Distribution of ⋅ for quantum states on a circle.

We first give the proper definition of the particle's position-momentum dot product, the so-called posmom x ⋅ p, to quantum states on a circular circle, in which the momentum turns out to be the geometric one that is recently intensively studied. Second, we carry out the posmom distributions for eigenstates of the free motion on the circle, i.e. , (m = 0, ±1, ±2, ...). The results are not only potentially experimentally testable, but also reflect a fact that the embedding of the circle S¹ in two-dimensional flat space R² is physically reasonable. [ABSTRACT FROM AUTHOR]