January 11, 2009
Adiabatic states and nonadiabatic couplings. Electronic
excitation and decay. Curve crossings and the breakdown of the
Born-Oppenheimer approximation. The non-crossing rule. The diabatic
representation. The conical intersections
The nuclear coordinates as parameters in the time-dependent
Schroedinger equation. Mean-field and surface hopping methods. Tully's
fewest switches algorithm. Quantum decoherence corrections. The
selection of initial conditions.
Many people find time-dependent quantum mechanics the most interesting and understandable part of quantum mechanics. However, standard courses in quantum mechanics devote little attention to this perspective, and its relationship to the rest of the syllabus is disjointed. Yet it is possible to develop quantum mechanics from beginning to end from a time-dependent perspective, using a small set of conceptual building blocks. The tutorial will have two parts. In the first part I will present the basic building blocks. I will begin with several animations that highlight the visual appeal of time-dependent quantum mechanics and its relationship with classical mechanics. Next, I will introduce the concept of a wavepacket time-correlation function and show how it is related to a spectrum via Fourier transform. I will then discuss the reciprocity of wavepackets and eigenstates — just as a wavepacket is a superposition of eigenstates, an eigenstate is a superposition of wavepackets. Finally, I will return to wavepacket time-correlation functions and show how they can be used to calculate reflection/transmission probabilities, emphasizing that barrier scattering and spectroscopy are two sides of the same coin. In the second part of the tutorial I will show that the time-dependent perspective provides a simple and unified interpretation of many of the frontier experiments in modern Chemical Physics — from femtochemistry to resonance Raman spectroscopy, from coherent control to photodissociation to reactive scattering — using the same small set of conceptual building blocks described in the first part.
In many physical systems it is believed that one can describe parts of the system by classical and parts of the system by quantum mechanics.
In this lecture I explain how to rigorously derive such effective "mixed quantum-classical" descriptions from the underlying quantum mechanics
for the whole system. The important mechanism is adiabatic decoupling and the method adiabatic perturbation theory. As the paradigmatic
example I will discuss the time-dependent Born-Oppenheimer approximation.