Chinese Physics Letters

Differential Completeness Relation in Two-Mode Mass-Weighted|ξ⟩ -|τ⟩ Representations

Hong-Yi Fan, Nai-Le Liu

Published Sep 1, 1999• 3 citations

Abstract

The explicit form of the common eigenstates| τ ⟩ of the mass-weighted relative coordinate μ 2 Q 1 - μ 1 Q 2 and the mass-combinatorial momentum μ 1 P 1 + μ 2 P 2 of two particles is given. On the basis of| τ ⟩ and their conjugate states| ξ ⟩, we find that there exist new differential completeness relations (DCR). The applications of the new DCR in deriving the normally ordered two-mode squeezing operators for the mass-related coordinates and momenta are demonstrated. Throughout the discussion the technique of differentiation within an ordered product of operators is fully employed.

DOI: 10.1088/0256-307x/16/9/001

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Chinese Physics Letters

Differential Completeness Relation in Two-Mode Mass-Weighted|ξ⟩ -|τ⟩ Representations

Hong-Yi Fan, Nai-Le Liu

Published Sep 1, 1999• 3 citations

Publisher

Abstract

DOI: 10.1088/0256-307x/16/9/001

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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.

Studying bi-partite entangled state representations via the integration over ket-bra operators in

\mathfrak{Q}

-ordering or

\mathfrak{P}

-ordering

\mathfrak{Q}

-ordering or

\mathfrak{P}

-ordering, where

\mathfrak{Q}

-ordering means all Qs are to the left of all Ps and

\mathfrak{P}

-ordering means all Ps are to the left of all Qs. In this way we newly derive

\mathfrak{P}

-ordered (or

\mathfrak{Q}

-ordered) expansion formulas of the two-mode squeezing operator which can show the squeezing effect on both the two-mode coordinate and momentum eigenstates. This tells that not only the integration over ket-bra operators within normally ordered, but also within

\mathfrak{P}

-ordered (or

\mathfrak{Q}

-ordered) are feasible and useful in developing quantum mechanical representation and transformation theory.