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Transitions

Ian R. Kenyon

Published Sep 10, 2019• 0 citations
Publisher

Abstract

Abstract A derivation of Fermi’s golden rule is given: this is the interface into which matrix elements from theory can be slotted to provide a prediction testable by experiment. The example of the prediction of the 2p→1s decay in hydrogen is worked through in detail. Selection rules, spectral line shapes (Breit–Wigner and Gaussian) and broadening processes are explained. The formula for the experimental cross-section in terms of the matrix element is produced. The Born approximation is presented and applied to Rutherford scattering. Then the decay rate for allowed β-decays is calculated in Fermi’s model and fitted to the observed rates. Low energy s-wave scattering is analysed in terms of phase shift and scattering length. The example of cold alkali metal atom scattering (≤10−6eV) is treated in preparation for use later with gaseous Bose–Einstein condensates. Ramsauer–Townsend effect explained.

DOI
10.1093/oso/9780198808350.003.0007

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Abstract A derivation of Fermi’s golden rule is given: this is the interface into which matrix elements from theory can be slotted to provide a prediction testable by experiment. The example of the prediction of the 2p→1s decay in hydrogen is worked through in detail. Selection rules, spectral line shapes (Breit–Wigner and Gaussian) and broadening processes are explained. The formula for the experimental cross-section in terms of the matrix element is produced. The Born approximation is presented and applied to Rutherford scattering. Then the decay rate for allowed β-decays is calculated in Fermi’s model and fitted to the observed rates. Low energy s-wave scattering is analysed in terms of phase shift and scattering length. The example of cold alkali metal atom scattering (≤10−6eV) is treated in preparation for use later with gaseous Bose–Einstein condensates. Ramsauer–Townsend effect explained.
Equivalent derivations of time-dependent pertubation theory; Fermi golden rule; the Born approximation for scattering from Fermi golden rule; survival probability of a state in a time-independent perturbation; positronium in static and oscillating magnetic fields; hydrogen atom in a time-dependent electric field; a model for inelastic scattering of a projectile with a target; semiclassical treatment of the electromagnetic field; ionization of the hydrogen atom by an electromagnetic wave; cross sections for stimulated absorption and emission in hydrogen; spontaneous emission and selection rules with an application to the 2p to 1s transition in hydrogen; theory of the line width; formal scattering theory; S- and T-operators.
Abstract Fermi's Golden Rule is one of the most important results in quantum mechanics, predicting transition rates in applications ranging from beta decay to optical transitions, autoionization and photoionization. Most textbooks derive the Golden Rule only for the canonical case of a discrete state coupled by a time-independent perturbation to a continuum of final states. Here we present a unified treatment of several physical situations, including light absorption in atoms and solids, that all lead to the same Golden Rule expression. The essential ingredients are the transition matrix element and a spectral weighting function, whose physical origin need not be the density of final states, nor necessarily anything to do with the matter.The conventional discrete-to-continuum derivation introduces pedagogical complications associated with continuum normalization and the interpretation of the density of states. Among the cases considered here, the derivation for discrete transitions driven by a noisy perturbation is no more difficult mathematically but conceptually clearer, while leading directly to practical calculations. This suggests that it provides a more natural first introduction to the Golden Rule, with conventional continuum derivations better viewed later as another realization of the same underlying mathematical structure, rather than the other way around.
In Chapter 3 the interaction Hamiltonian which determines the way light interacts with matter is derived. Simple perturbative analysis is applied to see if basic dynamics of atoms can be explained. The partial successes of perturbation theory are compared with predictions of an “exact” method of calculating the occupation probabilities of various atomic states. Even the “exact” method is shown to fail, establishing the need for improved approaches that yield correct results in later chapters. Some elementary results for the transition rates, including Fermi's golden rule are covered. Then the density matrix is introduced as a tool for describing not only the populations of atomic energy levels but coherence that can be created and lost during the dynamic evolution of atoms in time. A vector model based on the Bloch vector is presented as a useful way of picturing coherent atom–field interactions in an optical “spin” space, particularly multiple‐pulse interactions. Mechanisms are described that cause line‐broadening in optical spectroscopy, such as the Doppler effect. The chapter closes by showing that it is usually possible to reduce the complexity of real atoms on an experimental basis to facilitate direct comparisons with theoretical calculations based on simple models with only two or three energy levels.
A novel and readily understandable derivation of the Golden Rule of time dependent perturbation theory is presented. The derivation is based on adiabatic turning on of the perturbation as used, for instance, in some formal developments of scattering theory. Energy conservation is expressed in terms of an intuitively and physically appealing Lorentzian line shape rather than the artificial, oscillatory sin(x)=x type line shape that appears in conventional derivations. The conditions for the derivation's validity are compactly and conveniently expressed in the frequency/energy domain rather than in the usual time domain. The derivation also highlights how, along with energy conservation, the transition rate faithfully and instantaneously follows the variations in the square of the perturbing potential as one may expect in the adiabatic limit. In the first instance, the adiabatic turning on is achieved, as usual, by a single exponential time variation. But we demonstrate that the instantaneous following of the square of the perturbing potential by the transition rate is more general and that one can derive the Golden Rule for a general slowly varying time dependent perturbation. This allows one to derive generalisations of the simple decay law, originally derived in the classic paper by Weisskopf and Wigner; a tutorial exposition of the essence of this classic work is provided. The Oppenheimer method for applying the Golden rule to problems, such as the electric field ionisation of atoms, in which the perturbing potential can also create the final states, is reviewed. No use of an energy gap condition is needed to derive our results on adiabatic behaviour in contrast to the origianl derivations of the adiabatic theorem in quantum mechanics.
A quantum mechanical wave of a finite size moves like a classical particle and shows a unique decay probability. Because the wave function evolves according to the Schrodinger equation, it preserves the total energy but not the kinetic energy in the intermediate-time region of a decay process where those of the parent and daughters overlap. The decay rate computed with Fermi's golden rule requires corrections that vary with the distance between the initial and final states, and the energy distribution of the daughter is distorted from that of plane waves. The corrections have universal properties in relativistically invariant systems and reveal macroscopic quantum phenomena for light particles. The implications for precision experiments in beta decays and various radiative transitions are presented.
A quantum mechanical wave of a finite size moves like a classical particle and shows a unique decay probability. Because the wave function evolves according to the Schrödinger equation, it preserves the total energy but not the kinetic energy in the intermediate-time region of a decay process where those of the parent and daughters overlap. The decay rate computed with Fermi's golden rule requires corrections that vary with the distance between the initial and final states, and the energy distribution of the daughter is distorted from that of plane waves. The corrections have universal properties in relativistically invariant systems and reveal macroscopic quantum phenomena for light particles. The implications for precision experiments in beta decays and various radiative transitions are presented.