Rayleigh–ritz method for excited quantum states via nonlinear variations without constraints: Role of supersymmetry

Abstract

Quantum mechanical variation principle in the form of energy minimization is applicable only to ground states of systems, or, at best, states of lowest energies of given symmetries, provided the symmetry information is embedded in chosen trial functions. Thus, for bound quantum states with specified choices of trial functions involving nonlinear parameters, scope of the principle is severely restricted. A pedagogic way out is to enforce exact orthogonality of the chosen function with all exact lower energy states. In actual practice, this limits one to opt for linear variations where upper bound to each state is obtained in a single run. In this work, the motivation is to explore if there exists at all a way to determine optimized wave functions and energies for excited states via nonlinear variations but without any constraints, even for simple systems. Realizing that the major problem in excited-state nonlinear variations is concerned with the variations of nodal positions, at least for problems reducible to one dimension, we seek a route via which nodes could be fixed beforehand, so that the information gained may be subsequently utilized to construct a suitable nonlinear trial function and carry out a straightforward optimization. To achieve this end, the idea of supersymmetric quantum mechanics has been used quite profitably, yielding the nodal structure of the excited states. Workability of the strategy for several excited-state wave functions and their properties is demonstrated by choosing the problems of spherical Stark effect on hydrogen atom and anharmonic oscillator. V C 2011 Wiley Periodicals, Inc. Int J Quantum Chem 112: [960][961][962][963][964][965][966][967][968][969][970][971] 2012

A simulated annealing based study of the effect of Gaussian perturbation in quartic oscillator and quadratic double well potentials

“…Then the average energy of state θ k will be an upper bound to the true one [2][3][4]. In case of simple systems, it is notable that the crux of the problem lies in our lack of knowledge about the precise positions of nodes of excited-state wave functions [5]. Indeed, if we take a trial function with one or more variable nodal positions, it would turn out that an unconstrained minimization of energy is achieved only by placing the nodes farther beyond the classical turning points, thus getting closer and closer to the actual ground state.…”