The Variational Principle in Quantum Kinetics

Mitsuo Shimoji

Anatoly Yu. Zakharov

A system of quasi-particles with scattering on arbitrary static potential is considered by means of non-equilibrium Green function method in Kadanoff-Baym formulation. The problem of kinetic coefficients evaluation can be solved with Kadanoff-Baym quantum kinetic equations for the system in the presence of an external field. A variational method of solution for linearized over-external field quantum kinetic equations is proposed.

David E Khmel'nitskiĭ, B. A. Muzykantskiǐ

Kenneth R. Brownstein

A new variational principle for the energy eigenvalues of a confined quantum system is presented. Whereas the exact wave function \ensuremath{\Psi} must vanish on the bounding surface of the region, the trial function \ensuremath{\psi} in this principle need not obey any specific boundary condition. Calculationally, the method is similar to the conventional variational method except that kinetic energy turns out to be a weighted average of -F\ensuremath{\psi}${\mathrm{\ensuremath{\nabla}}}^{2}$\ensuremath{\psi} d\ensuremath{\tau} and F(\ensuremath{\nabla}\ensuremath{\psi})\ensuremath{\cdot}(\ensuremath{\nabla}\ensuremath{\psi})d\ensuremath{\tau} in the ratio 2:-1. Although the principle is not a definite (minimum) one, good results are obtained in several examples.

Albrecht Huber

Xingyu Zhang, Jinke Yu, Qingyong Meng

This work describes a geometric framework on molecular reaction dynamics based on the variational principle, where the Schr{ö}dinger equation must be solved to ``see'' how a reaction occurs. First, the mathematical preliminaries are given by discussing the principle of least action and the mountain pass theorem. Second, we discuss the physical preliminaries, including the principle of equivalence for deriving the kinetic energy operator (KEO) and artificial intelligence (AI) techniques to build the potential energy surface (PES) in general spacetime. Moreover, we simplified electromagnetic interactions in curved spacetime within the molecular system and consequently, we are able to construct the nuclear Hamiltonian in nonzero curvature spacetime. This indicates possibility to introduce gauge fields through the curvature, such as additional term in the nuclear KEO near a conical intersection. Third, the single-particle approximation provides a powful ansatz to solve the Schr{ö}dinger equation by variational principle. Thus, one can formulate the variational approaches for either electronic structure or quantum dynamics. In this work, based on previous discussions ({\it Phys. Chem. Chem. Phys.} {\bf 27} (2025), 20397) we unified them by a geometric description, where the geometric phase is naturally introduced. Finally, due to optimization characteristic of the present theory, further discussions on the present theory from optimization insight are also given, including two postulates, generative AI techniques, role of perturbation, and Markov process in optimization.