Applying the commonly used partial wave method in the quantum theory of collisions, the scattering cross-sections of the two- and three-dimensional attractive and repulsive potentials are expressed as sums of phase shifts. With the help of the Maple software program, the phase shifts and cross-sections are plotted as functions of the kinetic energy and potential height, and it is demonstrated that the systematic convergence of the cross-sections as higher angular-momentum phase shifts are included in the summations. It is found that the cross-sections of the attractive potential oscillate with both the kinetic energy and the potential strength, even though the phase shifts are monotonic. It is also shown that in the case of the repulsive potential, both the phase shifts and cross-sections change profoundly when the kinetic energy of the incident particle is lower than the potential height.
Applying the commonly used partial wave method in the quantum theory of collisions, the scattering cross-sections of the two- and three-dimensional attractive and repulsive potentials are expressed as sums of phase shifts. With the help of the Maple software program, the phase shifts and cross-sections are plotted as functions of the kinetic energy and potential height, and it is demonstrated that the systematic convergence of the cross-sections as higher angular-momentum phase shifts are included in the summations. It is found that the cross-sections of the attractive potential oscillate with both the kinetic energy and the potential strength, even though the phase shifts are monotonic. It is also shown that in the case of the repulsive potential, both the phase shifts and cross-sections change profoundly when the kinetic energy of the incident particle is lower than the potential height.
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.
Transmission and scattering problems involving complex potentials are important in physics, in particular in describing nuclear collisions. We describe many pedagogical features of transmission in one dimension and scattering and absorption cross sections in three dimensions for a rectangular absorptive potential and compare the results with scattering from real barriers and a hard sphere. For a given energy in one dimension we show that the absorption reaches a maximum for a critical value of the absorption strength and then monotonically decreases with an increase in the absorption strength. An infinitely absorptive well becomes almost fully reflective and thus is similar to an infinitely high barrier. Similar results are found in three dimensions. We show that the absorption cross section in three dimensions diverges at a threshold energy, which is of critical importance for exothermic reactions.
The immersing method is applied to solve the N-channel scattering problem for concrete potential. In particular, we consider the particle scattering on a two-dimensional potential barrier, which is constant in the scattering direction and arbitrary in the cross-section direction. For this case the scattering amplitudes t m and r m (1,2, m= ...,N) are determined. A transition from the obtained formulas to the case of ™-potential is performed. For this case transmission amplitudes t m and reflection amplitudes r m are obtained. It is also shown that the product of transmission and reflection amplitudes along the channel m does not depend on the scattering channel.
Synopsis: Students are introduced to the ideas of particle scattering in quantum mechanics in a one-dimensional context. e integral Schrödinger equation is deduced using a Wronskian technique, and reflection and transmission amplitudes are defined. Several simple scattering problems are solved, and the Born series for the energy-wave function is explained. A three-dimensional spherical basis is introduced, which allows the integral Schrödinger equation in this context to be formulated. e relation between plane waves and spherical waves is found from a solution to the unit-source Helmholtz equation, again using the Wronskian technique. Partial waves, phase shifts, and cross sections are defined, and finite range scattering is considered. e chapter concludes with a consideration of identical particle scattering and an application to the protonproton case, where a dramatic manifestation of the underlying substructure emerges at high energies.
The phase-space formulation of quantum scattering is discussed and the asymptotic quantities describing potential scattering in one and three dimensions (transmittances, cross sections) are analyzed. It is found that under certain conditions the transmittance in time-dependent scattering only depends on the momentum distribution of the incident packet, being independent of its particular shape in coordinate space. For three-dimensional scattering, generalized cross sections are defined for momentum coherences and a physical interpretation of the generalized optical theorem is proposed.
We study the near-threshold behavior of scattering phase shifts, quantum reflection amplitudes, and quantization functions for systems described by a circularly symmetric potential in two spatial dimensions. In contrast to the three-dimensional case, the centrifugal potential $({m}^{2}\ensuremath{-}\frac{1}{4}){\ensuremath{\hbar}}^{2}∕(2M{r}^{2})$ is nonvanishing and even attractive for s waves, m=0, and the leading near-threshold energy dependence of phase shifts and amplitudes for scattering and quantum reflection is logarithmic in this case. The near-threshold behavior of the s-wave phase shifts and amplitudes can nevertheless be characterized by a well-defined scattering length (for potentials falling off faster than 1∕r2) and an effective range (for potentials falling off faster than 1∕r4). For a potential with a bound state at energy ${E}_{n}=\ensuremath{-}{\ensuremath{\hbar}}^{2}{\ensuremath{\kappa}}_{n}^{2}∕(2M)$ very near threshold, the scattering length obeys $a\stackrel{{\ensuremath{\kappa}}_{n}\ensuremath{\rightarrow}0}{\ensuremath{\sim}}2{e}^{\ensuremath{-}{\ensuremath{\gamma}}_{\mathrm{E}}}∕{\ensuremath{\kappa}}_{n}+O({\ensuremath{\kappa}}_{n})$, with no term of order ${\ensuremath{\kappa}}_{n}^{0}$---in contrast to the three-dimensional case. Analytical results are derived for homogeneous potentials, and the necessary modification of the effective-range expansion is given for potentials proportional to 1∕r4. For $m\ensuremath{\ne}0$ we give analytical expressions for the near-threshold behavior of phase shifts as well as scattering and quantum reflection amplitudes, which are generally valid, even when m is neither integer nor half-integer.