Hall's variational principle for excited states

It is shown how excited state energies can be given upper bounds through the use of Hall's functional. Applications to atomic central field problems are worked out.     

Spectral variational principle for Green's functions

For a suitable approximation \(\tilde K\) (q, q′, τ) to the Dirac-Feynman Green's function of a quantummechanical system, the parameter \({\mathcal{L}\tilde K}\) is defined, where ?≡i?/?τ??. It is shown thatΔ≧0 andΔ→0 asK→K, the exact Green's function, thus providing a criterion on approximate Green's functions analogous in its role to the variational principle for wavefunctions. A second somewhat weaker criterion is also proposed, based on \(\Sigma \equiv \left[ {tr\tilde K*tr\mathcal{L}^2 \tilde K - |tr\mathcal{L}\tilde K|^2 } \right]_{\tau avg} \geqq 0\) . Recipes are given for projecting out continuum contributions toΔ or∑ and for analyzing for the discrete eigen-value spectrum.     

Husimi representation for stationary states

This paper considers a Husimi representation of quantum mechanics in which the (stationary) state of a system or ensemble is described by a Husimi function and an observable is described by a phase space function or distribution such that the expectation value of the observable is given by an integral over phase space of the product of that function or distribution and the Husimi function. The density matrix, Wigner function, and Husimi function are considered to be alternative ways of describing the state of a system or ensemble, and methods of recovering the Wigner function or density matrix from the Husimi function are discussed. The classical limits of the Wigner and Husimi functions and of the relationship between them are considered. © 1993 John Wiley & Sons, Inc.     

The wavefunction of non-stationary states in terms of complex coordinates and a related variational principle

The wavefunction of a decaying state is expressed in terms of complex coordinates as ψ(p) = a(θ)ψ _o(p) + b(θ)x(p), where the square integrable θ_o(p) and x(p) contain the localized and asymptotic information respectively. From Fano theory, we derive the relationship of a(?) and b(?) to the energy and width of the resonance state. This forms the basis for a new variatfonal principle for optimization of trial functions in the complex coordinate—energy plane.     

Conditional probability amplitudes and the variational principle

It is shown by counter-example, that when the theory of conditional probability amplitudes is applied to quantum mechanical systems no variational principle exists for the function U(R = fφ*(r/R)Hφ(r/R) dr, where φ(r/R) is the conditional probability amplitude and H is the total hamiltonian of the system.     

Time-dependent variational principle in quantum field theory

This paper proposes a direct, quantum-mechanical definition for Γ (effective action) using a two-step procedure for applying the variational principle to time-dependent quantum problems.     

Properties of variational transition states for association reactions

The relationships between the structures of microcanonical variational transition states for association reactions and potential energy surface properties are analyzed. The analysis is facilitated by the recent finding that in many cases the vibrational frequencies for the transitional modes in the association reaction vary exponentially with the reaction coordinate. An increase in the attenuation of the transitional mode frequencies results in a tighter transition state. Long-range minimum energy path potentials of the form −c/r″ have only one transition state at each energy. Certain types of association reactions may not have transition states.     

A variational principle for the excitation energy of a multielectron system

A variational principle is proposed for direct calculation of the excitation energy. The spectrum of collective excitations is derived for a homogeneous electron gas.     

Electron-molecule scattering with polarization using the schwinger variational principle

Electron-molecule scattering is formulated in an L² basis, using the Schwinger variational principle and including polarization via an ab initio optical potential. The necessary integrals are obtained in closed form. Electron-helium S-and P-wave phase shifts are calculated, and applications to molecules are mentioned.     

Novel variational definition of diabatic states

The requirement that diabatic states be dominated by a single electronic configuration is transformed into a variational principle. A very simple algorithm for the adiabatic—diabatic transformation is found. It is equally applicable to avoided and to true potential crossing situations.     

A variational principle for transition and density matrices and approximations to the equations-of-motion

A general variational principle for transition and density matrices is proposed. The principle is closely related to Rowe's variational treatment of the equations-of-motion method. It permits the simultaneous construction of coupled approximations for two eigenstates, and it is a straightforward extension of the usual variational method.     

A variational principle for weak stimulated absorptions with energy minimizing fields

Optimal time-dependent perturbations in agreement with the Bohr condition for spectroscopic transitions are obtained variationally for two-level systems via the first-order perturbation model. The criterion of optimality is that specified weak absorptions are achieved with minimal applied energy. Results obtain for fixed perturbation intervals of arbitrary length.     

Self-dual phase-space representation of quantum mechanics and the variational principle

The important duality problem of phase-space representations of quantum mechanics is discussed and the class of self-dual representations is established. It is shown that self-dual phase-space representations may be endowed with a Hilbert space structure resembling the standard picture of quantum mechanics. A stationary variational principle is then formulated, enabling one to solve eigenproblems of observables directly in phase space. A modified, more usable form of this variational principle is also given together with an example of practical application to the 1D Coulomb system. © 1994 John Wiley & Sons, Inc.     

Nonsingular constraints in time‐dependent variational principle for parametrized wave functions

In this work, we consider two conditions required for the nonsingularity of constraints in the time‐dependent variational principle (TDVP) for parametrized wave functions. One is the regularity condition which assures the static nonsingularity of the constraint surface. The other condition is the second‐class condition of constraints which assures the dynamic nonsingularity of the constraint surface with a symplectic metric. For analytic wave functions for complex TDVP‐parameters, the regularity and the second‐class conditions become equivalent. The second‐class condition for expectation values is reduced to the noncommutability of the corresponding quantum operators. The symplectic singularity of the equation of motion of TDVP is also shown to be a local breakdown of the second‐class condition in an extended canonical phase‐space. © 2012 Wiley Periodicals, Inc.