The time-independent bound-state perturbation theory derived for the quantum probability equation (Milward G C and Wilkin C 2000 J. Phys. A: Math. Gen. 33 7353 ) is extended to treat the scattering domain. Formulae for the changes in the partial wave phase shifts are derived to third order in the perturbing potential. The results are compared with those obtained for analytically soluble problems.
The time-independent bound-state perturbation theory derived for the quantum probability equation (Milward G C and Wilkin C 2000 J. Phys. A: Math. Gen. 33 7353 ) is extended to treat the scattering domain. Formulae for the changes in the partial wave phase shifts are derived to third order in the perturbing potential. The results are compared with those obtained for analytically soluble problems.
The partial-wave Schrödinger equation with real boundary conditions is recast as an equation for the probability density. When a small additional potential is included, the changes in the bound-state energy eigenvalues are obtained, up to third order in the perturbation, purely in terms of the perturbing potential and the unperturbed probability density. Although the approach is different, our results are equivalent to those derived by Bender (Bender C M 1978 Advanced Mathematical Methods for Scientists and Engineers (New York: McGraw-Hill) p 330). Knowledge of neither the unperturbed energy spectrum nor the wavefunctions of excited states is required. Evaluations of the second-order energy shift are given for some soluble S -wave problems.
There exists a simple relationship between a quantum-mechanical bound-state wave function and that of nearby scattering states, when the scattering energy is extrapolated to that of the bound state. This relationship is demonstrated numerically for the case of a spherical well potential and analytically for this and other soluble potentials. Provided that the potential is of a finite range and that the binding is weak, the theorem gives a useful approximation for the short-distance behavior of the scattering wave functions. The connection between bound and scattering-state perturbation theory is established in this limit.
Scattering theory studies the behaviour of (quantum) systems over time and length scales which are very large, compared with the time or length that are characteristic of the interaction which affects them. The chapter begins with an outline of the basic problems of scattering theory, i.e. the existence of scattering states, their uniqueness, weak asymptotic completeness the existence of the scattering transformation, S , the reduction of S due to symmetries, the analyticity of S , asymptotic completeness and the existence of wave operators. After considering the Schrödinger equation for stationary states, the integral equation for scattering problems is derived and studied. The Born approximation and the Born series are presented, and the conditions which ensure convergence of the Born series are also discussed. Further topics are partial wave expansion, its convergence and the uniqueness of the solution satisfying the asymptotic condition, the Levinson theorem. Lastly, in the first part of the chapter, scattering theory from singular potentials is introduced, with emphasis on the polydromy of the wave function, following early work in the literature, and the general problems in the theory of resonances are studied. In the second part, we examine in detail a separable potential model, the occurrence of bound states in the completeness relationship, an excitable potential model and the unitarity of the Möller wave operator. Lastly, we study the survival amplitude associated with quantum decay transitions. Aims and problems of scattering theory Scattering theory is the branch of physics that is 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 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.
A phase-shift analysis of quantum mechanical scattering problems in one dimension is developed. Many of the techniques and results of three dimensions remain even in one dimension, while the discussion is vastly simplified, and thus accessible to the student much earlier, due to the absence of special functions (in particular, Bessel and Legendre functions) with their complicated asymptotic formulas and orthogonality relations. A “partial wave” analysis leads to the familiar relation fl∼eiδl sinδl between the scattering amplitude in the lth partial wave and the phase shift δl. The optical theorem is evident as an immediate consequence of the conservation of probability. The method is applied to an analysis of the well-known square-well scattering resonances.
There are few potentials V (r) for which the Schrödinger equation is exactly solvable. We must therefore develop approximation techniques to obtain the eigenvalues and eigenfunctions for potentials that do not lead to exactly soluble equations. One of these methods is the perturbation theory, which was developed both for time independent and time dependent phenomena. We will discuss the time independent perturbation theory. The formulation of the time independent perturbation theory is different for nondegenerate and degenerate states.