERATOR

Shi-Min Xu, Xing-Lei Xu, Hongqi Li

We construct 2N operators for a N-particle system, namely one center-of-mass coordinate operator, N-1 relative coordinate operators, one total momentum operator and N-1 mass-weighted relative momentum operators, and give common eigenvectors of N compatible observables [Formula: see text], which are composed of N particles' coordinate [Formula: see text] and momentum [Formula: see text]. By compatible, we mean such observables can be simultaneously determined. Using the technique of integration within an ordered product of operators (IWOP), we prove that the common eigenvectors are complete and orthonormal, and hereby qualified for making up a representation. This new representation can be applied to solving some dynamic problems in quantum mechanics.We construct 2N operators for a N-particle system, namely one center-of-mass coordinate operator, N-1 relative coordinate operators, one total momentum operator and N-1 mass-weighted relative momentum operators, and give common eigenvectors of N compatible observables [Formula: see text], which are composed of N particles' coordinate [Formula: see text] and momentum [Formula: see text]. By compatible, we mean such observables can be simultaneously determined. Using the technique of integration within an ordered product of operators (IWOP), we prove that the common eigenvectors are complete and orthonormal, and hereby qualified for making up a representation. This new representation can be applied to solving some dynamic problems in quantum mechanics.

Modern Physics Letters BJul 10, 20087 citations

Eigenvectors of two particles’ relative position and total momentum

Fan Hong‐Yi, John R. Klauder

We give the explicit form of the common eigenvectors of the relative position ${\mathit{Q}}_{1}$ - ${\mathit{Q}}_{2}$ and the total momentum ${\mathit{P}}_{1}$ + ${\mathit{P}}_{2}$ , of two particles which were considered by Einstein, Podolsky, and Rosen [Phys. Rev. 47, 777 (1935)] in their argument that the quantum-mechanical state vector is not complete. Orthonormality and completeness of such eigenvectors, as well as their use in constructing the unitary operator for simultaneously squeezing ${\mathit{Q}}_{1}$ - ${\mathit{Q}}_{2}$ and ${\mathit{P}}_{1}$ + ${\mathit{P}}_{2}$ , are derived by using the technique of integration within an ordered product of operators.We give the explicit form of the common eigenvectors of the relative position ${\mathit{Q}}_{1}$-${\mathit{Q}}_{2}$ and the total momentum ${\mathit{P}}_{1}$+${\mathit{P}}_{2}$, of two particles which were considered by Einstein, Podolsky, and Rosen [Phys. Rev. 47, 777 (1935)] in their argument that the quantum-mechanical state vector is not complete. Orthonormality and completeness of such eigenvectors, as well as their use in constructing the unitary operator for simultaneously squeezing ${\mathit{Q}}_{1}$-${\mathit{Q}}_{2}$ and ${\mathit{P}}_{1}$+${\mathit{P}}_{2}$, are derived by using the technique of integration within an ordered product of operators.

Physical Review AFeb 1, 1994384 citations

Solving some two-body dynamical problems in 〈ζ|-〈η| representation

Fan Hong‐Yi, Chen Bozhan

We use the common eigenvectors \ensuremath{\Vert}\ensuremath{\zeta}〉 of two particles' center-of-mass position ${\mathit{X}}_{\mathit{c}}$ =${\mathrm{\ensuremath{\mu}}}_{1}$ ${\mathit{X}}_{1}$ +${\mathrm{\ensuremath{\mu}}}_{2}$ ${\mathit{X}}_{2}$ and mass-weighted relative momentum ${\mathit{P}}_{\mathit{r}}$ =${\mathrm{\ensuremath{\mu}}}_{2}$ ${\mathit{P}}_{1}$ -${\mathrm{\ensuremath{\mu}}}_{1}$ ${\mathit{P}}_{2}$ , as well as the eigenvectors \ensuremath{\Vert}\ensuremath{\eta}〉 of the other pair commutative operators X= ${\mathit{X}}_{1}$ - ${\mathit{X}}_{2}$ and P= ${\mathit{P}}_{1}$ + ${\mathit{P}}_{2}$ to solve the dynamics of two coupled oscillators. The coupling involves both interbond potential and kinetic coupling. This approach seems to be simple, convenient, and natural. \textcopyright{} 1996 The American Physical Society.We use the common eigenvectors \ensuremath{\Vert}\ensuremath{\zeta}〉 of two particles' center-of-mass position ${\mathit{X}}_{\mathit{c}}$=${\mathrm{\ensuremath{\mu}}}_{1}$${\mathit{X}}_{1}$+${\mathrm{\ensuremath{\mu}}}_{2}$${\mathit{X}}_{2}$ and mass-weighted relative momentum ${\mathit{P}}_{\mathit{r}}$=${\mathrm{\ensuremath{\mu}}}_{2}$${\mathit{P}}_{1}$-${\mathrm{\ensuremath{\mu}}}_{1}$${\mathit{P}}_{2}$, as well as the eigenvectors \ensuremath{\Vert}\ensuremath{\eta}〉 of the other pair commutative operators X=${\mathit{X}}_{1}$-${\mathit{X}}_{2}$ and P=${\mathit{P}}_{1}$+${\mathit{P}}_{2}$ to solve the dynamics of two coupled oscillators. The coupling involves both interbond potential and kinetic coupling. This approach seems to be simple, convenient, and natural. \textcopyright{} 1996 The American Physical Society.

Physical Review AMay 1, 199657 citations

Common eigenstates of many-particle compatible observables

Nian-Quan Jiang, Bai-Qi Jin, Yi Zhang +1 more

Europhysics Letters (EPL)Sep 22, 200832 citations

A New Kind of Bipartite Entangled State Representation and its Application

Shimin Xu, Xing-Lei Xu, Hongqi Li +1 more

Communications in Theoretical PhysicsFeb 1, 20091 citation

We give common eigenkets of three compatible observables {

{P}_{1}{+P}_{2}{+P}_{3},

({\ensuremath{\mu}}_{2}{Q}_{2}+{\ensuremath{\mu}}_{3}{Q}_{3})/({\ensuremath{\mu}}_{2}+{\ensuremath{\mu}}_{3})\ensuremath{-}{Q}_{1},

{Q}_{3}\ensuremath{-}{Q}_{2}

}, which are composed of three particles' coordinate

{Q}_{i}

and momentum

{P}_{i},

where

{\ensuremath{\mu}}_{i}{=m}_{i}{/(m}_{1}{+m}_{2}{+m}_{3}).

We give the explicit form of the common eigenstate \ensuremath{\Vert}\ensuremath{\xi}〉 of the center-of-mass coordinate ${\mathrm{\ensuremath{\mu}}}_{1}$

{\mathit{Q}}_{1}

+${\mathrm{\ensuremath{\mu}}}_{2}$

{\mathit{Q}}_{2}

and the mass-weighted relative momentum ${\mathrm{\ensuremath{\mu}}}_{2}$

{\mathit{P}}_{1}

-${\mathrm{\ensuremath{\mu}}}_{1}$

{\mathit{P}}_{2}

of two particles, which is more complicated than the common eigenstate of the other pair of commutative operators

{\mathit{Q}}_{2}

{\mathit{Q}}_{1}

and

{\mathit{P}}_{1}

{\mathit{P}}_{2}

. The orthonormal and completeness relation of \ensuremath{\Vert}\ensuremath{\xi}〉 are investigated easily by virtue of the technique of integration within an ordered product of operators. The normally ordered squeezing operator for ${\mathrm{\ensuremath{\mu}}}_{1}$

{\mathit{Q}}_{1}

+${\mathrm{\ensuremath{\mu}}}_{2}$

{\mathit{Q}}_{2}

and ${\mathrm{\ensuremath{\mu}}}_{2}$

{\mathit{P}}_{1}

-${\mathrm{\ensuremath{\mu}}}_{1}$

{\mathit{P}}_{2}

is also derived by using the \ensuremath{\Vert}\ensuremath{\xi}〉 representation.

We give the explicit form of the common eigenvectors of the relative position

{\mathit{Q}}_{1}

{\mathit{Q}}_{2}

and the total momentum

{\mathit{P}}_{1}

{\mathit{P}}_{2}

, of two particles which were considered by Einstein, Podolsky, and Rosen [Phys. Rev. 47, 777 (1935)] in their argument that the quantum-mechanical state vector is not complete. Orthonormality and completeness of such eigenvectors, as well as their use in constructing the unitary operator for simultaneously squeezing

{\mathit{Q}}_{1}

{\mathit{Q}}_{2}

and

{\mathit{P}}_{1}

{\mathit{P}}_{2}

, are derived by using the technique of integration within an ordered product of operators.

We use the common eigenvectors \ensuremath{\Vert}\ensuremath{\zeta}〉 of two particles' center-of-mass position

{\mathit{X}}_{\mathit{c}}

=${\mathrm{\ensuremath{\mu}}}_{1}$

{\mathit{X}}_{1}

+${\mathrm{\ensuremath{\mu}}}_{2}$

{\mathit{X}}_{2}

and mass-weighted relative momentum

{\mathit{P}}_{\mathit{r}}

=${\mathrm{\ensuremath{\mu}}}_{2}$

{\mathit{P}}_{1}

-${\mathrm{\ensuremath{\mu}}}_{1}$

{\mathit{P}}_{2}

, as well as the eigenvectors \ensuremath{\Vert}\ensuremath{\eta}〉 of the other pair commutative operators X=

{\mathit{X}}_{1}

{\mathit{X}}_{2}

and P=

{\mathit{P}}_{1}

{\mathit{P}}_{2}

to solve the dynamics of two coupled oscillators. The coupling involves both interbond potential and kinetic coupling. This approach seems to be simple, convenient, and natural. \textcopyright{} 1996 The American Physical Society.