Theoretical exploration of uncertainty in three-dimensional radial position and radial momentum using the circular orbit of hydrogen atoms
In a Cartesian coordinate system,linear position and linear momentum satisfy the uncertainty rela-tionship,ΔxΔp≥ћ/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 op-erator that satisfies the self-adjoint condition.Here,inspired by the two-dimensional hyperbolic momentum opera-tor,we theoretically study the uncertainty relationship between the three-dimensional hyperbolic momentum opera-tor and the radial position logarithm.Taking hydrogen atoms as an example,the products of the uncertainty of ra-dial position logarithmic and that of three-dimensional hyperbolic momentum,i.e.,Δln rΔPH,for different circu-lar 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.
three-dimensional hyperbolic momentumhydrogen atomHeisenberg uncertainty principleintelligent state
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