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 fo rce on the other determined by the pressure it exerts on every element of t he surface of separation, as required by the physics of an open system. In a quantum system, the force exerted or. the tip is the Ehrenfest force, a f orce that iu balanced by the pressure exerted on every element of its surfa ce, 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 observable s are governed by the equations of motion, the equation for the electronic momentum yielding the Ehrenfest force theorem. Thus the force measured in t he AFM I is exerted on a surface determined by the boundaries separating th e atoms in the tip from those in the sample, and its response is a conseque nce 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 liter ature that equate it to the Hellmann-Feyman forces exerted on the nuclei of the atoms in the tip.