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Physical Review A

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

Fan Hong‐Yi, Chen Bozhan

Published May 1, 1996• 57 citations

Abstract

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.

DOI: 10.1103/physreva.53.2948

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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 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 study the eigenstate problem of a two-coupled oscillator system. A new entangled state representation |γ composed of the common eigenvectors of operators (x1+p2) and (p1+x2) is established. Eigenvalues and eigenvectors of the Hamiltonian are obtained in |γ representations. The same problem is studied in the second-quantization representation. We find that the second-quantization representation can be used to derive the normally ordered product expression of eigenvector in Fock space. In particular, we find that the ground state of the Hamiltonian is a kind of generalized two-mode squeezed state.