Variational Principle for Schrödinger–Pauli Theory

Michele Cini

K. Dechoum, H. M. França, Coraci Pereira Malta

Abraham Klein, Toshio Marumori, Tsutomu Une

The time-dependent variational principle of the Schr\"odinger equation is applied to a formally exact solution of the Schr\"odinger equation whose variational elements are operators which define a collective subspace of the many-body system under study. This generalizes the procedure employed to derive time-dependent Hartree-Fock theory. Four distinct formally exact time-independent variational principles, including several familiar forms, are derived. Application to a class of exactly soluble models, studied in the vibrational regime, illustrates the different modes of implementation of two of the principles. The general theory of large amplitude collective motion is derived. It is shown that the principles can be applied equally well, using either boson or fermion pair degrees of freedom. Some aspects of the relation to the semiclassical limit are discussed.[NUCLEAR STRUCTURE Time-dependent variational principle for collective motion. Invariance principle of Schr\"odinger equation. Generalized coherent state trial function. Exact time-independent variational principles. Variational principle for intrinsic state. Cranking variational principle. Large amplitude collective motion. Semiclassical limit.]

Angel Ricardo Plastino, Christophe Vignat, Angelo Plastino

Abstract A classical field theory for a Schrödinger equation with a non-Hermitian Hamiltonian describing a particle with position-dependent mass has been recently advanced by Nobre and Rego–Monteiro (NR) [Phys. Rev. A 88 (2013) 032105 ]. This field theory is based on a variational principle involving the wavefunction Ψ(x,t) and an auxiliary field ϕ(x,t). It is here shown that the relation between the dynamics of the auxiliary field ϕ(x,i) and that of the original wavefunction Ψ(x,t) is deeper than suggested by the NR approach. Indeed, we formulate a variational principle for the aforementioned Schrödinger equation which is based solely on the wavefunction Ψ(x,t). A continuity equation for an appropriately defined probability density, and the concomitant preservation of the norm, follows from this variational principle via Noether's theorem. Moreover, the norm-conservation law obtained by NR is reinterpreted as the preservation of the inner product between pairs of solutions of the variable mass Schrödinger equation.

Viraht Sahni

Saul Altshuler

Variational principles are designed for the solution of the Schr\"odinger equation when a point source is placed in the presence of an inhomogeneous, absorbing medium represented by an arbitrary complex potential function. When the point source is allowed to recede to infinity, these stationary structures reduce to variational principles for the wave function in the standard scattering problem, namely the outgoing solution to the Schr\"odinger equation for an incident plane wave. Finally in the asymptotic region, the well-known bifunctional variational principles for the transition amplitudes arise automatically from the stationary forms for the wave function describing the standard scattering problem. A few examples leading to variationally improved wave functions are discussed.