Chapter III. Bound States

Massimiliano Sassoli de Bianchi

In this Thesis we study several aspects of potential scattering in non-relativistic quantum mechanics. In the first part, we study the concept of time delay. More precisely, after an elementary introduction to the theory (chapter I), we clarify in chapter II the role played by the localizing regions in the definition of global time delay. In chapter III, we generalize the concept to arbitrary conditions of observation for the scattered particle (conditional time delay). In chapters IV and V, we address the problem of the measure of time delay by physical clocks and, more specifically, by the Larmor clock which exploit the mechanism of precession of a spin in a magnetic field. In the second part, we are interested in the spectral property of time delay, its connection with Levinson's theorem, and the application of the latter to one-dimensional systems. We derive Levinson's theorem in chapter VI, using as a single ingredient the completeness of physical states. In chapter VII, we apply Levinson's result to determine the number of bound-states of a finite periodic potential, as a function of its period. For this, we use the factorization property of the scattering matrix which we derive in chapter VIII in the more general situation of a particle with position-dependent mass. In the third part, we are concerned by time-dependent potentials. In chapter IX, we generalize the concept of time delay, and for the particular case of a periodic variation we derive a Levinson theorem. In chapter X, we consider potentials with very slow or very rapid variations in time. The low and high frequency limits are derived as well as their first corrections, and their physical significance discussed.

J Hamilton, Amira Salam

Charles G. Wohl

Juan Sotelo-Campos

In this thesis we adapt the Davies and Lewis’ operational approach to quantum probability to the Op*-algebra framework. In Chapter I we introduce our formalism of quantum mechanics. We define the space of pure states U as a nuclear-Frechet space generated by a self-adjoint operator M whose inverse is nuclear. The algebra of observables is taken to be L+, the adjoint-stable set of linear maps from U into itself. This algebra and its dual under the uniform topology form a dual pair. We derive several algebraic and topological properties of this dual pair and show that it forms a suitable structure for the adaptation of Davies and Lewis’ theory to the Op*-algebra framework. In Chapter II we set the basis of the Davies and Lewis’ operational approach to quantum probability within the framework developed in chapter I. Here we define the fundamental concepts of an expectation and an instrument and discuss their relationship. We show that instruments are bounded Radon measures and that the compose of two of them is not in general a stable operation on the universe of all instruments. We conclude the chapter discussing the general Robertson-Heisenberg uncertainty relations. In Chapter III we build a special class of instruments to measure Q, P and H on S; these are based on Davies’ approximate position measurements. In particular we show that the compose of any two, hence any finite number, of these instruments is an instrument.

Yi‐Cai Zhang

Abstract In this work, we investigate the bound states in a one-dimensional spin-1 flat band system with a Coulomb-like potential of type III, which has a unique non-vanishing matrix element in basis ∣1〉. It is found that, for such a kind of potential, there exists infinite bound states. Near the threshold of continuous spectrum, the bound state energy is consistent with the ordinary hydrogen-like atom energy level with Rydberg correction. In addition, the flat band has significant effects on the bound states. For example, there are infinite bound states which are generated from the flat band. Furthermore, when the potential is weak, the bound state energy is proportional to the potential strength α. When the bound state energies are very near the flat band, they are inversely proportional to the natural number n (e.g., E n ∝ 1/n, n = 1, 2, 3, …). Further we find that the energy spectrum can be well described by quasi-classical approximation (WKB method). Finally, we give a critical potential strength α c at which the bound state energy reaches the threshold of continuous spectrum. After crossing the threshold, the bound states in the continuum (BIC) would exist in such a flat band system.