Abstract: In a Cartesian coordinate system, linear position and linear momentum satisfy the uncertainty relationship, ΔxΔp≥2. Recent years also witnessed the research interest as to whether the radial position and radial momentum satisfies the similar uncertain relationship, in which one of the key points is how to construct a radial operator that satisfies the self-adjoint condition. Here, inspired by the two-dimensional hyperbolic momentum operator, we theoretically study the uncertainty relationship between the three-dimensional hyperbolic momentum operator and the radial position logarithm. Taking hydrogen atoms as an example, the products of the uncertainty of radial position logarithmic and that of three-dimensional hyperbolic momentum, i.e., Δln rΔPH, for different circular orbits are calculated. It is found that as the main quantum number increases, the wavefunctions of circular orbits are more approaching to the intelligent states.

Key words:  three-dimensional hyperbolic momentum, hydrogen atom, Heisenberg uncertainty principle, intelligent state