Communications in Theoretical Physics

Differential Forms of Completeness Relation in Two-Mode 〈ξ|-〈η| Representations

Fan Hong-Yi, Liu NaiLe

Published Dec 15, 1999• 0 citations

Abstract

On the basis of the common eigenvectors 〈η| and 〈ξ|, which respectively are eigenvectors of two particles' relative position and total momentum as well as center-of-mass coordinate and relative momentum, we find that there exist differential forms of completeness relation. The applications of the new completeness relation are pointed out. Throughout our discussion the technique of differentiation within an ordered product of operators is fully employed.

DOI: 10.1088/0253-6102/32/4/611

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Communications in Theoretical Physics

Differential Forms of Completeness Relation in Two-Mode 〈ξ|-〈η| Representations

Fan Hong-Yi, Liu NaiLe

Published Dec 15, 1999• 0 citations

Publisher

Abstract

DOI: 10.1088/0253-6102/32/4/611

Related papers

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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 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.