A quantum-mechanical scattering theory is developed for the scattering by a potential of which strength is randomly fluctuating between two values with a switching rate γ/2. Within this model the T matrix averaged over the fluctuation is derived exactly, from which the averaged scattering cross section (ACS) is obtained by using the optical theorem. Explicit calculations of the ACS are carried out for three cases; (i) within the Born approximation, (ii) for a separable potential, (iii) for a short-range potential in a lattice. It is found that the low-energy ACS is enhanced in comparison with the static case for \(E/\hbar{\lesssim}\gamma{\lesssim}\hbar/(2ma^{2})\) because of the large inelastic cross section, where E is the energy of the incident particle with a mass m and a is the range of the potential.
A quantum-mechanical scattering theory is developed for the scattering by a potential of which strength is randomly fluctuating between two values with a switching rate γ/2. Within this model the T matrix averaged over the fluctuation is derived exactly, from which the averaged scattering cross section (ACS) is obtained by using the optical theorem. Explicit calculations of the ACS are carried out for three cases; (i) within the Born approximation, (ii) for a separable potential, (iii) for a short-range potential in a lattice. It is found that the low-energy ACS is enhanced in comparison with the static case for (E/\hbar{\lesssim}\gamma{\lesssim}\hbar/(2ma^{2})) because of the large inelastic cross section, where E is the energy of the incident particle with a mass m and a is the range of the potential.
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.
The theory of scattering is a very important part of the quantum mechanics.The Born approximation and Eikonal approximation of the Scattering cross-section are calculated with the Gauss potential and Lenz-jensen potential under a particular incident energy.The results show that the bigger the Electronic energy is,the smaller the total scattering crossection is and the weaker the electron scattering of potential field is.There is little influence of potential to the energetic particle about the Lenz-jensen potential.And the majority of scattered particles are concentrated on a narrow range.
The Schrödinger equation for two-particle scattering with random potential is considered, including both the cases of a local and a nonlocal potential with convolution kernel. It is shown that the statistical fluctuations of the potential give rise to additional terms which increase the scattering cross section, and an evaluation of the additional terms in the Born approximation is given. A brief discussion about analyticity properties of the scattering amplitude under suitable assumptions is included.
We have analytically explored the phenomenon of quantum scattering for unfixed quantum scatterers. We have considered short ranged (Fermi-Huang δp3 [1]) interaction between the incident `particle' and the scatterer with no interactions among the scatterers, and the scatterers as (i) particle(s) in a 1-D box, (ii) particle(s) in a 1-D double box, (iii) particle(s) in a 1-D grating, (iv) particle(s) in a 2-D rectangular box, (v) particle(s) in a 3-D harmonic trap, (vi) Bose-Einstein condensates in a double well, (vii) Bose-Einstein condensates in an optical lattice, etc. Coherent scattering from all possible positions of the quantum scatterers in the finite geometry gives rise to rich physics. We have predicted differential scattering cross-section for `particle' scattering, not only for the above cases, but also for 3-D harmonically trapped Bose and Fermi gases in thermodynamic equilibrium. Our predictions can be tested within the present day experimental setups.
We are concerned with the justification of the statement, commonly (explicitly or implicitly) used in quantum scattering theory, that for a free non-relativistic quantum particle with initial wave function Ψ0(x), surrounded by detectors along a sphere of large radius R, the probability distribution of the detection time and place has asymptotic density (i.e., scattering cross section) σ(x,t)=m3ℏ−3Rt−4∣Ψ0(mx/ℏt)∣2 with Ψ0 the Fourier transform of Ψ0. We give two derivations of this formula, based on different macroscopic models of the detection process. The first one consists of a negative imaginary potential of strength λ>0 in the detector volume (i.e., outside the sphere of radius R) in the limit R→∞,λ→0,Rλ→∞. The second one consists of repeated nearly-projective measurements of (approximately) the observable 1∣x∣>R at times T,2T,3T,… in the limit R→∞,T→∞,T/R→0; this setup is similar to that of the quantum Zeno effect, except that there one considers T→0 instead of T→∞. We also provide a comparison to Bohmian mechanics: while in the absence of detectors, the arrival times and places of the Bohmian trajectories on the sphere of radius R have asymptotic distribution density given by the same formula as σ, their deviation from the detection times and places is not necessarily small, although it is small compared to R, so the effect of the presence of detectors on the particle can be neglected in the far-field regime. We also cover the generalization to surfaces with non-spherical shape, to the case of N non-interacting particles, to time-dependent surfaces, and to the Dirac equation.
The classical cross section for scattering by a 1/r2 potential is linear in the strength of the potential, while from the Born approximation to quantum mechanics one might expect it to be quadratic. To resolve this puzzle, we explicitly carry out the passage to the classical limit of the quantum-mechanical partial-wave series for this potential. In the process, we discover the conditions under which quantum scattering becomes classical; under these conditions the Born approximation is not applicable.