An analysis of potential scattering in terms of a Gaussian wavepacket is given. It is used to derive rigorously a bound alpha on the ratio of the modulus of the difference between the exact and Born transition amplitudes for any given momentum transfer to the modulus of the Born amplitude in the forward direction. Bounds on alpha are given in the case of a square-well potential, a screened Coulomb potential and an exponential potential; the last two are applied to elastic electron-helium scattering.
An analysis of potential scattering in terms of a Gaussian wavepacket is given. It is used to derive rigorously a bound alpha on the ratio of the modulus of the difference between the exact and Born transition amplitudes for any given momentum transfer to the modulus of the Born amplitude in the forward direction. Bounds on alpha are given in the case of a square-well potential, a screened Coulomb potential and an exponential potential; the last two are applied to elastic electron-helium scattering.
We treat the nonrelativistic quantum mechanical scattering of a wavepacket (for a single, charged, spinless particle) of well-resolved momentum incident on a fixed Coulomb potential. To implement this, we calculate the incoming and outgoing probability amplitudes for normalized state vectors on the basis of eigenvectors of total momentum magnitude and total angular momentum. We show that the method of partial wave analysis becomes applicable for this long range potential with the use of wavepackets. We find a relation between the finite probability and the differential cross section that is seen to hold sufficiently far from the singularity of the latter in the forward direction. The resulting differential cross section reduces to the familiar Rutherford form for scattering angles sufficiently far from the forward direction. We find that the interaction causes small time shifts in the detection probabilities. For our particular scattering geometry, we see, at low energies, a shadow zone of low probability for small angles around the forward direction, a phenomenon that cannot be predicted by the standard treatment.
We consider scattering by screened Coulomb and screened short-range potentials. We prove, in Born approximation, that the cross section for a sharply cutoff Coulomb potential does not converge to the Rutherford cross section as the screening radius ρ→∞. On the other hand, for certain ’’smooth’’ screening functions we show that the screened Coulomb cross section does approach the Rutherford cross section as ρ→∞. In the case of short-range potentials, we prove (exactly, not just in Born approximation) that the screened cross section approaches the unscreened cross section as ρ→∞, whether the screening function is smooth or not.
We investigate possible characterizations of ``short-range'' interactions by means of conditions on the total transition probability ∥Tψ∥2. We desire to require that the total transition probability decrease exponentially as the displacement of the wavepacket ψ increases. It is shown that this can only be done for certain types of sequences of wavepackets, and a criterion is developed for selecting these ``proper'' wavepackets. We then show for all such interactions that what is essentially the square of the scattering amplitude is analytic, in the cosine of the scattering angle, in an ellipse which always includes the physical region. We then compare these results with the Schrödinger potential theory and thereby relate the type of the exponential decrease of a potential to the type of the exponential decrease of the corresponding transition probability. Finally, our results are compared with similar results previously obtained by others working on this problem.
Several problems of nonrelativistic scattering by analytic potentials are studied, where both the energy and the momentum transfer are large. Specifically, we find asymptotically the reflection coefficients from the one-dimensional potentials $V{e}^{x}{(1+{e}^{x})}^{\ensuremath{-}2}$, $V{e}^{\frac{x}{2}}{(1+{e}^{x})}^{\ensuremath{-}1}$, and $V\mathrm{exp}(\ensuremath{-}{x}^{2})$, and the scattering amplitudes for their three-dimensional generalizations $V{e}^{r}{(1+{e}^{r})}^{\ensuremath{-}2}$, $V{e}^{\frac{r}{2}}{(1+{e}^{r})}^{\ensuremath{-}1}$, and $V\mathrm{exp}(\ensuremath{-}{r}^{2})$. The results are compared with these obtained from Born approximation, and it is found that the first Born approximation gives the correct answer asymptotically in the cases $V{e}^{\frac{x}{2}}{(1+{e}^{x})}^{\ensuremath{-}1}$ and $V{e}^{\frac{r}{2}}{(1+{e}^{r})}^{\ensuremath{-}1}$, but not in the other cases. Conjectures about more general cases are also given.
In this chapter we discuss the quantum theory of scattering of incident particles from the target where an incident beam has point particles, scattering is elastic, the scattering potential is time independent, there is no multiple scattering, and the interaction between an incident particle and a scattering center is described by the potential that depends only on relative positions of particles. Scattering cross-section and frame of reference, asymptotic expansion of wave function and formal expression for scattering amplitude, partial wave analysis, integral equation, and the Born approximation are covered in the chapter, in a broader sense. Free particle and asymptotic solutions, scattering amplitude and phase shift, optical theorem, scattering length, phase shift and its interpretation, scattering by a square well potential and by hard sphere potential, and interpretation of the phase shift and its expression are explained with the use of partial wave analysis. Scattering from bare and screened potentials within the Born approximation, validity of the Born approximation, and the transformation from the center of a mass coordinate system to a laboratory coordinate system are written towards the end of the chapter.
Quantum-mechanical scattering states are energy eigenstates obeying particular boundary conditions, whose behavior at infinity encodes the S-matrix which defines the outcoming of scattering experiments. With an eye toward numerical algorithms for computing nonrelativistic S-matrices, we present a family of stable minimum principles for scattering states. States that approximately satisfy these minimum principles are shown to have a bounded difference with the true scattering states. These minimum principles and stability estimates can be used to obtain rigorous bounds on scattering amplitudes. We show that these minimum principles are applicable to momentum-dependent potentials, long-range (Coulomb) interactions, and elastic or inelastic scattering of bound states.