Chapter 5. Aspects of Time-Dependent Quantum Mechanics

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.

Zoltán Papp

Walter Wilcox

This textbook offers a unique introduction to quantum mechanics progressing gradually from elementary quantum mechanics to aspects of particle physics. It presents the microscopic world by analysis of the simplest possible quantum mechanical system (spin 1/2). A special feature is the author’s use of visual aids known as process diagrams, which show how amplitudes for quantum mechanical processes are computed. The second edition includes a new chapter and problems on time-dependent processes, in addition to new material on quantum computing and improved illustrations. Key Features: Provides a completely updated text with expanded contents. Includes a brand new chapter on time-dependent processes and expanded coverage of recent developments in particle physics. Emphasizes a visual approach employing process diagrams and utilizing new figures. Incorporates quantum information theory in a new appendix, with other helpful supplements on notation, lattice models, weak flavor mixing, and numerical simulations.

J. W. Halleý

In the early to mid-twentieth century, physicists found that they could not describe the observed behavior of the electrons orbiting the nuclei of atoms by use of Newtonian physics. A new mathematical description of that behavior was developed. Called quantum mechanics, it has been tremendously successful in accounting for widely diverse phenomena at both the qualitative and quantitative level and is universally regarded as generally applicable to all matter. However quantum mechanics has some peculiar counter intuitive features and some aspects of its interpretation remain in dispute. This chapter focuses on how quantum mechanics changes the Newtonian picture of the past, present and future and of time itself. The mathematical apparatus governing the time dependence of the wave functions which describe states in quantum mechanics is completely time reversible, in a way closely analogous to the time reversibility of Newtonian mechanics, though the state descriptions are different.

Daniel R. Bes