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 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.

Physical Review AApr 1, 1995124 citations

Spectral Decomposition

Daniel M. Greenberger, Klaus Hentschel, Friedel Weinert

Compendium of Quantum PhysicsJan 1, 20090 citations

Perturbative method for generalized spectral decompositions

Roberto Laura, Mario Castagnino, R. Id Betán

arXiv:quant-ph/0005073Physica A Statistical Mechanics and its ApplicationsSep 1, 199915 citations

Eigenvectors of two particles’ relative position and total momentum

Fan Hong‐Yi, John R. Klauder

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.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.

Physical Review AFeb 1, 1994384 citations

Spectral representation

Aloïs Würger

Springer tracts in modern physicsJan 1, 19971 citation

Phase-Space Formulation of the Dynamics of Canonical Variables

C. L. Mehta

Statistical reformulation of quantum mechanics in terms of phase-space distribution functions as given by Moyal using Weyl's correspondence rule between classical functions and operators has been extended to various different correspondence rules. The dynamical bracket in the Weyl correspondence (the ``Moyal'' or the ``sine'' bracket) is shown to be a Lie bracket. It is further shown that if the theory is restricted to Lie brackets of the form [u(p, q),υ(p, q)]={f(∂2∂q1 ∂p2−∂2∂q2 ∂p1)u(p1, q1)υ(p2, q2)},evaluated for p1 = p2 = p; q1 = q2 = q after differentiation, then the only admissible functional form of f is f(x) = β[(sin αx)/α], where α and β are constants. A law of multiplication which is associative and distributive with respect to addition is also introduced in each case. It gives a correct correspondence between operator multiplication and the multiplication of classical functions. The dynamical brackets obtained in each case are also found to be Lie brackets. Conditions on the phase-space distribution functions to describe pure states are also given.Statistical reformulation of quantum mechanics in terms of phase-space distribution functions as given by Moyal using Weyl's correspondence rule between classical functions and operators has been extended to various different correspondence rules. The dynamical bracket in the Weyl correspondence (the ``Moyal'' or the ``sine'' bracket) is shown to be a Lie bracket. It is further shown that if the theory is restricted to Lie brackets of the form [u(p, q),υ(p, q)]={f(∂2∂q1 ∂p2−∂2∂q2 ∂p1)u(p1, q1)υ(p2, q2)},evaluated for p1 = p2 = p; q1 = q2 = q after differentiation, then the only admissible functional form of f is f(x) = β[(sin αx)/α], where α and β are constants. A law of multiplication which is associative and distributive with respect to addition is also introduced in each case. It gives a correct correspondence between operator multiplication and the multiplication of classical functions. The dynamical brackets obtained in each case are also found to be Lie brackets. Conditions on the phase-space distribution functions to describe pure states are also given.

Journal of Mathematical PhysicsMay 1, 1964154 citations

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.