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Journal of Signal Processing

Circuit Theory Based on New Concepts and Its Application to Quantum Theory

Nobuo Nagai, Takashi Yahagi

Published Jan 1, 2017• 0 citations
Publisher

Abstract

In Session 8, we demonstrated that circuit theory can be applied to the Schrödinger equation, a basic equation in quantum mechanics. In this session, we attempt to apply circuit theory to the Dirac equation using the Riccati differential equation and demonstrate that physical phenomena such as spin 1/2 can be explained on the basis of circuit theory. To this end, we obtain the one-dimensional Dirac equation for the complex equivalent circuit for a physical phenomenon. We show that the Dirac equation can be regarded as a type of low-energy effective field theory in quantum theory and that circuit theory can be applied to the Dirac equation.

DOI
10.2299/jsp.21.225

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In Session 8, we demonstrated that circuit theory can be applied to the Schrödinger equation, a basic equation in quantum mechanics. In this session, we attempt to apply circuit theory to the Dirac equation using the Riccati differential equation and demonstrate that physical phenomena such as spin 1/2 can be explained on the basis of circuit theory. To this end, we obtain the one-dimensional Dirac equation for the complex equivalent circuit for a physical phenomenon. We show that the Dirac equation can be regarded as a type of low-energy effective field theory in quantum theory and that circuit theory can be applied to the Dirac equation.
In Session 10, we showed that the telegrapher's equations can be extended using the Riccati differential equation and that a new circuit element can be obtained. In Session 11, we showed that the Dirac equation, which is used in quantum theory, can be expressed as the extended telegrapher's equations and that circuit theory can be applied to the Dirac equation. In this session, we demonstrate that a new circuit element can be obtained by applying reactive power to the Dirac equation. In this case, energy given by Einstein's well-known formula E = mc2 should be considered for the discussion on an ordinary world that does not require the theory of relativity.
The Schrödinger equation is derived from the concept that particles of matter such as electrons have a wave nature, and it is used to express phenomena specific to quantum mechanics, such as the advancing wave and the quantum tunneling effect. In this session, we apply circuit theory to the Schrödinger equation and define voltage and current using wave functions that satisfy the Schrödinger equation. By this method, we show that a lossless transmission line or circuit can be obtained. Thus, a circuit that satisfies the Schrödinger equation is lossless and is not directly related to heat, but the reactive power is related to the regulation of energy. Therefore, such a circuit can be discussed in a theoretical system that does not rely on probability theory.
In the electromagnetic field formed by coaxial cables, electromagnetic waves of different frequencies are shown to propagate at the same speed (the speed of light c). Therefore, the variational method, which is based on the fact that electromagnetic waves of different frequencies have different propagation speeds, cannot be applied. In this session, the characteristics of light propagating through an electromagnetic field of glass, suitable for applying the variational method and for constructing lenses, are determined. Although the electromagnetic field is derived from the properties of dielectrics, it has been demonstrated to be equivalent to a circuit that represents the Schrödinger equation used in quantum mechanics. In other words, it may be possible to apply electromagnetic fields from classical physics (circuit theory) to quantum mechanics. Quaternions and wave functions that are considered to be related to complex functions will be the topic of future studies because the Schrödinger equation is a complex function.
Circuit theory, which explains the behavior of electromagnetic waves, concerns reflection waves caused by the difference in impedance. In this session, we demonstrate that electromagnetic waves with the angular frequency of nπ or nπ + π/2 exhibit interesting behavior in circuits with a high impedance ratio and a large amount of reflection. The physical phenomenon of increasing the amount of reflection is considered to be the same as the tunneling effect in quantum mechanics. This demonstration shows that the application of the telegrapher's equations expressing the electromagnetic fields and waves of the Maxwell equations to quantum mechanics may provide a clue to seeing the light at the end of the mysterious tunnel in quantum mechanics.