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arXiv:2006.13180

New Perspectives on the so-called Fermi's Golden Rule in Quantum Mechanics including Adiabatic Following

Mike G. Burt

Published Jun 23, 2020• 0 citations
arXivHTMLPDF

Abstract

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.

DOI
10.48550/arxiv.2006.13180

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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.
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.
A time-dependent perturbation theory which is based upon the U-matrix approach is presented using the interaction e-αt(1 + e-βt)−1V(x) with β real, α complex and β > Re α > 0. Thus the adiabatic or time-independent approximation (β → 0) and the sudden approximation (β → ∞) can be obtained using just one formalism. The usual series for the U matrix is derived and shown to converge for a semi infinite range if the interaction is of the form |t|-δ, δ > 1, for large times t. Two interesting results are (1) The derivation of the ``golden rule'' from the discrete-state time-dependent perturbation theory presented in most text books of quantum mechanics leads to erroneous physical interpretations because the energy-level shift is ignored. (2) When considering scattering between states in the continuum, it is found that a characteristic feature of time-dependent interactions is a discrete momentum spectrum of final states which, in the relativistic case, leads to a mass spectrum. These spectra cannot be obtained using the S matrix.
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.
For a perturbed quantum system initially in the ground state, the coefficient ck(t) of excited state k in the time-dependent wave function separates into adiabatic and nonadiabatic terms. The adiabatic term ak(t) accounts for the adjustment of the original ground state to form the new ground state of the instantaneous Hamiltonian H(t), by incorporating excited states of the unperturbed Hamiltonian H0 without transitions; ak(t) follows the adiabatic theorem of Born and Fock. The nonadiabatic term bk(t) describes excitation into another quantum state k; bk(t) is obtained as an integral containing the time derivative of the perturbation. The true transition probability is given by bk(t)2, as first stated by Landau and Lifshitz. In this work, we contrast bk(t)2 and ck(t)2. The latter is the norm-square of the entire excited-state coefficient which is used for the transition probability within Fermi’s golden rule. Calculations are performed for a perturbing pulse consisting of a cosine or sine wave in a Gaussian envelope. When the transition frequency ωk0 is on resonance with the frequency ω of the cosine wave, bk(t)2 and ck(t)2 rise almost monotonically to the same final value; the two are intertwined, but they are out of phase with each other. Off resonance (when ωk0 ≠ ω), bk(t)2 and ck(t)2 differ significantly during the pulse. They oscillate out of phase and reach different maxima but then fall off to equal final values after the pulse has ended, when ak(t) ≡ 0. If ωk0 < ω, bk(t)2 generally exceeds ck(t)2, while the opposite is true when ωk0 > ω. While the transition probability is rising, the midpoints between successive maxima and minima fit Gaussian functions of the form a exp[−b(t − d)2]. To our knowledge, this is the first analysis of nonadiabatic transition probabilities during a perturbing pulse.
A simple way to derive Bardeen’s tunneling theory is introduced. With a low rate of tunneling of, for example, electrons from one electrode to another through a wide vacuum gap, the appearance of the second electrode can be regarded as a perturbation. Then Bardeen’s formula for tunneling follows exactly from Fermi’s golden rule, supposing that this perturbation appears suddenly at some instant of time. Bardeen’s theory is illustrated by assuming a smoothly variable barrier shape, including the rectangular one as a limiting case. This exactly solvable example clarifies the sensitivity of the tunneling probability to the barrier shape and the energy of the tunneling particle.
Abstract In this chapter we review time-independent and time-dependent perturbation theory in quantum mechanics. We begin with Rayleigh-Schrödinger perturbation theory for small corrections to the Hamiltonian for non-degenerate and degenerate spectra, which leads us to a discussion of avoided crossings. We then apply these methods to calculate the fine-structure corrections for the energy levels of the hydrogen atom, including the relativistic correction to the kinetic energy, the spin-orbit correction, and the Darwin term. We then turn to the effect of weak constant magnetic fields: for the resulting Zeeman effect we distinguish between the anomalous Zeeman effect and the Paschen-Back effect. In the context of the corresponding Stark effect for electric fields we introduce the Wigner-Eckart theorem and calculate perturbations to first and second order. We then turn to time-independent problems and discuss the Born-von Neumann series for the propagator, time-ordered integrals, the interaction picture, before explaining Fermi’s golden rule.