Variational procedure for time-dependent processes.

Area of Science:

• Quantum mechanics
• Statistical mechanics
• Theoretical physics

Background:

• Describing the time evolution of quantum systems is crucial.
• Existing methods can be complex for arbitrary density matrices.
• Connecting quantum dynamics with macroscopic phenomena like hydrodynamics is challenging.

Purpose of the Study:

• To propose a simple variational Lagrangian for density matrix time evolution.
• To demonstrate the formalism's applicability to diverse physical systems.
• To validate the method using a two-level system.

Main Methods:

• A variational Lagrangian is formulated using density factorization.
• The Lagrangian includes only kinetic energy terms.
• The method is tested on a two-level system interacting with an oscillator and in a dissipative mode.

Main Results:

• The proposed variational Lagrangian successfully describes the time development of arbitrary density matrices.
• The formalism unifies concepts from quantum mechanics, hydrodynamics (Navier-Stokes equations), and transport theory.
• It naturally recaptures the Rayleigh-Onsager least dissipation function condition.

Conclusions:

• The variational Lagrangian offers a flexible and unifying framework for studying time evolution in various physical systems.
• This approach provides a simplified yet powerful tool for theoretical and applied research.
• The method's success on the two-level system demonstrates its practical utility.