Atomic force microscope as an open system and the Ehrenfest force

Two systems in contact, such as the tip of an atomic force microscope (AFM) and a sample, share a common surface. Each exerts an equal and opposite force on the other determined by the pressure it exerts on every element of the surface of separation, as required by the physics of an open system. In a quantum system, the force exerted on the tip is the Ehrenfest force, a force that is balanced by the pressure exerted on every element of its surface, as determined by the quantum stress tensor. The surface separating the tip from the sample is one of local zero flux in the gradient vector field of the electron density, the surface that separates two neighboring atoms. A zero-flux surface also defines a proper open system, one whose observables are governed by the equations of motion, the equation for the electronic momentum yielding the Ehrenfest force theorem. Thus the force measured in the AFM is exerted on a surface determined by the boundaries separating the atoms in the tip from those in the sample, and its response is a consequence of the atomic form of matter. This approach to the determination of the force measured in the AFM is contrasted with results reported in the literature that equate it to the Hellmann-Feyman forces exerted on the nuclei of the atoms in the tip.