Lagrangian and Hamiltonian, Lorentz Force in Quantum Mechanics

Pierre Amiot, James J. Griffin

Thomas Durt, Pierre Pelcé

C E Smith

Methods are presented for extending a promising formalism that incorporates dissipative forces into Quantum Mechanics. Hamiltonians and Lagrangians are constructed that have additional force and frictional terms while still being consistent with the rules of the formalism. The simple systems of a damped particle in a force field and a quadratically damped oscillator are discussed.

P. A. M. Dirac

Gabriele Carcassi, C. Aidala, David J Baker +1 more

The aim of this work is to show that particle mechanics, both classical and quantum, Hamiltonian and Lagrangian, can be derived from few simple physical assumptions. Assuming deterministic and reversible time evolution will give us a dynamical system whose set of states forms a topological space and whose law of evolution is a self-homeomorphism. Assuming the system is infinitesimally reducible—specifying the state and the dynamics of the whole system is equivalent to giving the state and the dynamics of its infinitesimal parts—will give us a classical Hamiltonian system. Assuming the system is irreducible—specifying the state and the dynamics of the whole system tells us nothing about the state and the dynamics of its substructure—will give us a quantum Hamiltonian system. Assuming kinematic equivalence, that studying trajectories is equivalent to studying state evolution, will give us Lagrangian mechanics and limit the form of the Hamiltonian/Lagrangian to the one with scalar and vector potential forces.