The probabilistic interpretation of the quantum scattering cross section in the case of potential scattering is discussed in terms of Poisson random measures on the impact parameter plane and the sphere of outgoing directions.
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.
Introduction.We give three applications of probability to scattering theory.The first question we consider is that of the "scattering length."We obtain a probabilistic expression for scattering length and use this expression to prove that in the case of an impenetrable obstacle ("hard" potential) the scattering length T is the same as the electrostatic capacity.Next, we briefly touch upon a problem which is inherently probabilistic-namely, scattering in a medium with random boundary.We find that if a large number of small hard spheres ("dust particles") are distributed at random in the medium, the effect is to shift all the eigenvalues by a constant.Our final topic is the so-called "inverse problem" of scattering theory: given the scattering operator, to compute the potential.For certain spherically symmetric potentials, this problem was solved in a famous paper of GelTand and Levitan.Here we show that their result can be included in the circle of potential-theoretic problems which can be solved by means of Brownian motion and Wiener integrals.We quickly sketch some of the physical background.We are concerned with the Schrodinger equation of non-relativistic quantum mechanics.In its stationary or time-independent form, this equation isThe "wave function" I/J depends on both the position vector fiEfi 3 and the wave number k.The real parameter k 2 is proportional to the energy.The function q(R) is called the "potential."It represents the interaction of the quantum-mechanical "particle" with some force field.The "scattering problem" is, roughly speaking, to determine the effect of this interaction asymptotically for large R.More precisely, we seek a solution of (1) which at |fl| = » is of the asymptotic form e iV2kz + fk(e)e iV ^k> ^l\\R\\ where e is the unit vector in the direction of R. The first term, e iV2kz , is an incident wave in the direction of the zaxis; the second term /fc(e)e iV ^I^H/||R||, is the "scattered wave."
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.
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.