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Modern Physics Letters B

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

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

Published Jul 10, 2008• 7 citations
Publisher

Abstract

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.

DOI
10.1142/s0217984908016327

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We give common eigenkets of three compatible observables {P1+P2+P3,{P}_{1}{+P}_{2}{+P}_{3},P1​+P2​+P3​, $({\ensuremath{\mu}}_{2}{Q}_{2}+{\ensuremath{\mu}}_{3}{Q}_{3})/({\ensuremath{\mu}}_{2}+{\ensuremath{\mu}}_{3})\ensuremath{-}{Q}_{1},$ ${Q}_{3}\ensuremath{-}{Q}_{2}$}, which are composed of three particles' coordinate Qi{Q}_{i}Qi​ and momentum Pi,{P}_{i},Pi​, 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.
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We give the explicit form of the common eigenvectors of the relative position Q1{\mathit{Q}}_{1}Q1​-Q2{\mathit{Q}}_{2}Q2​ and the total momentum P1{\mathit{P}}_{1}P1​+P2{\mathit{P}}_{2}P2​, 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 Q1{\mathit{Q}}_{1}Q1​-Q2{\mathit{Q}}_{2}Q2​ and P1{\mathit{P}}_{1}P1​+P2{\mathit{P}}_{2}P2​, are derived by using the technique of integration within an ordered product of operators.
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 (IWOP) of operators is employed to prove that those common eigenvectors are complete and orthonormal. Therefore, a new two-mode intermediate momentum-coordinate representation which involves quantum entanglement for a two-particle system is proposed and applied to some two-body dynamic problems. Moreover, the pure-state density matrix |ξ 1 , ξ 2 〉 C,D C,D 〈ξ 1 , ξ 2 | is a Radon transform of Wigner operator.