Physical Review

Variational Principles for the Wave Function in Scattering Theory

Saul Altshuler

Published Mar 1, 1958• 10 citations

Abstract

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.

DOI: 10.1103/physrev.109.1830

Related papers

Variational Principle for Potential Scattering

G. H. Goedecke, E. Brown

Potential scattering is treated by a minimum variational principle. The method, known as the error method, is applicable to problems which can be cast as linear inhomogeneous equations. The method makes use of a non-negative functional which reduces to zero for the exact solution. The magnitude of the functional for approximate solutions provides an indication of the accuracy of the wave function. The method is illustrated by some examples, and compared with the Schwinger and Kohn principles.

Physical ReviewMar 15, 19622 citations

Principles of Limiting Absorption and Limiting Amplitude in Scattering Theory. I. Schrödinger's Equation

Farouk M. Odeh

The outgoing solution of the time-independent Schrödinger equation, with a suitably restricted real potential, is shown to be the uniform limit of the square-integrable solutions of the same equation with complex energy as the imaginary part of the energy tends to zero. Under further restrictions on the potential, it is also shown that the solution to the initial-value problem for the time-dependent Schrödinger equation tends to the outgoing solution as time increases indefinitely.

Journal of Mathematical PhysicsNov 1, 19618 citations

Quantum scattering in 1D revisited

L. R. Ram‐Mohan

Abstract A new approach to Quantum Scattering using the Theory of Sources and Absorbers is developed in these chapters. Existing theories of quantum scattering usually calculate the scattered wavefunction and the scattering cross-section by obtaining the solution to Schrödinger’s equation and ignoring the possibility of representing the problem in terms of the principle of stationary action. The action integral approach has several advantages, including quadratic convergence to the accurate results in the variational approach; also, the symmetry of the Lagrangian is readily manifest. Furthermore, the “near-field” wavefunctions and evanescent wave contributions are invariably ignored. Computational methods using discretization of space require the equivalent of “radiation boundary conditions” at large distances, and these are not cleanly implementable.

Apr 1, 20260 citations

Variational Principle for Derivation Equations

G�nter Sauermann, H. Turschner

It is pointed out that the Schwinger variational principle of scattering theory applies to the case of linear and nonlinear relaxation problems in quantum statistics. By means of this principle it is possible to derive closed sets of equations for expectation values. To illustrate this variational method and to clarify the connection to other standard approaches some simple examples are treated for which the equations of motion are already known.

Jan 1, 19810 citations

VARIATIONAL PRINCIPLES FOR NONSTATIONARY QUANTUM MECHANICAL PROBLEMS AND PERTURBATION THEORY

Yu. N. Demkov

A variational principle is formulated which is invariant with respect to normalization of wave functions and which can be used in the case of nonorthogonal wave functions of the initial and final states. The problera of uniqueness of perturbation theory in this case is considered. The possibility of employing the variational principle for solution of problems concerning decay of the quasi-stationary state and charge exchange is discussed. The main equations for charge exchange at large impact parameters are derived from the variational principle. (auth)

Jun 1, 19600 citations

Variational principles and second-order Lagrangian densities

G J N Brown, S F C O'Rourke, Derrick S. F. Crothers

The functionals of the Kohn and Sil variational principles of atomic collision theory contain second-order spatial derivatives of the wave-function. Ipso facto the "second-order" generalised Euler-Lagrange differential equation is a necessary condition for these functionals to yield a stationary value. To illustrate this we provide a classical prescription for the time-dependent and time-independent quantum potential scattering problems, generalising the latter to the three-body problem. As practical examples where the presence of second-order derivatives in the Lagrangian density makes a non-trivial contribution to the equations of motion, we consider the optimisation of translation factors in a trial wave-function using the Sil variational principle and the optimisation of a variable charge using the Kohn variational principle. As further examples of Lagrangian densities where second-order derivatives may occur we briefly mention electromagnetism and relativistic quantum mechanics.

Physica ScriptaFeb 1, 19943 citations