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Journal of the University of Shanghai for Science and Technology

Double wave description of the time-dependent mesoscopic LC electric circuit

Liang Mai-lin

Published Jan 1, 2003• 0 citations
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Abstract

The quantum double wave theory is applied to the time-dependent mesoscopic LC electric circuit with time-dependent capacitance and inductance. The averages and quantum fluctuations of the electric charge and magnetic flux (electric current) are given for a single time-dependent LC electric circuit. The obtained averages of the electric charge and magnetic flux satisfy the classical equation of motion. After making statistical average over the initial phases, the average and quantum fluctuation of every quantity reduce to that derived by the standard quantum mechanics. These results show that the double wave theory can be used to describe a single time-dependent mesoscopic electric circuit.

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The quantum double wave theory is applied to the time-dependent mesoscopic LC electric circuit with time-dependent capacitance and inductance. The averages and quantum fluctuations of the electric charge and magnetic flux (electric current) are given for a single time-dependent LC electric circuit. The obtained averages of the electric charge and magnetic flux satisfy the classical equation of motion. After making statistical average over the initial phases, the average and quantum fluctuation of every quantity reduce to that derived by the standard quantum mechanics. These results show that the double wave theory can be used to describe a single time-dependent mesoscopic electric circuit.
By extending the thermal field dynamics to the time dependent system, quantum fluctuations of the time dependent mesoscopic LC electric circuit at finite temperature were studied. While the up going temperature will increase the quantum fluctuations, time changes of the physical quantities such as the inductance and capacitance can suppress the quantum fluctuations. The electric source will not affect the quantum fluctuations, but will contribute to the averages of the charge and magnetic flux. It was also found that if there is a source, the averages will depend on the temperature; if there is no source, the averages will be zero and so temperature independent.
We discuss the problem of a mesoscopic LC circuit with a negative inductance ruled by a time-dependent Hermitian Hamiltonian. Classically, we find unusual expressions for the Faraday’s law and for the inductance of a solenoid. Quantum mechanically, we solve exactly the time-dependent Schrödinger equation through the Lewis and Riesenfeld invariant operator method and construct Gaussian wave packet solutions for this time-dependent LC circuit. We also evaluate the expectation values of the charge and the magnetic flux in these Gaussian states, their quantum fluctuations and the corresponding uncertainty product.
The authors took the LC circuit as an example to discuss the relationship between the time evolution formula for the charge mean value in mesoscopic circuit and the Kirchoffs law,put forward the equation satisfying the wave function of coherent state,the wave function of coherent state of LC circuit and the quasi-classical wave function satisfying the Schrdinger equation,the current intensity,the quantum fluctuation and the quantum phase,and discussed the motion characteristics of wave packet.
For the general time-dependent inductance-capacitance coupled circuits, quantum-classical correspondence and the quantum fluctuations are studied by using the Heisenberg correspondence principle. Through the wave functions and exact solution of equations of motion in Heisenberg picture, the classical solution is derived from quantum solution in the limit of large quantum numbers carrying out the average over the initial phase in the matrix elements, quantum fluctuations of the charges and magnetic fluxes are obtained. When the inductances in the circuit increase and the capacitances decrease exponentially with time, quantum fluctuations of the charge and current decrease exponentially with time; When the inductances and capacitances in the two component circuits are equal to each other and time-independent, it is found that the coupling capacitance tends to reduce the quantum fluctuations of the electric current and the coupling inductance tends to reduce the charge quantum fluctuations.