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Physical Review A

Variationallowerbound on the scattering length

Leonard Rosenberg, Larry Spruch

Published Oct 1, 1975• 10 citations
Publisher

Abstract

The scattering length $A$ characterizes the zero-energy scattering of one system by another. It was shown some time ago that a variational upper bound on $A$ could be obtained using methods, of the Rayleigh-Ritz type, which are commonly employed to obtain upper bounds on energy eigenvalues. Here we formulate a method for obtaining a variational lower bound on $A$. Once again the essential idea is to express the scattering length as a variational estimate plus an error term and then to reduce the problem of bounding the error term to one involving bounds on energy eigenvalues. In particular, the variational lower bound on $A$ is rigorously established provided a certain modified Hamiltonian can be shown to have no discrete states lying below the level of the continuum threshold. It is unfortunately true that necessary conditions for the existence of bound states are not available for multiparticle systems in general. However, in the case of positron-atom scattering the adiabatic approximation can be introduced as an (essentially) solvable comparison problem to rigorously establish the nonexistence of bound states of the modified Hamiltonian. It has recently been shown how the validity of the variational upper bound on $A$ can be maintained when the target ground-state wave function is imprecisely known. Similar methods can be used to maintain the variational lower bound on $A$. Since the bound is variational, the error in the calculated scattering length will be of second order in the error in the wave function. The use of the adiabatic approximation in the present context places no limitation in principle on the accuracy achievable.

DOI
10.1103/physreva.12.1297

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The scattering length AAA characterizes the zero-energy scattering of one system by another. It was shown some time ago that a variational upper bound on AAA could be obtained using methods, of the Rayleigh-Ritz type, which are commonly employed to obtain upper bounds on energy eigenvalues. Here we formulate a method for obtaining a variational lower bound on AAA. Once again the essential idea is to express the scattering length as a variational estimate plus an error term and then to reduce the problem of bounding the error term to one involving bounds on energy eigenvalues. In particular, the variational lower bound on AAA is rigorously established provided a certain modified Hamiltonian can be shown to have no discrete states lying below the level of the continuum threshold. It is unfortunately true that necessary conditions for the existence of bound states are not available for multiparticle systems in general. However, in the case of positron-atom scattering the adiabatic approximation can be introduced as an (essentially) solvable comparison problem to rigorously establish the nonexistence of bound states of the modified Hamiltonian. It has recently been shown how the validity of the variational upper bound on AAA can be maintained when the target ground-state wave function is imprecisely known. Similar methods can be used to maintain the variational lower bound on AAA. Since the bound is variational, the error in the calculated scattering length will be of second order in the error in the wave function. The use of the adiabatic approximation in the present context places no limitation in principle on the accuracy achievable.
A minimum principle for the nonrelativistic scattering length was derived some years ago under the assumption that only a finite number of discrete states exist below the scattering threshold. The variational bound is applicable even when the bound-state wave functions are imprecisely known--they need only be accurate enough to give binding in a Rayleigh-Ritz calculation. The method is generalized here to apply to potential scattering described by the Dirac equation. An apparent difficulty associated with the existence of a continuum of negative-energy states, that is, the problem of ``variational collapse,'' is removed through the inclusion of second-order terms in the variational expression involving matrix elements of the square of the Dirac Hamiltonian. In the course of the derivation a relativistic version of the Hylleraas-Undheim theorem is developed. Applications of this theorem are described that provide a sufficient condition for the existence of a given number of bound states, lower bounds on the energy eigenvalues, and a systematic procedure for improving the accuracy of trial bound-state wave functions. A very simple model calculation was performed to illustrate the minimum property and the stability of the numerical procedure.
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A minimum principle for the calculation of the scattering length, applicable when the ground-state wave function of the target system is known precisely, has been available for some time. When, as is almost always the case, the target wave function is imprecisely known, a minimum principle is available but the simple minimum principle noted above is not applicable. Further, as recent calculations show, numerical instabilities usually arise which severely limit the utility of even an ordinary variational approach. The difficulty, which can be traced to the appearance of singularities in the variational construction, is here removed through the introduction of a minimum principle, not for the true scattering length, but for one associated with a closely connected problem. This guarantees that no instability difficulties can arise as the trial scattering wave function and the trial target wave function are improved. The calculations are little different from those required when the target ground-state wave function is known, and, in fact, the original version of the minimum principle is recovered as the trial target wave function becomes exact. A careful discussion is given of the types of problems to which the method can be applied. In particular, the effects of the Pauli principle, and the existence of a finite number of composite bound states, can be accounted for.
Previous work on the application to scattering theory of the Feynman path-integral formulation of nonrelativistic quantum mechanics is extended. By using the regularization $\mathrm{Tr}({e}^{\ensuremath{-}\ensuremath{\beta}H}\ensuremath{-}{e}^{\ensuremath{-}\ensuremath{\beta}{H}^{0}})$ of the divergent partition function $\mathrm{Tr}{e}^{\ensuremath{-}\ensuremath{\beta}H}$, where H0{H}^{0}H0 is the free-particle Hamiltonian, H=H0+VH={H}^{0}+VH=H0+V, and VVV is a short-range potential of the form $V(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}})$ which cannot support a bound state, we obtain a variational upper bound on the scattering length. This bound is expressed in terms of path integrals containing a trial functional $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{S}$, and reduces to the natural analog of the Rayleigh-Ritz result when $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{S}$ is of the standard form $\ensuremath{-}\ensuremath{\int}\frac{[{E}_{K}+\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{V}]dt}{\ensuremath{\hbar}}$, with EK{E}_{K}EK​ the kinetic energy and $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{V}$ a trial potential. Since $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{S}$ need not be of this form, the bound contains new features.
In recent work, the existence of stationary bounds on bound-state matrix elements for rather broad classes of quantum-mechanical operators was used to obtain a stationary upper bound on the scattering length for the collision of positrons with atoms whose target wave functions are not precisely known. This was done by obtaining calculable stationary bounds on the unknown matrix elements which appear in the formal expression for the bound on the scattering length. This method is extended in the present paper to electron-atom scattering for which it is necessary to obtain-in addition to bounds of the previous type-stationary bounds on the various exchange integrals. These latter cannot be obtained by means of techniques previously described. New techniques are presented for deriving such bounds, which are then used to obtain rigorous stationary upper bounds on the scattering length for the collisions of electrons with helium and with atoms heavier than helium.
Recently developed variational techniques for determining upper bounds on scattering lengths are applied to singlet and triplet scattering of zero-energy electrons by atomic hydrogen. The results obtained are not only rigorous but are in fact somewhat lower and therefore somewhat better than those previously obtained by variational methods. We find that the triplet and singlet scattering lengths, AT{A}_{T}AT​ and AS{A}_{S}AS​ respectively, satisfy the inequalities ${A}_{T}\ensuremath{\leqq}1.91{a}_{0}$ and ${A}_{S}\ensuremath{\leqq}6.23{a}_{0}$, where a0{a}_{0}a0​ is the Bohr radius. The only assumptions involved in the deduction of these results are that there be no bound triplet state and one and only one bound singlet state. The singlet trial function determined during the course of the calculation generates a singlet effective range, r0S{r}_{0S}r0S​, of about 2.7a02.7{a}_{0}2.7a0​. The triplet trial functions which were obtained were not sufficiently accurate to be useful in a determination of the triplet effective range, r0T{r}_{0T}r0T​.