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CRC Press eBooks

Scattering Theory II

Nandita Rudra

Published Aug 19, 2019• 0 citations
Publisher

Abstract

In this chapter we deal with formal theory of scattering. We start with the scattering process in a time independent formulation. The Hamiltonian is assumed to be given as usual by
DOI
10.1201/9780429325762-15

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Cambridge University Press eBooks•Feb 17, 2011•1 citation
The time-dependent theory of scattering is reformulated. In §1, it is developed on the basis of a new limiting process which is self-consistent, and the equivalence is shown of Heisenberg's S-matrix and Dyson's one, when the total Hamiltonian permits the existence of bound states. In §2, a theory of scattering of wave packet is proposed in conformity with the physical picture. The damping factor e−ε|t| is derived from the amplitude of the wave packet. In §3, the rearrangement scattering is treated on the basis of the wave packet formalism.
Scattering theory deals with the evolution of states of a system and, in particular, attempts to relate the state of a system as t → +∞ with the state of that system as t → −∞. In Chapter 3 we indicated that such a comparison could be made in terms of so-called wave operators. In this chapter we shall introduce these quantities rather more properly and outline some of their more important properties. This we can do most conveniently in an abstract setting; we shall consider specific cases later.
We begin by considering the following scattering experiment. A projectile, e.g., an atom A in a given electronic state, collides with a target which we will take as a diatomic molecule BC in its ground electronic, vibrational and rotational state. For a short period of time an activated complex [ ABC ] # is generated. As time passes the activated complex can break into different products. For instance, in our example these products will be A + BC , B + AC, C + AB and A + B + C . Each one of the possible products can be in different electronic, vibrational and rotational quantum states. The total energy which is originally the sum of the electronic and translational energies of the projectile A and the electronic, vibrational, rotational and translational energies of target BC is conserved during the scattering process. Time-independent scattering theory enables one to calculate the probability of obtaining the specific products in given quantum states and the kinetic energy distribution of the products as a function of the total energy of the system without the need to solve the time-dependent Schrödinger equation. The time-independent formulation of scattering theory is based on the ability to propagate analytically an initial given wavepacket, Φ(0), to infinite times. That is, we need to get a closed form expression for lim t →±∞ e −iĤ t /ħ |Φ( t = 0)〉. To quote from the introduction of the excellent book on scattering theory written by Taylor: “The most important experimental technique in quantum physics is the scattering experiment. That this is so is clear from even the briefest review of modern physics”.