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Cambridge University Press eBooks

Dissipation, level broadening and radiation

Alfred Brian Pippard

Published Nov 16, 1989• 0 citations
Publisher

Abstract

After this excursion into the field of non-linear vibrators, we now return to the harmonic vibrator and take up a point which had begun to reveal itself at the end of chapter 13, where we found that under the influence of a uniform, but arbitrarily time-varying, force the vibrator never forgets its initial state. If it started in the ground state, for ever afterwards its response can be described by the movement of a compact distribution in the σ-representation of equivalent classical vibrators; only the centroid of the distribution responds to the applied force. The harmonic vibrator is thus extraordinarily resistant to randomization. To be sure, if the force is not uniform, but depends on the displacement of the oscillating particle, the result just summarized is no longer true. Nevertheless, in the most important application, where an oscillator of atomic dimensions is influenced by electromagnetic vibrations, the force due to the electric field is as nearly uniform as makes no difference, since the wavelength of electromagnetic waves at a typical atomic resonant frequency is a thousand times the size of an atom. The disturbing aspect of the resistance of a vibrator to randomization is that in all theories of black-body radiation, before and after Planck, it is assumed that material oscillators and electromagnetic vibrations in a cavity will eventually share the chaotic state that allows statistical mechanics to be applied.

DOI
10.1017/cbo9780511622908.019

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After this excursion into the field of non-linear vibrators, we now return to the harmonic vibrator and take up a point which had begun to reveal itself at the end of chapter 13, where we found that under the influence of a uniform, but arbitrarily time-varying, force the vibrator never forgets its initial state. If it started in the ground state, for ever afterwards its response can be described by the movement of a compact distribution in the σ-representation of equivalent classical vibrators; only the centroid of the distribution responds to the applied force. The harmonic vibrator is thus extraordinarily resistant to randomization. To be sure, if the force is not uniform, but depends on the displacement of the oscillating particle, the result just summarized is no longer true. Nevertheless, in the most important application, where an oscillator of atomic dimensions is influenced by electromagnetic vibrations, the force due to the electric field is as nearly uniform as makes no difference, since the wavelength of electromagnetic waves at a typical atomic resonant frequency is a thousand times the size of an atom. The disturbing aspect of the resistance of a vibrator to randomization is that in all theories of black-body radiation, before and after Planck, it is assumed that material oscillators and electromagnetic vibrations in a cavity will eventually share the chaotic state that allows statistical mechanics to be applied.
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As is appropriate in an exposition of the quantum mechanics of vibrators, we started with a quantal calculation and have finished with one; but we have never strayed far from the classical models. We began, indeed, in chapter 13 the task of establishing limits to the validity of classical reasoning, and the very last example of this chapter has been used to demonstrate, somewhat perversely it might be thought, that the method of quantum mechanics can occasionally be applied to systems which physicists and engineers would instinctively regard as classical. So long as the discussions give insight into physical processes and reveal the strengths and weaknesses of the analytical tools available, no apology is needed and no defence is offered except the evidence of the book itself. Even the rather simple systems which provide material for the whole of the volume demonstrate clearly the increased power available to those who can handle both classical and quantal reasoning. The reader who wishes to apply his skill to complex vibratory processes will find his tasks eased if he can use whichever seems advantageous at each stage, and be confident that his understanding of simple processes will enable him to recognize the dangers inherent in both approaches – the danger of carrying classical reasoning too far into the quantum domain, and the danger of forcing over-simplifications on physical systems to make them amenable to the unforgiving methodology of quantum mechanics.
We begin with some examples of one-dimensional vibration in a nonparabolic potential chosen to permit complete analytical solution of Schrödinger's time-independent equation. These examples are of isochronous vibrators which classically have a frequency independent of amplitude, and which might be expected therefore to have energy levels equally spaced at a separation of ħω 0 . This expectation turns out to be very nearly right and inspires a certain confidence in the semi-classical procedure developed by (among others) Bohr, Wilson and Sommerfeld. We therefore apply this procedure to some non-isochronous systems and find once more rather good agreement with the results of exact quantum mechanics. Periodic systems in fact can often be treated semi-classically with adequate accuracy, and significant economy of effort in comparison with strict quantum-mechanical analysis. This approach pays handsomely when we turn in the next chapter to the quantization of electron cyclotron orbits which, as already discussed in chapter 8, are closely related to harmonic oscillators. Conduction electrons in semi-conductors, and still more in metals, have their behaviour modified by the lattice through which they move, and a complete quantal treatment has never been achieved. It is clear, however, from approximate calculations, often of great complexity, that the semiclassical method describes most of the interesting physical processes correctly and very simply. In chapter 15 we shall describe in outline some of the effects which can be treated quite well enough for most purposes without even writing down Schrödinger's equation.
Abstract This paper is a study of a statistical ensemble of classical harmonic oscillators which is stationary in time, and whose position and momentum distribution functions are those of the corresponding quantum-mechanical oscillator in its ground state. If the oscillating particle is charged, then in order to maintain the distribution stationary a certain random electromagnetic field must be present. The intensity distribution of the radiation field is calculated, and it is found to be identical with the ‘photon vacuum’ of quantum electrodynamics. It is suggested, therefore, that this radiation field, which in quantum field theory is treated entirely formally, might exist in the classical sense. The method is extended to the excited states. It is found that the classical ensembles corresponding to these have probability distributions which may be negative. However, when attention is shifted to the quantum-mechanical ‘mixture’, this is no longer the case. Furthermore, the intensity distribution of the radiation field at temperature T is shown to be simply the sum of the Planck distribution and the previously obtained zero-temperature field. The application of these results to statistical mechanics is discussed. In this paper the treatment is non-relativistic throughout, but radiative corrections of the type which give rise to the Lamb shift are an integral part of the theory.
In a generic quantum mechanical description of a molecule interacting with its thermal environment, the molecule is represented as a few level system (in the simplest description just two, for example, ground and excited states) and the environment is often modeled as a bath of harmonic oscillators. The resulting theoretical framework is known as the spin–boson model, a term that seems to have emerged in the Kondo problem literature (which deals with the behavior of magnetic impurities in metals) during the 1960s, but is now used in a much broader context. Indeed, it has become one of the central models of theoretical physics, with applications in physics, chemistry, and biology that range far beyond the subject of this book. Transitions between molecular electronic states coupled to nuclear vibrations, environmental phonons, and photon modes of the radiation field fall within this class of problems. The present chapter discusses this model and some of its mathematical implications. The reader may note that some of the subjects discussed in Chapter 9 are reiterated here in this more general framework. In Sections 2.2 and 2.9 we have discussed the dynamics of the two-level system and of the harmonic oscillator, respectively. These exactly soluble models are often used as prototypes of important classes of physical system. The harmonic oscillator is an exact model for a mode of the radiation field and provides good starting points for describing nuclear motions in molecules and in solid environments. It can also describe the short-time dynamics of liquid environments via the instantaneous normal mode approach. In fact, many linear response treatments in both classical and quantum dynamics lead to harmonic oscillator models: Linear response implies that forces responsible for the return of a system to equilibrium depend linearly on the deviation from equilibrium—a harmonic oscillator property! We will see a specific example of this phenomenology in our discussion of dielectric response in Section 16.9.