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Mathematics

Improved Quantization Method of Coupled Circuits in Charge Discrete Space

Jinying Ma, Weiran Zhao, Weilin Wang, Zhan-Yuan Yan

Published Nov 28, 2024• 1 citation
Publisher

Abstract

A quantum theory of mesoscopic circuits, based on the discreteness of electric charges, was recently proposed. However, it is not applied widely, mainly because of the difficulty of the mathematical solution to the finite-difference Schrödinger equation. In this paper, we propose an improved perturbation method to calculate Schrödinger equations of the inductance and capacity coupling mesoscopic circuit. With a unitary transformation, the finite differential Schrödinger equation of the system is divided into two equations in generalized momentum representation. The concrete value of the parameter in circuits is important to solving the equation. Both the perturbation method suitable case and the improved perturbation method suitable case, the wave functions, and energy level of the system are achieved. As an application, the current fluctuations are calculated and discussed. The improved mathematical method would be helpful for the popularization of the quantum circuits theory.

DOI
10.3390/math12233753

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Using the quantum theory for a mesoscopic circuit based on the discretenes of electric charges, the finite-difference Schrödinger equation of the non-dissipative mesoscopic inductance and capacity coupling circuit is achieved. The Coulomb blockade effect, which is caused by the discreteness of electric charges, is studied. Appropriately choose the components in the circuits, the finite-difference Schrödinger equation can be divided into two Mathieu equations in representation. With the WKBJ method, the currents quantum fluctuations in the ground states of the two circuits are calculated. The results show that the currents quantum zero-point fluctuations of the two circuits are exist and correlated.
Using the quantum theory for a mesoscopic circuit based on the discreteness of electric charges, the finite-differenced Schrodinger equation of the non-dissipative mesoscopic capacity coupling circuit are achieved. The quantum effect not only comes from the quanumnization of circuit but also that of charge. Only after the properties of charge quanumnization is considered can the Coulomb blockade effect of a mesoscopic circuit be discussed. The Schrodinger equation can be divided into two Mathieu equations in p-representation. With the WKBJ method, the currents quantum fluctuations in the ground states of the two circuits are calculated. And the fluctuations are the main noise sources of quantum in the mesoscopic circuit.
In the framework of an advanced quantum theory for mesoscopic electric circuits in accord with the discreteness of electric charges, a mesoscopic quartz piezoelectric crystal equivalent circuit is quantized. To resolve the finite difference Schrödinger equation, an improved parameter perturbative method is proposed when WKB and perturbative method are inapplicable. With this method, the energy spectrum and wavefunctions of the system are obtained and used to calculate current quantum fluctuation as an application. The new method would be helpful to the application of the mesoscopic circuits quantum theory. Besides, the detail characters of energy spectrum and wavefunctions in the system would be helpful to the design of integrate circuit.
For the improvement of quantum theory of mesoscopic circuits,the finite-difference Schrdinger equation of a diode included nonlinear mesoscopic circuit is achieved in the framework of advanced quantum theory of mesoscopic circuits,which based on the fundamental fact that the electric charge takes discrete value,it is a four-order difference equation in p^ representation.Using the perturbation method,the detail energy spectrum and wave functions are obtained,as an application of the results,the current quantum fluctuation is calculated.
The research work on the quantum effects in mesoscopic circuits has undergone a rapid development recently, however the whole quantum theory of the mesoscopic circuits should consider the discreteness of the electric charge. In this paper, based on the fundamental fact that the electric charge takes discrete values, the finite-difference Schrödinger equation of the mesoscopic RLC circuit with a source is achieved. With a unitary transformation, the Schrödinger equation becomes the standard Mathieu equation, then the energy spectrum and the wave functions of the system are obtained. Using the WKBJ method, the average of currents and square of the current are calculated. The results show the existence of the current fluctuation, which causes noise in the circuits. This paper is an application of the whole quantum mesoscopic circuits theory to the fundamental circuits, and the results will shed light on the design of the miniation circuits, especially on the purpose of reducing quantum noise coherent controlling of the mesoscopic quantum states.
The mesoscopic nonlinear inductance-capacitance circuit is a typical anharmonic oscillator, due to diodes included in the circuit. In this paper, using the advanced quantum theory of mesoscopic circuits, which based on the fundamental fact that the electric charge takes discrete value, the diode included mesoscopic circuit is firstly studied. Schrödinger equation of the system is a four-order difference equation in representation. Using the extended perturbative method, the detail energy spectrum and wave functions are obtained and verified, as an application of the results, the current quantum fluctuation in the ground state is calculated. Diode is a basis component in a circuit, its quantization would popularize the quantum theory of mesoscopic circuits. The methods to solve the high order difference equation are helpful to the application of mesoscopic quantum theory.