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American Journal of Physics

A simple model for inelastic scattering

Jennifer G. Loeser, HERSCHEL A. RABITZ, Andrew E. DePristo, Eric A. Rohlfing

Published Nov 1, 1981• 1 citation
Publisher

Abstract

A simple model for inelastic scattering may be obtained by suitably generalizing scattering from a square well. The generalization introduces matrices into the quantum-mechanical scattering equations. These equations may be solved exactly to give an explicit expression for the scattering matrix. In this paper we first describe and solve the model, and then discuss the results it predicts for a simple example. We will see that even a simple inelastic system can exhibit rich scattering behavior.

DOI
10.1119/1.12580

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A simple model for inelastic scattering may be obtained by suitably generalizing scattering from a square well. The generalization introduces matrices into the quantum-mechanical scattering equations. These equations may be solved exactly to give an explicit expression for the scattering matrix. In this paper we first describe and solve the model, and then discuss the results it predicts for a simple example. We will see that even a simple inelastic system can exhibit rich scattering behavior.
We present a simple theory for obtaining the semiclassical approximation for inelastic scattering. The theory gives explicit expressions for the expectation values of a wide class of quantum operators, generally valid up to O(\ensuremath{\Elzxh}), which may be used to characterize the final distribution of a scattered particle. We illustrate the method with a one-dimensional model consisting of a particle scattering from a harmonic oscillator. In addition, we apply the theory to examine under what conditions the so-called trajectory approximation is and is not valid.
Abstract We discuss a general description of the solutions to the 1D time-independent Schrödinger equation that does not a priori distinguish between scattering states and bound states and emphasizes and reinforces their relationship and connection to each other. This manuscript also introduces the concept of transfer matrices, which it presents as a logical extension of the traditional approach to evaluating 1D potentials. Using the transfer matrix method and a finite step approximation allows for a simple and straight-forward numerical solution of arbitrary 1D potentials. It also separates the process of solving the Schrödinger equation from selecting physically relevant solutions, which is a critical skill in quantum mechanics and is at the core of physics problems in general.
We use the framework setup recently to compute nonperturbatively inelastic scattering from quantum impurities [G. Zar\'and et al., Phys. Rev. Lett. 93, 107204 (2004)] to study the energy dependence of the single-particle SSS matrix and the inelastic scattering cross section for a number of quantum impurity models. We study the case of the spin S=1∕2S=1∕2S=1∕2 two-channel Kondo model, the Anderson model, and the usual S=1∕2S=1∕2S=1∕2 single-channel Kondo model. We discuss the difference between non-Fermi-liquid and Fermi-liquid models and study how a crossover between the non-Fermi-liquid and Fermi-liquid regimes appears in the case of channel anisotropy for the S=1∕2S=1∕2S=1∕2 two-channel Kondo model. We show that for the most elementary non-Fermi-liquid system, the two-channel Kondo model, half of the scattering remains inelastic even at the Fermi energy. Details of the derivation of the reduction formulas and a simple path integral approach to connect the TTT matrix to local correlation functions are also presented.
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.