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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 Sep 14, 2019• 0 citations
Publisher

Abstract

In quantum theory, a wave function expresses the state of matter waves and is not considered to physically exist in general. Equations that are satisfied by a wave function are called wave equations. In classical physics, wave equations are well known and are derived from the Maxwell equations. However, the Maxwell equations do not define important physical properties, such as the lossless property of circuits. The lossless property is represented by the cascade matrix defined in analog circuit theory. The definition of the lossless property in wave equations became possible owing to the telegrapher's equations introduced by Heaviside. The cascade matrix can be determined from uniform transmission lines that can be obtained from the telegrapher's equations and those extended using the Riccati differential equation.

DOI
10.2299/jsp.23.227

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In quantum theory, a wave function expresses the state of matter waves and is not considered to physically exist in general. Equations that are satisfied by a wave function are called wave equations. In classical physics, wave equations are well known and are derived from the Maxwell equations. However, the Maxwell equations do not define important physical properties, such as the lossless property of circuits. The lossless property is represented by the cascade matrix defined in analog circuit theory. The definition of the lossless property in wave equations became possible owing to the telegrapher's equations introduced by Heaviside. The cascade matrix can be determined from uniform transmission lines that can be obtained from the telegrapher's equations and those extended using the Riccati differential equation.
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.
Heaviside proposed the telegrapher's equations as an alternative to Maxwell equations and obtained the solution of the telegrapher's equations by the operator method. Indeed, the solution was a one-dimensional wave function. One-dimensional wave equations are partial differential equations and are conventionally solved as initialvalue and boundary-value problems in mathematics. Heaviside's solution was found to be related to the algebraic invariants studied in mathematics in the 19th century. From algebraic invariants, lossless circuits were obtained as Junitary matrix. These research findings are still under study. To utilize the research achievements effectively, the application of lossless circuits obtained from algebraic invariants to quantum physics is one of the issues to be studied in the future.
To apply circuit theory to quantum theory, we review the problems of circuit theory described in the past sessions. In concrete, the physical properties of circuits, such as losslessness, steady-state response, and reactive power, are reexamined. It is shown that the Laplace transform is more advantageous than Heaviside's operational calculus and that active power can be conserved when a lossless circuit is used. It is also demonstrated that a forward wave exp(-jβx) travels clockwise because exp(jωt) in the Laplace transform is required for separating the variables of partial differential equations related to waves.
The electromagnetic waves derived from the Maxwell equations are characterized by being lossless. It is known that such lossless electromagnetic waves are obtained in the electromagnetic fields of coaxial cables and the electromagnetic fields generated using coils and capacitors. Because the physical behavior and effects of electromagnetic waves differ depending on frequency, the electromagnetic waves propagating through the electromagnetic fields of coaxial cables and the electromagnetic fields generated using coils and capacitors are limited to very high frequency (VHF) and ultrahigh frequency (UHF). Electromagnetic waves of other frequencies are left as problems for future research. Also, as stated in Section 8 of this article, a "lossless circuit" is also derived from the equations of motion for molecules (Newtonian mechanics). Namely, because the oscillations of a periodic system consisting of N molecules are expressed in the representation equivalent to the circuit representation of lossless electromagnetic waves, it has been demonstrated that a lossless circuit also arises from the motion of molecules with mass derived from Newtonian mechanics. This is the key physical phenomenon that links quantum mechanics to mathematics. We will leave this point as an important problem for future research.
Physical phenomena showing both particle and wave natures at the same time exhibit resonance. In addition, resonance is a physical phenomenon occurring in the steady state. Transient and steady-state responses can be discriminated when transmission lines are used. We demonstrate that (1) extended telegrapher's equations can be obtained using the Riccati differential equation, (2) lossless transmission lines called Pasteur, Tellegen, and bi-isotropic (BI) media can be characterized using the extended telegrapher's equations, and (3) the resonance conditions can be determined for each of these media.
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.