I present a derivation of the many qualitative changes to basic scattering theory that result when a (partially) coherent matter beam inelastically interacts with a quantum target. The weak beam limit, where only one object in the beam interacts with the quantum target, is developed. Some of the basic changes to scattering theory are as follows. (1) The transition probability can increase as the square of the number of particles in the beam. (2) The transition probability is not proportional to the convolution of the total inelastic cross section with the momentum distribution of the beam. (3) The transition probability to different final states of the target does not depend on the population in the initial states but depends on the amplitudes for the initial states. (4) The total transition probability depends on the differential cross section integrated over all final angles and on the differential cross section only in the forward scattering direction. (5) The transition probability can be nonzero even when the beam misses the target. (6) In the perturbative limit, the transition probability can be related to classical, large scale properties of the beam (such as the average density).
I present a derivation of the many qualitative changes to basic scattering theory that result when a (partially) coherent matter beam inelastically interacts with a quantum target. The weak beam limit, where only one object in the beam interacts with the quantum target, is developed. Some of the basic changes to scattering theory are as follows. (1) The transition probability can increase as the square of the number of particles in the beam. (2) The transition probability is not proportional to the convolution of the total inelastic cross section with the momentum distribution of the beam. (3) The transition probability to different final states of the target does not depend on the population in the initial states but depends on the amplitudes for the initial states. (4) The total transition probability depends on the differential cross section integrated over all final angles and on the differential cross section only in the forward scattering direction. (5) The transition probability can be nonzero even when the beam misses the target. (6) In the perturbative limit, the transition probability can be related to classical, large scale properties of the beam (such as the average density).
Inelastic scattering from N coherent spins, or any two-state system, scales as N2, invalidating the Born approximation at large N even at weak coupling. We develop a nonperturbative theory of this superradiant scattering and its backreaction on the spins. Since one particle can exchange many quanta, the target's energy-transfer and energy-variance rates keep their favorable N2 scaling even after the cross section saturates, unless a conservation law limits each encounter to exchange of one quantum. We apply the theory to millicharged-particle searches, as well as neutron and charged-particle beams that can test quantum protocols for detecting cosmic relics.
A simple model coupling a one-dimensional beam particle to a one-dimensional harmonic oscillator is used to explore complementarity and entanglement. This model, well-known in the inelastic scattering literature, is presented under three different conceptual approaches, with both analytical and numerical techniques discussed for each. In a purely classical approach, the final amplitude of the oscillator can be found directly from the initial conditions. In a partially quantum approach, with a classical beam and a quantum oscillator, the final magnitude of the quantum-mechanical amplitude for the oscillator's first excited state is directly proportional to the oscillator's classical amplitude of vibration. Nearly the same first-order transition probabilities emerge in the partially and fully quantum approaches, but conceptual differences emerge. The two-particle scattering wavefunction clarifies these differences and allows the consequences of quantum entanglement to be explored.
Aims and problems of quantum scattering theory Scattering theory is the branch of physics concerned with interacting systems on a scale of time and/or distance that is large compared with the scale of the interaction itself, and it provides the most powerful tool for studying the microscopic nature of the world. In quantum mechanics, a typical scattering process involves a beam of particles prepared with a given energy and with a more or less defined linear momentum. One then studies either the collision with a similar wave packet, or the collision of the given wave packet against a fixed target. In particular, for a two-body elastic scattering problem with conservative forces, we work in the centre of mass frame, hence reducing the original problem to the analysis of a particle in an external field of forces. Indeed, in Section 7.2 we already used such a technique in the investigation of the hydrogen atom. The crucial difference with respect to Section 7.2 is that, in scattering problems, the continuous spectrum and its perturbations are studied, whereas the Balmer formula has to do with bound states. In the following section we outline the basic aspects of what is called time-dependent scattering. Time-dependent scattering The physical situation we would like to study is as follows: a beam of particles is prepared in a state that is approximately free and allowed to evolve towards the scatterer. After the collision we detect the scattered beam far away from the interaction region.
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.
Stationary states are unavoidable from the practical viewpoint, since all the computations of observables, such as cross sections and other experimentally measurable physical quantities, are carried out within timeindependent scattering theory. In a scattering experiment, one attempts to determine the momentum of the impact projectile as precisely as possible. However, in the standard formulation of scattering theory, one tries to reach the same ideal goal by identifying the initial state in the remote past with the plane wave. The problem of the asymptotic convergence of scattering states cannot be formulated in the same way for both proper and improper state vectors. The optical theorem is a convenient way of computing the total cross section Q, when one knows the value of the imaginary part of the scattering amplitude in the forward direction. The scattered particles are moved away from the incident beam for a certain amount which is proportional to the value of the total cross section Q.
The conventional notion of elastic, coherent atom-surface scattering originates from the scattering particles acting as a quantum-mechanical matter wave, which coherently interfere to produce distinct Bragg peaks which persist at finite temperature. If we introduce inelastic scattering to this scenario, the result is that the surface particles become displaced by the scattering atoms, resulting in emission or absorption of phonons that shift the final energy and momentum of the scatterer. As the lowest-lying phonons are gapless excitations, the ability to measure these phonons is very difficult and this difficulty is exacerbated by the roughly 1-eV resolution found in high-energy helium scattering experiments. Even though the surface has in effect measured the presence of the scatterer which decoheres the particle, we retain the diffraction spots which are referred to as coherent scattering. How do we reconcile these disparate viewpoints? We propose an experiment to more precisely examine the question of coherence in atom-surface scattering. We begin with an initially coherent superposition of helium particles with equal probabilities of interacting with the surface or not interacting with the surface. The beams are directed so that after the scattering event, the atoms are recombined so that we can observe the resulting interference pattern. The degree to which phonons are excited in the lattice by the scattering process dictates the fringe contrast of the interference pattern of the resulting beams. We use semiclassical techniques to simulate and test the viability of this experiment and show that for a wide range of conditions, despite the massive change in the momentum perpendicular to the surface, we can still expect to have coherent (in the superposition sense) scattering.