Semiclassical Dynamics and Relaxation

This text concerns 'semiclassical' within various meanings. These include the familiar JWKB approximation and its phase-integral generalizations in Chapter 2 to two and four transition points with or without one or two poles: by corollary, crossing and non-crossing nonadiabatic collision theory. Above and below threshold Wannier ionization is covered in Chapter 3 where the large parameters are the inverses of the variation of the hyperspherical angles from their ridge values. The more familiar impact parameter treatment, in which the possibly relativistic heavy-particle relative motion is treated classically and the electrons quantally, is well covered in Chapter 4. Diffusion in solids and liquids is described in Chapter 5 where typically the large parameter is the height of the barrier which is overcome by thermal agitation. Hypergeometric functions are introduced in Chapter 1 and Mittag-Leffler functions in Appendix B.

Autorentext

The author is an expert in quantum dynamics and the processes of atomic collisions over 42 years and he has developed the mathematics of the semiclassical theory over 18 years and has applied this to condensed-matter physics. He is currently a Professor of Theoretical Physics in the Department of Applied Mathematics and Theoretical Physics at The Queen's University of Belfast. He has also been appointed Local Chair for the International Conference for Photonic, Electronic and Atomic Collisions in Belfast. Dr. Crothers other honors and memberships include:

Personal Chair in Theoretical Physics Queens University (1985- )

Readership in Applied Mathematics and Theoretical Physics Queen's University (1976-1985)

Lecturer in Applied Mathematics Queen's University (1966-1976)

Fellow of American Physical Society (1994)

Fellow of Institute of Physics (1973)

Fellow of Institute of Mathematics and its Applications (1973)

Fellow of Institution of Electrical Engineers (1997)

Member of the Royal Irish Academy (1991)

Honorary Professor, Physics, St. Petersburg State University (2003)

Belfast Local Chair, ICPEAC XXVII, 2011 (2005)

Honorary Fellow, Trinity College Dublin (2006)

Zusammenfassung

Condensed-matter physics plays an ever increasing role in photonics, electronic and atomic collisions research. Dispersion (Dynamics and Relaxation) includes scattering/collisions in the gaseous phase. It also includes thermal agitation, tunneling and relaxation in the liquid and solid phases. Classical mechanics, classical statistical mechanics, classical relativity and quantum mechanics are all implicated. 'Semiclassical' essentially means that there is a large or asymptotic real parameter. 'Semiclassical' can also mean 'classical with first-order quantal correction', based on an exponentiated Liouville series commencing with a simple pole in the -plane, being Planck's reduced constant and coming with all the attendant connection problems associated with the singularity at the turning or transition point and with the Stokes phenomenon. Equally,' semiclassical' can mean 'electrons described quantally and the heavy particles classically'. This latter gives rise to the so-called impact parameter method based on a pre-assigned classical trajectory.

With evermore sophisticated experiments, it has become equally more important to test theory over a wider range of parameters. For instance, at low impact energies in heavy-particle collisions, the inverse velocity is a large parameter; in single-domain ferromagnetism, thermal agitation (including Brownian motion and continuous-time random walks) is faced with a barrier of height 'sigma', a possibly large parameter. Methods of solution include phase-integral analysis, integral transforms and change-of-dependent variable. We shall consider the Schrödinger time-independent and time-dependent equations, the Dirac equation, the Fokker Planck equation, the Langevin equation and the equations of Einstein's classical general relativity equations.

There is an increasing tendency among physicists to decry applied mathematics and theoretical physics in favour of computational blackboxes. One may say appliedmathematics concerns hard sums and products (and their inverses) but unless one can simplify and sum infinite series of products of infinite series, can one believe the results of a computer program? The era of the polymath has passed; this book proposal aims to show the relevance to, and impact of theory on, laboratory scientists.

Inhalt
Mathematics for the Semiclassicist.- Semiclassical Phase Integrals.- Semiclassical Method for Hyperspherical Coordinate Systems.- IonAtom Collisions.- Diffusion in Liquids and Solids.