Physical Review A

Generalized center-of-mass coordinate and relative momentum operators studied through unitary transformations

Shen-xi Yu

Published Aug 1, 1996• 3 citations

Abstract

A method for studying the generalized center-of-mass coordinate and relative momentum operators is provided. This method is based on unitary transformations. The eigenvectors of these operators can be easily obtained. Their properties and relationship with one-body system eigenvectors are also discussed. \textcopyright{} 1996 The American Physical Society.

DOI: 10.1103/physreva.54.1161

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Physical Review A

Generalized center-of-mass coordinate and relative momentum operators studied through unitary transformations

Shen-xi Yu

Published Aug 1, 1996• 3 citations

Publisher

Abstract

DOI: 10.1103/physreva.54.1161

Related papers

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

Fan Hong-Yi, Ye Xiong

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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 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.