Phase-dependent characteristics of a superconducting junction by using the Schroedinger wave function

As a macroscopic quantum system, superconducting junction devices are considered both analytically and numerically with the help of the Schroedinger wave equation. Considering the Cooper pair wave function ψ in a time-dependent nature, the Hamiltonian of the system is constructed. The phase difference θ is assumed to be a unique independent variable of the superconducting junction in order to evaluate the characteristics of the system. The probability ρ(θ) and super current j(θ) densities are determined for different applied voltages V. It is observed that both the amplitude and the period of the wave function ψ(θ) have indicated different behaviors as functions of θ for various cases of V. Similarly, ρ and j reflect drastic changes. It is shown that numerical results are in good agreement with the analytic results. Our results indicate consistency with some earlier experimental studies for different superconducting devices