Eigenvectors of two particles’ relative position and total momentum

Abstract

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.