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.
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.
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.
The static approximation, which neglects the distortion of the (complex) target, that is, virtual excitations, during the collision of a particle with the target, was shown previously to give an upper bound on the scattering length when the number of bound states supported by the static and true Hamiltonians are the same. It represents an underadjustment of the target to the field of the scattering particle. It is perhaps then to be expected that the adiabatic approximation, in which the target wave function is calculated for each value of the incident particle coordinate as if the incident particle were fixed at the given point and which represents therefore an overadjustment of the target to the field of the incident particle, might provide the opposite bound, the lower bound, on the scattering length. This possibility is explored for the determination of scattering lengths. The bound property is shown to hold in cases where both the true and adiabatic Hamiltonians support the same number of bound states; some (reasonable) subsidiary conditions must also be satisfied. Possible extensions of the formulation to scattering with nonzero energy and with exchange are briefly mentioned. The theory is then applied to e+H and e+He scattering at zero incident energy; a reasonably accurate bound for the latter system is obtained, but the adiabatic approximation gives only a very crude, though rigorous, bound on the e+H scattering length. A connection between the generalized optical potential and the adiabatic potential is derived using projection-operator techniques, thus further clarifying the physical content of the adiabatic picture. Bounds are also obtained for atom-atom and ion-atom collisions; they are interesting only if the electron-proton mass ratio is not thought of as vanishingly small. Some consideration is given to pH scattering. Lower bounds on electron-atom scattering lengths are more difficult to obtain.
A trial wavefunction, Psi 1 , which contains both linear and nonlinear parameters is constructed using an approximate target wavefunction, Psi 1 is related to a quantity, a v , which becomes the Kohn scattering length if the target wavefunction is exact, through an integral expression corresponding to a slight modification of the usual Kohn variational method. It is conjectured that if the values of a v display a local minimum when all the nonlinear parameters of Psi 1 are varied, then this local minimum value of a v is an upper bound on the exact scattering length. Calculations on the zero-energy positron-helium scattering system using Hylleraas-type approximations to the helium ground-state wavefunction are analysed using this criterion. The use of either a six-parameter or a ten-parameter Hylleraas-type approximation to the helium ground-state wavefunction can give good results in scattering calculations for the positron-helium system at zero energy.
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.
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 and AS respectively, satisfy the inequalities ${A}_{T}\ensuremath{\leqq}1.91{a}_{0}$ and ${A}_{S}\ensuremath{\leqq}6.23{a}_{0}$, where 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, of about 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.
Simple analytical expressions are derived for the bound-state energy as well as for the resonance energy and width in the case of the Gaussian potential. These accurate formulas may find applications in nuclear physics, in studies of low-energy electron–atom collisions, and in simulations of few-body properties in Bose–Einstein condensates and Fermi gases. An important feature of the present approach is that the formulas rely solely on the knowledge of a set of real bound-state energies. In addition, an analogous expression is obtained for the s -wave scattering length. This formula is constructed so as to yield highly accurate values of the scattering length in the ground-state region, where the scattering length becomes large.