The Schrödinger Equation for a Spherically Symmetric System, Its Structure and Solutions#

The Wheeler–DeWitt geometrodynamics, as the first attempt to develop a quantum theory of gravity, faces certain challenges, including the problem of time and interpretation of the wave function. In this paper, we present the extended phase space approach to quantization of gravity as an alternative approach to the Wheeler–DeWitt quantum geometrodynamics. For a space-time with a nontrivial topology, the Wheeler–DeWitt equation loses its sense, but we can derive the Schrödinger equation. Until now, the Schrödinger equation was derived for systems with a finite number of degrees of freedom, and we need to generalize the procedure for field models. The simplest field model is a spherically symmetric one. We derive the integro-differential Schrödinger equation for this model, examine its structure, and find its solution.