Electric Circuit Models of the Schrödinger Equation

Abstract

Equivalent circuits are developed to represent the Schr\"odinger amplitude equation for one, two, and three independent variables in orthogonal curvilinear coordinate systems. The networks allow the assumption of any arbitrary potential energy and may be solved, within any desired degree of accuracy, either by an a.c. network analyzer, or by numerical and analytical circuit methods. It is shown that by varying the impressed frequency on a network of inductors and capacitors (or by keeping the frequency constant and varying the capacitors), it is possible to find by measurements the eigenvalues, eigenfunctions, and the statistical mean of various operators belonging to the system represented. The electrical model may, of course, be replaced by an analogous mechanical model containing moving masses and springs. At first the network for the one-dimensional wave equation for a single particle in Cartesian coordinates is developed in detail, then the general case. A companion paper contains results of a study made on an a.c. network analyzer of one-dimensional problems: a potential well, a double barrier, the harmonic oscillator, and the rigid rotator. The curves show good agreement, within the accuracy of the instruments, with the known eigenvalues, eigenfunctions, and "tunnel" effects.