College Physics

Derivation of the common eigenvectors of two particles' compatible observables

Shi-Min Xu

Published Jan 1, 2008• 0 citations

Abstract

The common eigenvectors |η〉 of two particles' compatible observables x^1-x^2 and p^1+p^2 are derived in the Fock's space by virtue of the IWOP technique.An effective method for solving this kind of problem is offered.

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College Physics

Derivation of the common eigenvectors of two particles' compatible observables

Shi-Min Xu

Published Jan 1, 2008• 0 citations

OpenAlex

Abstract

Related papers

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.

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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.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.

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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.

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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

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.

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 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.