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Physica Scripta

Exact solution of a quantizedLCcircuit coupled to a power source

Jeong-Ryeol Choi

Published May 15, 2006• 21 citations
Publisher

Abstract

By means of a two-step unitary transformation, the exact quantum mechanical solution of two-dimensional mesoscopic circuit coupled via inductance is derived. The wavefunction, propagator and expectation value of Hamiltonian are evaluated. The fluctuations of charges and currents are calculated and uncertainty products between charges and their conjugate currents are obtained. We apply our results to the system with special cases of the power source such as the sinusoidal and the sawtooth power source and show that the probability densities oscillate with time due to the power source.

DOI
10.1088/0031-8949/73/6/010

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By means of a two-step unitary transformation, the exact quantum mechanical solution of two-dimensional mesoscopic circuit coupled via inductance is derived. The wavefunction, propagator and expectation value of Hamiltonian are evaluated. The fluctuations of charges and currents are calculated and uncertainty products between charges and their conjugate currents are obtained. We apply our results to the system with special cases of the power source such as the sinusoidal and the sawtooth power source and show that the probability densities oscillate with time due to the power source.
Exact quantum solutions of two-dimensional dissipative mesoscopic circuit coupled to a power source are derived using two-step unitary transformation approach. With the choice of appropriate unitary operators, the original Hamiltonian of the system is transformed to a simple form whose quantum behavior is well known. From inverse transformation of the quantum solutions associated to transformed system, we obtained the full wave functions in the original system. Thermofield dynamics is employed in order to investigate thermal states of the system. The vacuum correlations between the so called tilde and non-tilde variables produce the temperature effects. The fluctuations and uncertainty relations for charges and currents are derived in thermal states. Due to the effect of energy dissipation generated in the resistances of circuit, the fluctuation of charges decreases exponentially with time whereas that of currents exhibits exponential increase. Accordingly, their products which gives the uncertainty relation does not depend on such exponential functions and always larger than or equal to ℏ/2\hbar/2ℏ/2 so that the uncertainty principle holds.
The quantum fluctuations of charges and currents of mesoscopic capacitance coupled circuit with mutual inductance and power source were investigated by using canonical transformation and unitary transformation. The results show that the quantum fluctuations of charges and currents are independent of power source. When the components of the circuit are given, if the value of L_1/L_2 is very small or very large, the effect of coupling coefficient on quantum fluctuations of charges and currents are great. From no mutual inductance to having mutual inductance, the quantum fluctuation of charge of loop 2 and the quantum fluctuation of current of loop 1 are obvious. We can conclude that it play a very important role whether the mutual inductance exists or not.
In this paper, we use Hermitian linear invariants and the Lewis and Riesenfeld invariant method to obtain the general solution of the Schrödinger equation for a mesoscopic RLC circuit with time-dependent resistance, inductance, capacitance and a power source and represent it in terms of an arbitrary weight function. In addition, we construct Gaussian wave packet solutions for this electromagnetic oscillation circuit and employ them to calculate the quantum fluctuations of the charge and the magnetic flux as well as the associated uncertainty product. We also show that the width of the Gaussian packet and the fluctuations do not depend on the external power.