Common Eigenvectors of Four Particles’ Compatible Observables

Abstract

We construct eight operators for a four-particle system, namely one center-of-mass coordinate operator, three relative coordinate operators, one total momentum operator and three mass-weighted relative momentum operators, and give common eigenvectors of four compatible observables \(\{\sum_{i=1}^{4}\hat{p}_{i},\hat{x}_{1}-\hat{x}_{2},\hat{x}_{2}-\hat{x}_{3},\hat{x}_{3}-\hat{x}_{4}\}\) , which are composed of four particles’ coordinate \(\hat{x}_{i}\) and momentum \(\hat{p}_{i}\) . By compatible we mean such observables can be simultaneously determined. Using the technique of integration within an ordered product (IWOP) of operators, we prove that the common eigenvectors are complete and orthonormal, and hereby qualified for making up a representation.