Common eigenstates of two particles’ center-of-mass coordinates and mass-weighted relative momentum

Abstract

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.