Chapter 21: The Schrödinger Equation & Superconductivity

The Schrödinger Equation & Superconductivity bridges the gap between fundamental quantum mechanics and macroscopic phenomena by examining the behavior of superconducting systems through the lens of the Schrödinger equation in a classical context. The lecture begins by applying the time-dependent Schrödinger equation to a charged particle moving in a magnetic field, which is formally described using the vector potential A. A crucial concept explored is the meaning of the wave function psi, where the product psi multiplied by its complex conjugate psi-star represents the probability density for a single particle. When dealing with large numbers of particles, such as the electron pairs in a superconductor, this density relates directly to the observable charge density rho. The conservation of probability naturally leads to the equation of continuity, which defines the current density J. The analysis distinguishes between the kinetic momentum (m times v) and the canonical momentum, often referred to as p-momentum (m times v plus q times A), demonstrating how magnetic fields influence particle dynamics in quantum systems. Applying these quantum principles to superconductivity explains classical observations, such as the Meissner effect, where external magnetic fields are expelled from the bulk of the material. Furthermore, the chapter details the critical phenomenon of flux quantization, proving that magnetic flux trapped within a superconducting ring must occur only in discrete, integral multiples of a fundamental quantum unit, where the relevant charge q is that of a pair of superconducting electrons. By expressing the wave function using a density rho and a phase theta, the governing Schrödinger equation for superconducting pairs is transformed into two dynamic equations that mathematically mirror the equations of motion for an incompressible, charged fluid, establishing a profound connection between quantum behavior and hydrodynamics. Finally, the principle of quantum tunneling is demonstrated through the Josephson junction, where superconducting electron pairs tunnel across a thin insulating barrier. The resulting coupled equations predict a direct current dependent on the phase difference across the junction, as well as an alternating current when a voltage is applied, detailing how quantum phases lead to observable electrical current phenomena.