Appendix 6: Derivation of Fermi's Golden Rule

Abstract

Fermi's Golden Rule provides the rate at which atomic or electronic transitions take place between two states. It applies to a wide range of optical and electronic processes for which the initial and final states can be described by wave functions. The various surface and interface analysis techniques may involve one or more atomic transitions. The rates are calculated from probabilities determined by transition matrix elements in quantum mechanical, first-order perturbation theory. This appendix follows the derivation presented in L.C. Feldman and J.C. Mayer, Fundamentals of Surface and Thin Film Analysis (Prentice Hall, NJ, 1986), pp. 189-207. For a Hamiltonian, H = H 0 + H (A6.1) where H 0 is a Hamiltonian for which the Schrödinger equation can be solved and H represents an additional potential due to, for example, an applied electric field. Then, H 0 satisfies the time-dependent Schrödinger equation: i δψ 0 δt = H 0 ψ 0 (A6.2) for a wavefunction ψ 0 = u x, y, z e −iE 0 t/ (A6.3) or, more generally, ψ 0 = n a 0 n u 0 n e −iE o n t/ (A6.4) where the unperturbed state described by H 0 has a set of energy eigenvalues such that H 0 u 0 n = E 0 n u n (A6.5) For an orthogonal set of eigenvectors u n and prefactors a n 0 are independent of time. Then, Hψ = i δψ δt = H 0 + H ψ (A6.6) Surfaces and Interfaces of Electronic Materials. Leonard J. Brillson