A study on quantum similarity in the phase space

•We discuss a recently proposed definition of quantum similarity in the phase space.•We investigate the dependence of this quantum similarity on an integer parameter.•We apply this study to the hydrogen atom in the ground and excited states.

Quantum similarity represents an important concept in the context of many applied disciplines such as physical and quantum chemistry. Nowadays, two definitions exist based, respectively, on the real and the phase   spaces. In this paper, we focus on the second one, which was presented recently, and investigate it. In particular, being its mathematical definition dependent on a given integer ss, we study the influence of this parameter on the similarity between two systems. To keep this investigation comprehensible, while still meaningful, we focus on a very simple quantum system represented by a hydrogen atom in the ground and excited states corresponding to the quantum numbers (n,l,m)=(1,0,0)(n,l,m)=(1,0,0) and (n,l,m)=(2,0,0)(n,l,m)=(2,0,0).