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International Journal of Quantum Chemistry

Quantum theory of radiative transition phenomena

Peter W. Langhoff, W. R. Heffner

Published Jun 18, 2009• 0 citations
Publisher

Abstract

An elementary account is given of the absorption and emission of radiation by molecules employing a particular formulation of Schrödinger theory which facilitates the treatment of radiative line widths and level shifts. The familiar Fermi golden rule for transition rates, the Rabi Bloch-Siegert absorption profile, and the Weisskopf-Wigner approximation for spontaneous decay are obtained under appropriate conditions in the context of a common perspective. Comparisons are made with other recently devised perturbative and nonperturbative formulations of electromagnetic transition phenomena which correctly include radiative line widths and level shifts, and the importance of proper treatment of the normalization and phase of the Schrödinger wave function in this connection is emphasized.

DOI
10.1002/qua.560090856

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An elementary account is given of the absorption and emission of radiation by molecules employing a particular formulation of Schrödinger theory which facilitates the treatment of radiative line widths and level shifts. The familiar Fermi golden rule for transition rates, the Rabi Bloch-Siegert absorption profile, and the Weisskopf-Wigner approximation for spontaneous decay are obtained under appropriate conditions in the context of a common perspective. Comparisons are made with other recently devised perturbative and nonperturbative formulations of electromagnetic transition phenomena which correctly include radiative line widths and level shifts, and the importance of proper treatment of the normalization and phase of the Schrödinger wave function in this connection is emphasized.
Abstract “Spontaneous emission of radiation” calculates the rate of spontaneous electric-dipole emission of a photon by an excited atom or molecule. The calculation proceeds by using basic quantum mechanics (i.e. not using the short cuts of Chapter 19); it uses quantum electrodynamics but is not, on that account, particularly difficult. A 2p–1s transition in hydrogen is used as exemplar; the radiation is elliptically polarized. The spectral line profile (lineshape function) is approximately Lorentzian, but has a high-frequency cut-off, needed to prevent the power radiated from diverging. A radiation-induced frequency shift is negligible. The width of the line profile agrees with the Einstein A-coefficient. A high-frequency cut-off is shown to apply similarly in the derivation of Golden Rule Number Two.
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Radiation can change the energy state of matter via three optical processes: spontaneous emission, absorption and stimulated emission. In spontaneous emission, an excited atomic or molecular state decays to a lower energy state with the emission of a photon of energy equal to the difference in the quantum state energies. Photon absorption is the reverse process, with a photon causing excitation from a lower to an excited quantum state. Stimulated emission was first proposed by Einstein in a paper published in 1917. He attempted to balance spontaneous emission and photon absorption, but could not get the correct balance without the postulated process of stimulated emission (see Section 4.2). With stimulated emission, an excited quantum state is stimulated to decay to a lower quantum state by an impinging photon. The existing photon survives with another identical photon being created (so that energy is conserved). The rates at which radiative transitions occur can be calculated using quantum mechanics. The time-dependent Schrodinger equation is used with the system wavefunction comprising time-varying linear combinations of the two wavefunctions for the lower and upper quantum states. In semi-classical treatments, an atom is treated quantum mechanically with the radiation field treated classically. There is a perturbation in energy associated with, say, the electric field of the radiation interacting with the electric dipole of the atom, though other interactions can be significant if the electric dipole interaction energy is small. In absorption, for example, a differential equation representing the change in ‘weighting’ of the wavefunction from the lower to upper state is obtained. A ‘time constant’ for the speed of change represents the rate of the radiative process. We examine the quantum mechanical nature of radiation absorption in Section 10.1 before discussing (Section 10.3) another important parameter arising with radiative transitions between discrete bound quantum states: the lineshape function. Quantum Theory of the Atom–Radiation Interaction An expression for the Einstein B-coefficient for photo-absorption is derived using quantum mechanics in this section. Einstein determined the detailed balance relationship between the rates of spontaneous emission, photo-absorption and stimulated emission when light interacts with an atom by considering an atomic system of two energy levels which are in equilibrium with a black-body radiation field. We use a semi-classical treatment where the atom is treated quantum mechanically and the radiation field is treated as a classical oscillating electric field.
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.
Perturbation theory for the interaction of light with a binary collisional system is applied to the calculation of line profiles, spontaneous emission rates and absorption or stimulated emission coefficients. For the usual radiation fields are perturber densities, this approximation is valid at all detunings Delta omega except the very narrow resonance region. Appropriate expansions of the total binary system wavefunctions give the emission and absorption profiles or rates, for atom-electron systems as well as for atom-atom systems, from the impact regime at small Delta omega up to the Born-Oppenheimer regime at very large Delta omega in collision-induced radiative transitions, the transitions being asymptotically allowed or forbidden. Emphasis is given to the proper averages and sums over initial and final states of the binary system which give rise to non-adiabatic effects. One shows that it is of crucial importance to include all dynamical effects during one collisional event before undertaking any calculation at small Delta omega where simultaneous collisions may occur.
The development of the theory of the interaction of electromagnetic radiation with atoms and molecules is outlined. A fully classical analysis of absorption and emission is formulated in which particular attention is paid to questions of detailed balance and to the sum rules obeyed by the susceptibility. Collision broadening is introduced through time-dependent dipole moment correlation functions, and brief discussions of interruption broadening and various frequency modulation models are given. The corresponding quantum mechanical analysis is presented with emphasis on the points in common with the classical approach. The importance of correspondence principles in bridging the gap between classical and quantum mechanical theories is stressed.