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Journal of Physics A Mathematical and General

Improved constraints on states bound by a class of potentials

Alan K. Common, A Martin

Published Sep 11, 1987• 8 citations
Publisher

Abstract

Recent work on states bound by a central potential whose Laplacian is positive is extended to cover a wider class of potentials for which energy level ordering theorems can be proved. This class includes power law potentials in particular. Constraints on the moments of the radial distance are improved and similarly for the kinetic energy. These constraints are shown to provide tight bounds on the energies of ground states for power law potentials, and on their variation with angular momentum. They are also used to bound the position of the maximum of the wavefunction for these states.

DOI
10.1088/0305-4470/20/13/028

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Recent work on states bound by a central potential whose Laplacian is positive is extended to cover a wider class of potentials for which energy level ordering theorems can be proved. This class includes power law potentials in particular. Constraints on the moments of the radial distance are improved and similarly for the kinetic energy. These constraints are shown to provide tight bounds on the energies of ground states for power law potentials, and on their variation with angular momentum. They are also used to bound the position of the maximum of the wavefunction for these states.
We derive eigenstate comparisons from ordered Schrödinger operators, extending energy bounds to overlaps, spatial distributions and nodes. For three form-ordered operators with discrete comparison ground states, suppose the upper ground-state energy lies in the lower operator's ground-state gap. Three comparison spectral values then bound all three squared ground-state overlaps and guarantee a simple discrete intermediate ground state. Related gap estimates control low-energy subspaces and individual excited-state overlaps under additional gap conditions. In one dimension and radially, a single crossing of the potential difference relative to the energy shift orders ground-state likelihood ratios and monotone observables; a nondecreasing difference suffices. The results cover asymmetric wells and shortened right Dirichlet walls, with coordinatewise comparisons for submodular reference potentials and coordinatewise nondecreasing perturbations. A two-sided Sturm argument orders corresponding internal nodes for nondecreasing potential differences, even between orthogonal one-dimensional excited states. For Yukawa ground states, Hulthén comparisons bound overlaps, radial moments and the density at the nucleus; the latter two-sided bounds hold for screening 0<δ<2/60<δ<2/\sqrt60<δ<2/6​. Optimizing scaled Hulthén majorants gives a lower bound of approximately 1.136111.136111.13611 for n2δc(n,0)n^2δ_c(n,0)n2δc​(n,0), where δc(n,0)δ_c(n,0)δc​(n,0) is the critical screening for the nnnth sss-wave bound state. This bound is optimal within that two-parameter comparison family.
The Rényi and Shannon entropies are information-theoretic measures, which have enabled to formulate the position–momentum uncertainty principle in a much more adequate and stringent way than the (variance-based) Heisenberg-like relation. Moreover, they are closely related to various energetic density functionals of quantum systems. Here we derive upper bounds on these quantities in terms of the second-order moment 〈r2〉 for general central potentials. This improves previous results of this type. The proof uses the Rényi maximization procedure with a covariance constraint due to Costa et al. [in Proceedings of the Fourth International Workshop on Energy Minimization Methods in Computer Vision and Pattern Recognition (EMMCVPR), edited by A. Rangarajan, M. A. T. Figueiredo, and J. Zerubia (Springer-Verlag, Lisbon, 2003), [Lect. Notes Comput. Sci. 52, 211 (2003).]] The contributions to these bounds coming from the radial and angular parts of the physical wave functions are taken into account. Finally, the application to the d-dimensional (d ⩾ 3) hydrogenic and oscillator-like systems is provided.
Rigorous and universal bounds on frequency moments of one-particle densities in terms of radial expectation values in the conjugate space are obtained. The results, valid for any d-dimensional quantum-mechanical system, are derived by using Rényi-like position-momentum inequalities in an information-theoretical framework. Especially interesting are the upper bounds on the Dirac exchange and Thomas-Fermi kinetic energies, as well as the disequilibrium or self-similarity of both position and momentum distributions. A variety of bounds for these functionals in a given space are known, but most usually in terms of quantities defined within the same space. Very few results including a density functional on one space, and expectation values on the conjugate one, are found in the literature. A pioneering bound on the disequilibrium in terms of the kinetic energy is improved in this work. A numerical study of the aforementioned relationships is carried out for atomic systems in their ground state. Some results are given in terms of relevant physical quantities, including the kinetic and electron-nucleus attraction energies, the diamagnetic susceptibility and the height of the peak of the Compton profile, among others.
As an illustration of the problem of handling classical constraints in wave mechanics, the correct wave equation is obtained for a model consisting of two particles, one of them constrained to have a fixed radial distance from the origin and the other free save for an angular constraint confining the two radius vectors to be coincident. The particle with fixed radius is here equivalent to a thin rigid rod. The wave equation is obtained by starting with a system of two unconstrained particles in which the constraints are replaced by restoring forces, writing the wave equation in appropriate polar coordinates by transforming the Laplacian from Cartesian coordinates, and then going to the limit of strong restoring forces. If, in this problem of a rotating rigid rod collinear with the position of an otherwise free particle, one follows the rather usual procedure of using the derivative with respect to the conjugate angular coordinate as the operator for a canonical angular momentum, one obtains in the radial wave equation in good approximation a "centrifugal potential" term with the familiar coefficient l(l+1)l(l+1)l(l+1). The limiting procedure shows that the correct result instead has a similar term but with coefficient l(l+1)+1l(l+1)+1l(l+1)+1. Also the relation between the wave function and the solution of the radial equation is not the same as for a single particle. It is thereby shown that an application sometimes made to angular coordinates of a metric transformation in analogy to one from Cartesian to curvilinear coordinates is not strictly correct. The result derived by the limiting procedure is related to recent discussions of 16O^{16}\mathrm{O}16O-16O^{16}\mathrm{O}16O scattering.