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Journal of Mathematical Physics

In general, the less degeneracy the less transition. A principle for time-dependent Hamiltonian systems in quantum mechanics

H. Gingold

Published Oct 1, 1987• 8 citations
Publisher

Abstract

A principle in quantum mechanics is proposed: ‘‘In general, the less degeneracy the less transition.’’ Mathematical support of this principle is given in a setting of a slowly varying time-dependent Schrödinger equation via a theorem of asymptotic decomposition. Formulas that quantitatively relate transition and degeneracy are developed. Ramifications of those formulas are discussed.

DOI
10.1063/1.527778

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A principle in quantum mechanics is proposed: ‘‘In general, the less degeneracy the less transition.’’ Mathematical support of this principle is given in a setting of a slowly varying time-dependent Schrödinger equation via a theorem of asymptotic decomposition. Formulas that quantitatively relate transition and degeneracy are developed. Ramifications of those formulas are discussed.
In a quantum-mechanical problem with a slowly varying Hamiltonian, such as appears in a semiclassical treatment of inelastic atomic collisions, it is common to restrict attention to a small number of states among which transitions are anticipated. This procedure is justified by an extension of the conventional adiabatic theorem to nearly degenerate states. The correct mixing within the restricted basis is determined by a time-dependent variational principle. The contribution of states outside the basis vanishes as the time scale for variation of the Hamiltonian becomes large. A further generalization deals with pseudoeigenfunctions, such as appear in the theory of the Stark effect.
We expose the Schrödinger quantum mechanics with traditional applications to Hydrogen atom. We discuss carefully the experimental and theoretical background for the introduction of the Schrödinger, Pauli and Dirac equations, as well as for the Maxwell equations. We explain in detail all basic theoretical concepts. We explain all details of the calculations and mathematical tools: Lagrangian and Hamiltonian formalism for the systems with finite degree of freedom and for fields, Geometric Optics, the Hamilton-Jacobi equation and WKB approximation, Noether theory of invariants including the theorem on currents, four conservation laws (energy, momentum, angular momentum and charge), Lie algebra of angular momentum and spherical functions, scattering theory (limiting amplitude principle and limiting absorption principle), the Lienard-Wiechert formulas, Lorentz group and Lorentz formulas, Pauli theorem and relativistic covariance of the Dirac equation, etc. We give a detailed oveview of the conceptual development of the quantum mechanics, and expose main achievements of the ``old quantum mechanics'' in the form of exercises. One of our basic aim in writing this book, is an open and concrete discussion of the problem of a mathematical description of the following two fundamental quantum phenomena: i) Bohr's quantum transitions and ii) de Broglie's wave-particle duality. Both phenomena cannot be described by autonomous linear dynamical equations, and we give them a new mathematical treatment related with recent progress in the theory of global attractors of nonlinear hyperbolic PDEs.