The geometry of electron wave functions (Q2842986)

The properties of the wave function of a single electron are studied from a geometric point of view. A submanifold of codimension-two in Euclidian space is associated to the wave function of the electron. It is shown that the geometric properties of this submanifold are closely related to the physical properties of the elementary particle corresponding to the respective wave function. The cases of the wave functions of a single electron in the hydrogen atom or in other hydrogen-like atoms are considered in detail. The following connected objects are studied in this paper: (1) the submanifold corresponding to the wave function of a free particle; (2) the internal geometry of the electron wave function in the hydrogen atom, the metric and the scalar curvature; (3) integral equations for the scalar curvature; (4) the external geometry of the wave function, the mean curvature vector and the curvature tensor of the normal connection; (5) the geometric generalizations of the Schrödinger equation. Some theorems describing the behaviour of the scalar and sectional curvatures of the constructed submanifold and depending on the quantum numbers are proven. The relationship between Schrödinger's equation and the variational geometric problem is also considered for the case when the magnetic quantum number vanishes.