Constrained Hamiltonian dynamics for electrons in magnetic field and additional forces besides the Lorentz force acting on electrons

TL;DR

This paper develops a constrained Hamiltonian framework for electrons in magnetic fields, incorporating quantum effects like Berry connection, revealing additional forces beyond the Lorentz force relevant to superconductivity and electrical phenomena.

Contribution

It introduces a novel Hamiltonian approach that includes quantum mechanical effects and additional forces acting on electrons in magnetic fields, extending classical force models.

Findings

Identification of forces beyond Lorentz force, including energy gradient and topological forces.

Quantum effects like Berry connection significantly influence electron dynamics.

Additional forces are crucial in understanding superconductivity and electrical behavior.

Abstract

We consider the forces acting on electrons in magnetic field including the constraints and a condition arising from quantum mechanics. The force is calculated as the electron mass, $m_e$, multiplied by the total time-derivative of the velocity field evaluated using the quantum mechanical many-electron wave function. The velocity field includes a term of the Berry connection from the many-body wave function; thereby, quantum mechanical effects are included. It is shown that additional important forces besides the Lorentz force exist; they include the gradient of the electron velocity field kinetic energy, the gradient of the chemical potential, and the `force' for producing topologically protected loop currents. These additional forces are shown to be important in superconductivity, electric current in metallic wires, and charging of capacitors.