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Canadian Journal of Physics

AN AUDIO-FREQUENCY CIRCUIT MODEL OF THE ONE-DIMENSIONAL SCHROEDINGER EQUATION AND ITS SOURCES OF ERROR

J. H. Blackwell, D. R. Fewer, L. J. Allen, R. S. Cass

Published Aug 1, 1955• 1 citation
Publisher

Abstract

An audio-frequency circuit model of the one-dimensional Schroedinger equation has been constructed, based on original suggestions by G. Kron. The previous theory of such devices has been re-examined thoroughly and errors due to the basic finite-difference approximation separated from those due to electrical causes. It is found that the former type of error is likely to be dominant in practice and that in the past discrepancies due to errors of this kind have actually been ascribed to experimental causes.

DOI
10.1139/p55-058

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An audio-frequency circuit model of the one-dimensional Schroedinger equation has been constructed, based on original suggestions by G. Kron. The previous theory of such devices has been re-examined thoroughly and errors due to the basic finite-difference approximation separated from those due to electrical causes. It is found that the former type of error is likely to be dominant in practice and that in the past discrepancies due to errors of this kind have actually been ascribed to experimental causes.
The model presented by Gabriel Kron in 1945 is an example of an analog computer simulating quantum phenomena on a hardware level. It uses passive RLC elements to construct a hardware solver for the problem of quantum particles confined by rectangular or other classes of potential. The analytical and numerical validation of Kron’s second model is conducted for different shapes of particle-confining potentials in the one-dimensional case using an LTspice simulator. Thus, there remains potential for obtaining solutions in two- and three-dimensional cases. Here, a circuit model representing a linearized Ginzburg–Landau equation is given. Kron’s second model is generalized by the introduction of linear and non-linear resistive elements. This transforms the deformed Schrödinger equation into a linear dissipative Schrödinger equation and its non-linear form. The quantum mechanical roton problem is the main result of this work and is formulated by means of classical physical states naturally present in the LC classical circular electrical transmission line. The experimental verification of Kron’s model is confirmed.
Synopsis: At this point we are equipped to solve some simple but fundamental one-dimensional problems with the Schrödinger equation. After setting the stage with an introduction, four quantum mechanical problems are specified and solved. Along the way, we encounter important concepts such as parity in the case of the infinite well and harmonic oscillator problems, transmission and reflection coefficients in the case of the finite barrier, and energy bands and gaps in a model of conduction in solids. Raising and lowering ladder operators are introduced in the harmonic oscillator and square well cases. I conclude with some qualitative comments regarding expected types of solutions in one-dimensional problems.