On the characterization of bound states and scattering states in quantum mechanics

B. G. Sidharth

Andrew M. Essin, David J. Griffiths

In quantum mechanics a localized attractive potential typically supports a (possibly infinite) set of bound states, characterized by a discrete spectrum of allowed energies, together with a continuum of scattering states, characterized (in one dimension) by an energy-dependent phase shift. The 1∕x2 potential on 0<x<∞ confounds all of our intuitions and expectations. Resolving its paradoxes requires sophisticated theoretical machinery: regularization, renormalization, anomalous symmetry-breaking, and self-adjoint extensions. Our goal is to introduce the essential ideas at a level accessible to advanced undergraduates.

Dževad Belkić

This chapter examines the most critical properties, which make the collision phenomenon substantially different from the corresponding problem of bound states in quantum mechanics. In order to elaborate the treatments further in the course of the presented analysis, it was necessary to employ the quantum-mechanical entities, such as the equations of time evolution, as well as the state vectors and the Hilbert space of eigenstates. A given picture of quantum mechanics can be more practical than formalism for the examined physical phenomenon. Hence, the choice of the optimal picture of quantum mechanics becomes a matter of considerable practical importance. The initial state Ψ0i is experimentally prepared, i.e. characterized by means of the impulse, polarization, spin, isospin and other internal quantum numbers of the constituent parts of the scattering system.

David Pierre Ruelle

W. W. Bell

Stephen L. Adler

Abstract We proceed in this chapter to discuss scattering theory and related aspects of the bound-state problem in quaternionic quantum mechanics. We begin with a detailed discussion of a simple one-dimensional example, the delta function potential model. From this model will emerge the result (Adler, 1988) that will be the principal theme of this and the succeeding two chapters: with the standard ray representative choice of Sec. 4.2, for short-range potentials the asymptotic scattering states and the S-matrix in quaternionic quantum mechanics are always complex ℂ(1, i).