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Harbin Ligong Daxue xuebao

Exact Solutions to the Feynman Model Establishing the Josephson Equations

Xue Wang

Published Jan 1, 2000• 0 citations
OpenAlex

Abstract

The Feynman model is a famous theoretical model establishing the Josephson equations. Feynman et al. based on the Schrodinger equations describing the two states system of superconductor junction, have derived the differential equations of the amplitudes and phases of the two wave functions versus time, and solved them with a slightly rough method, and established the Josephson equation. It is worth considering carefully that mathematical treatment in Feynman model was not exact. In this paper we have solved the differential equations in Feynman model exactly, the results we have gotten show that A. C. Josephson equations can be established only under certain conditions..

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The Feynman model is a famous theoretical model establishing the Josephson equations. Feynman et al. based on the Schrodinger equations describing the two states system of superconductor junction, have derived the differential equations of the amplitudes and phases of the two wave functions versus time, and solved them with a slightly rough method, and established the Josephson equation. It is worth considering carefully that mathematical treatment in Feynman model was not exact. In this paper we have solved the differential equations in Feynman model exactly, the results we have gotten show that A. C. Josephson equations can be established only under certain conditions..
By starting from a microscopic quantum mechanical description of Josephson dynamics of a one-dimensional array of N coupled superconductors, we obtain a set of linear differential equations for the system order parameter and for additional macroscopic physical quantities. With opportune considerations, we adapt this description to two coupled superconductors, obtaining the celebrated Feynman model for Josephson junctions. These results confirm the correspondence between the microscopic picture and the semi-classical Ohta’s model adopted in describing the superconducting phase dynamics in multi-barrier Josephson junctions.
The Josephson equations are derived by means of the weakly coupled two-level quantum system model given by Feynman. Adopting a simplified version of Ohta's model, starting from Feynman's model, the strict voltage–frequency Josephson relation is derived. The contribution of Ohta's approach to the comprehension of the additional term given by the classical external system is recognized. Extension of the model to double-barrier Josephson junctions and multilayered systems is finally briefly commented on.
In all the applications of the formalism of quantum theory, we have considered mainly time-independent potential functions, V ( r , t ) = V ( r ). Such potentials made it easier to solve exactly the Schrodinger's equation or by solving the Schrodinger's equation approximately by using perturbation theory. The solution of the Schrodinger equation for time-independent potentials is obtained by using method of separable variables. The solution of time-independent Schrodinger gives stationary states, and the time dependence of the wave function is obtained by adding a phase factor to the stationary state. The schematic that follows describes such an approach ( Figure 7.1 ).
An alternative method for obtaining analytical solutions of the time-independent Schrödinger equation (TISE) is developed and applied to 1-dimensional model quantum systems of interest in atomic and molecular physics. The method is based on proposing a wave function that is a product of two functions, one of them an integrating factor. The differential equation resulting from operating the TISE on the product wave function is compared with a differential equation with a known solution. This comparison produces a system of differential equations for the integrating factor. This system of equations is easily solved for model quantum systems such as the harmonic oscillator, the hydrogen atom and the Morse oscillator. The method presented simplifies some approaches shown in textbooks, based on asymptotic analyses of the TISE, and power series methods with recurrence relations. In addition, the method presented here facilitates the understanding of the relationship between the ordinary differential equations of the mathematical physics and the TISE of physical models as the harmonic oscillator, the Hydrogen atom, and the Morse oscillator.
Abstract A rigorous formulation of the boson method in superconductivity is presented. Assuming the local gauge invariance and presence of a phase‐dependent order parameter and utilizing the path‐integral formalism, we derive a set of macroscopic equations which control all the superconducting states. These equations are model‐independent. They contain certain functions and parameters which should be calculated for a given model. Using this rigorous formulation, we study what kind of approximations are involved in the usual formulation of the boson method and how these approximations can be improved.
The quantum dynamics of a one-dimensional bosonic Josephson junction is studied by solving the time-dependent many-boson Schrödinger equation numerically exactly. Already for weak interparticle interactions and on short time scales, the commonly employed mean-field and many-body methods are found to deviate substantially from the exact dynamics. The system exhibits rich many-body dynamics such as enhanced tunneling and a novel equilibration phenomenon of the junction depending on the interaction, which is attributed to a quick loss of coherence.