Behavior of an electron in the vicinity of a tridimensional charged polar nanoparticle through a classical and quantum constant of motion

We present a classical and a quantum constant of motion used to study the behavior of an electron moving in the vicinity of a tridimensional charged nanostructure with electric dipole. It is the generalization of the bidimensional version studied in a previous work. The classical case is found by using two approaches: the Hamilton Jacobi equation and a Newtonian treatment. The quantum case is obtained by separating the Schrödinger equation in spherical coordinates. We use three quantum numbers to classify the states. The angular part is solved by using an expansion of spherical harmonics to get an eigenvalue equation for the new constant of motion. The solution shows that the probability density of the electron shifts toward the values where $\theta <\frac{\pi }{2}$. The eigenvalue λ is near the value l(l + 1) of the angular momentum, but their difference grows if the electric dipole increases, so that the effect is small for a cluster like (GaAs)₃ but very important for fullerenes like RbC₆₀ and LiC₆₀. The radial part is solved using the shooting method and we find the ground state, and first and second excited states.