Communications in Theoretical Physics

Two-Mode Wigner Operator in 〈ξ| Representation and 〈τ| Representation

Song Tong-Qiang, Fan Hong-Yi

Published Dec 15, 2001• 0 citations

Abstract

We derive the two-mode Wigner operator in the 〈ξ| representation and 〈τ| representation, where |ξ〉 is the common eigenvector of the mass-weighted relative coordinate and the mass-combinatorial momentum, and |τ〉 is the common eigenvector of the mass-weighted relative coordinate and the mass-combinatorial momentum. As an application, we calculate the Wigner function of some two-mode state.

DOI: 10.1088/0253-6102/36/6/695

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Communications in Theoretical Physics

Two-Mode Wigner Operator in 〈ξ| Representation and 〈τ| Representation

Song Tong-Qiang, Fan Hong-Yi

Published Dec 15, 2001• 0 citations

Publisher

Abstract

DOI: 10.1088/0253-6102/36/6/695

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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.