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Nonadiabatic transition is a highly multi-disciplinary concept and phenomenon, constituting a fundamental mechanism of state and phase changes in various dynamical processes of physics, chemistry and biology. This book is intended to be readable to a broad audience so that they can deepen their understanding of the basic concepts of both time-independent and time-dependent nonadiabatic transitions. Quantum mechanically intriguing phenomena such as complete reflection and nonadiabatic tunneling are emphasized. The Zhu-Nakamura theory that can deal with non-negligible classically forbidden transitions is explained. Furthermore, by controlling nonadiabatic transitions induced by an external field such as laser, designing chemical reaction dynamics as we desire is shown to be theoretically possible.

In this article the authors study Hamiltonian flows associated to smooth functions $H:\mathbb R^4 \to \mathbb R$ restricted to energy levels close to critical levels. They assume the existence of a saddle-center equilibrium point $p_c$ in the zero energy level $H^{-1}(0)$. The Hamiltonian function near $p_c$ is assumed to satisfy Moser's normal form and $p_c$ is assumed to lie in a strictly convex singular subset $S_0$ of $H^{-1}(0)$. Then for all $E \gt 0$ small, the energy level $H^{-1}(E)$ contains a subset $S_E$ near $S_0$, diffeomorphic to the closed $3$-ball, which admits a system of transversal sections $\mathcal F_E$, called a $2-3$ foliation. $\mathcal F_E$ is a singular foliation of $S_E$ and contains two periodic orbits $P_2,E\subset \partial S_E$ and $P_3,E\subset S_E\setminus \partial S_E$ as binding orbits. $P_2,E$ is the Lyapunoff orbit lying in the center manifold of $p_c$, has Conley-Zehnder index $2$ and spans two rigid planes in $\partial S_E$. $P_3,E$ has Conley-Zehnder index $3$ and spans a one parameter family of planes in $S_E \setminus \partial S_E$. A rigid cylinder connecting $P_3,E$ to $P_2,E$ completes $\mathcal F_E$. All regular leaves are transverse to the Hamiltonian vector field. The existence of a homoclinic orbit to $P_2,E$ in $S_E\setminus \partial S_E$ follows from this foliation.

Light one-electron atoms are a classical subject of quantum physics. The very discovery and further progress of quantum mechanics is intimately connected to the explanation of the main features of hydrogen energy levels. Each step in the development of quantum physics led to a better understanding of the bound state physics. The Bohr quantization rules of the old quantum theory were created in order to explain the existence of the stable discrete energy levels. The nonrelativistic quantum mechanics of Heisenberg and Schr¨ odinger provided a self-consistent scheme for description of bound states. The re- tivistic spin one half Dirac equation quantitatively described the main - perimental features of the hydrogen spectrum. Discovery of the Lamb shift [1], a subtle discrepancy between the predictions of the Dirac equation and the experimental data, triggered development of modern relativistic quantum electrodynamics, and subsequently the Standard Model of modern physics. Despite its long and rich history the theory of atomic bound states is still very much alive today. New importance to the bound state physics was given by the development of quantum chromodynamics, the modern theory of strong interactions. It was realized that all hadrons, once thought to be the elementary building blocks of matter, are themselves atom-like bound states of elementary quarks bound by the color forces.