Physical Review A

Eigenvectors of two particles’ relative position and total momentum

Fan Hong‐Yi, John R. Klauder

Published Feb 1, 1994• 384 citations

Abstract

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.

DOI: 10.1103/physreva.49.704

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

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Physical Review A

Eigenvectors of two particles’ relative position and total momentum

Fan Hong‐Yi, John R. Klauder

Published Feb 1, 1994• 384 citations

Publisher

Abstract

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.

DOI: 10.1103/physreva.49.704

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.

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Modern Physics Letters BJul 10, 20087 citations

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Hong-Yi Fan, Ying Zhang

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We demonstrate that a common eigenvector |η> (Einstein, Podolsky and Rosen, EPR states) of two particles' relatively coordinate and total momentum possesses a property that the projection (a Bell measurement) of two pairs of EPR states |η>1,2⊗| η'>3,4 onto the subspace spanned by 1,4<η″|, yields a EPR state |η″-η*-η′* >2,3 for particles 2 and 3. Entanglement swapping operator is also derived by virtue of the technique of integration within an ordered product of operators.We demonstrate that a common eigenvector |η> (Einstein, Podolsky and Rosen, EPR states) of two particles' relatively coordinate and total momentum possesses a property that the projection (a Bell measurement) of two pairs of EPR states |η>1,2⊗| η'>3,4 onto the subspace spanned by 1,4<η″|, yields a EPR state |η″-η*-η′* >2,3 for particles 2 and 3. Entanglement swapping operator is also derived by virtue of the technique of integration within an ordered product of operators.

Modern Physics Letters BFeb 10, 20010 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

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}

{\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 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 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.