The hydrogen atom confined by one and two hard cones

The bound states of the H atom in a semi-infinite space limited by one or two conical boundaries are studied. The exact solution when the nucleus is located at the apex of the conical boundaries is obtained. A rapid increase of the energy when the cone angle opens and tends to π/2 is found. A second situation with the atom separated from the summit of the cone is considered. The changes on the energy and the electronic structure are analyzed. The quantum force is evaluated by calculating the energy derivative versus the distance to the cone vertex. One of the forces exerted on the tip of an Atomic Force Microscope can be modelized by a hard cone probing the electron cloud in the contact mode. Our numerical results show that the quantum force present an important dependence with the cone angle and it vanishes rapidly as the distance increases.