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Journal of Physics A Mathematical and General

Bertlmann-Martin inequality for a weakly bound particle

Monique Lassaut, M Keriaki, R.J. Lombard

Published Feb 13, 1998• 2 citations
Publisher

Abstract

The Bertlmann-Martin (BM) inequality for the rms radius is checked against the exact value in the case of a particle in a weak central potential having a single bound state. A number of usual potentials are considered in one and three dimensions. The BM inequality is poorly saturated but the correction factor needed to recover the exact rms radius is weakly depending on the shape of the potential, except for the Hulthén potential. For potentials having a hard-core component, we found that , which depends on the bound-state energy, tends to a universal curve as the hard core radius increases.

DOI
10.1088/0305-4470/31/6/011

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The Bertlmann-Martin (BM) inequality for the rms radius is checked against the exact value in the case of a particle in a weak central potential having a single bound state. A number of usual potentials are considered in one and three dimensions. The BM inequality is poorly saturated but the correction factor needed to recover the exact rms radius is weakly depending on the shape of the potential, except for the Hulthén potential. For potentials having a hard-core component, we found that , which depends on the bound-state energy, tends to a universal curve as the hard core radius increases.
Reconstructing a radial (1D) quantum potential, [Formula: see text], from a few bound-state energies is a long-standing inverse problem because limited spectral data must constrain an entire potential. We present a Laplace-moment reconstruction pipeline that links the Bertlmann–Martin gap bound to generalized Bertlmann–Martin (GBM) even-moment ladders, continues the Laplace transform with Padé approximants, and inverts the transform to recover [Formula: see text] and [Formula: see text]. Odd moments are supplied by a physically consistent interpolation scheme. Benchmark settings and diagnostics for Coulomb, harmonic oscillator, Hulthén, Kratzer, and hyperbolic-well cases are stated so each approximation stage can be assessed under a common empirical basis. The conclusions are therefore limited to the reported benchmark settings rather than offered as universal method claims.
We study the radial Schr\"odinger equation for a particle of mass mmm in the field of a singular attractive g2∕r4{g}^{2}∕{r}^{4}g2∕r4 potential with particular emphasis on the bound-states problem. Using the regularization method of Beane et al. [Phys. Rev. A 64, 042103 (2001)], we solve analytically the corresponding ``renormalization-group flow'' equation. We find in agreement with previous studies that its solution exhibits a limit cycle behavior and has infinitely many branches. We show that a continuous choice for the solution corresponds to a given fixed number of bound states and to low-energy phase shifts that vary continuously with energy. We study in detail the connection between this regularization method and a conventional method modifying the short-range part of the potential with an infinitely repulsive hard core. We show that both methods yield bound-states results in close agreement even though the regularization method of Beane et al. does not include explicitly any new scale in the problem. We further illustrate the use of the regularization method in the computation of electron bound states in the field of neutral polarizable molecules without dipole moment. We find the binding energy of sss-wave polarization bound electrons in the field of C60{\mathrm{C}}_{60}C60​ molecules to be 17 meV for a scattering length corresponding to a hard-core radius of the size of the molecule radius $(\ensuremath{\sim}3.37\phantom{\rule{0.3em}{0ex}}\mathrm{\AA{}})$. This result can be further compared with recent two-parameter fits using the Lennard-Jones potential yielding binding energies ranging from 3 to 25 meV.
It is shown that for nonrelativistic central potentials of specific forms, the bound state wavefunctions u (r) have the property that G (r) ≡ℱ r0dx u (x)(∂u (x)/∂μ) ⩽0 for all r. Here μ is the reduced mass. Thus, for such potentials, the probability that a particle lies within a spherical shell of radius r is a monotonically increasing function of μ. The forms for which this property has been established are (a) V (r) =Crε, −2<ε<∞; (b) V (ir) =C lnr, and (c) S waves in the potential, V (r) =−V0 (0⩽r<a), V (r) =0 (a⩽r<∞). For all monotonically increasing potentials, G (r) ≳0 for nodeless states. It is shown explicitly that G (r) cannot be nonnegative for all bound states in an arbitrary monotonically increasing potential. The question remains open regarding the widest class of potentials for which G (r) is nonnegative for all bound states.
Relativistic effects on the relation between the root-mean-square radius rD{\mathit{r}}_{\mathit{D}}rD​ of the bound state and the scattering length a are examined for models which simulate the triplet S state of the two-nucleon system. Two models are considered; one is based on the one-body Dirac equation and the other on the two-body Dirac equation. The relation is expressed as (rD{\mathit{r}}_{\mathit{D}}rD​/a)2{)}^{2})2=(c0{\mathit{c}}_{0}c0​+c1{\mathit{c}}_{1}c1​x1{\mathit{x}}^{1}x1+c2{\mathit{c}}_{2}c2​x2{\mathit{x}}^{2}x2+...)/8, where x==\ensuremath{\alpha}r0{\mathit{r}}_{0}r0​; \ensuremath{\alpha} is related to the binding energy and r0{\mathit{r}}_{0}r0​ is the effective range. This expansion is compared with the corresponding expansion of the nonrelativistic Schr\"odinger case. The first three terms are shape independent. The c0{\mathit{c}}_{0}c0​ (=1) and c1{\mathit{c}}_{1}c1​ (=0) are the same as those of the nonrelativistic case but a relativistic correction appears in c2{\mathit{c}}_{2}c2​.
A new method is proposed for treating the hard core potential with a small core radius a. This is an alternative to the method based on Fermi's pseudopotential. Various shortcomings of the latter method are removed with the present method. These shortcomings include the penetrability of hard spheres and incompatibility with the completeness of the unperturbed states, and they result in the invalidity of the pseudopotential method even in applications to effects that are of low order in powers of a. The present method is developed taking account of the three-particle correlation, which is a higher-effect plagued by problems of divergence and the nonanalyticity of the relevant energy as a function of the potential strength. Treatment of the three-particle correlation requires a careful preliminary study of the two-particle interaction in many-body systems, particularly when a hard core potential is involved. In this report, we present basic ideas and obtain improvement over the methods appearing in the literature.
In most textbooks on quantum mechanics, it is asserted that the bound states of a localized potential have negative energies. To stress that this view should be taken with caution, we give several examples of scalar potentials that tend to zero at infinite distance and yet possess one or several discrete positive bound states for the Schrödinger single-particle wave equation. The consequence that usual tunneling may not occur, though it should according to a superficial view, is of interesting pedagogical value.