Geometry of a quantal system

Abstract

AbstractClassically every physical thing has some geometry or other, but in the quantum theory the notions of spatial structure, shape, and size seem to become hazy if not outright inapplicable. In particular, elementary entities are by definition structureless and seem to have no size or shape other than those imposed by their environmént—as represented, e.g., by the boundary conditions. However, this does not entail that systems of interacting components, such as molecules, lack a geometry as well, as has been argued in recent times. The purpose of this paper is to investigate this claim. Since the solution to the problem whether a quantum‐mechanical system possesses a geometry depends critically upon the definition of the latter, we elucidate the notions of extension, shape, and spatial configuration of a system. We do so first classically in terms of the energy density, then quantum mechanically in terms of a suitably constructed position distribution function that depends upon the spatial coordinates (and is thus a field quantity) rather than on the particle coordinates. In this way we construct both classical and quantal concepts of the geometry of a system. We also define the notions of spatial structure and chemical structure of a quantal system. Our quantum‐mechanical concepts are quite general and constitute smooth extensions of the corresponding classical concepts. The upshot of this investigation is that a quantal system does have a geometry, albeit not as clearcut a one as a classical system, that is primarily determined by the interactions among its components. (Thus, physics determines geometry of physical systems, not the other way around.) This conclusion is obtained within nonrelativistic quantum mechanics without adding any extra assumptions. Our result vindicates the ball‐and‐spoke models of molecules and thus shows once again that quantum chemistry, although different from classical chemistry, stays close to the chemist's work bench.