Quantum Theory of Absorption and Emission

We will return to the shapes of absorption and emission spectra in Chaps. 10 and 11. 

Although Einstein s theory accounts well for the relative amplitudes of absorption, fluorescence and stimulated emission, the notion that fluorescence occurs spontaneously is fundamentally inconsistent with the assertion we made in Chap. 2 that an isolated system is stable indefinitely in any one of its eigenstates. If the latter principle is correct, fluorescence must be caused by some perturbation we have neglected. The quantum theory of radiation provides a way out of this conundrum. As we discussed in Chap. 3, a radiation field has an eigenstate in which the number of photons is zero. Spontaneous fluorescence can be ascribed to perturbation of the excited molecule by the zero-point radiation field [26,27]. Let s examine this rather xmsettling idea. 

We saw in Chap. 3 that electromagnetic radiation can be described by a vector potential V that is a periodic function of time and position, along with a scalar potential that can be made equal to zero by a judicious choice of the guage of the potentials. For linearly polarized radiation in a vacuum, the vector potential can be 

Now consider the interactions of a radiation field with an electron. In the absence of the radiation field, the Hamiltonian for the electron would be 

Note that the vector potential combines vectorially with the momentum operator in Eq. (5.31a), rather than simply adding to the ordinary potential. The derivation of this formulation of the Hamiltonian is not straightforward, but the result can be justified by showing that it leads to the correct forces on a charged particle [26,28, 29]. Combining V V and V V into a single term in Eq. (5.31c) is justified by the fact that V and V commute, which follows from Eq. B3.1.16 [26]. 

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The transformation of the time-dependent function Pj into a momentum operator is consistent with Einstein s description of light in terms of particles (photons), each of which has momentum hv (Sect. 1.6 and Box 2.3). We can interpret the quantum number rij in Eq. (3.50) either as the particular excited state occupied by oscillator j, or as the number of photons with frequency Vj. The oscillating electric and magnetic fields associated with a photon can stiU be described by Eqs. (3.44) and (3.45) if the amplitude factor is scaled appropriately. However, we will be less concerned with the spatial properties of photon wavefunctions themselves than with the matrix elements of the position operator Q. These matrix elements play a central role in the quantum theory of absorption and emission, as we ll discuss in Chap. 5. 

With a traveling fellowship awarded by Harvard, Slater spent his first postdoctoral year at Cambridge. There, he developed a theory on radiative transitions in atoms. On discussing this idea with Neils Bohr and Hans Kramers, a joint paper on the quantum theory of radiation was published in 1924. However, Bohr and Kramers altered Slater s original idea by ascribing a virtual existence to the photons in the transitions— not the real photons that Slater believed in. In early 1925, Slater was back at Harvard and published further work of his own on radiative transitions. He presented a picture of absorption and emission of real photons coupled with energy conservation in transition processes. He also established a relationship between the width of spectral lines and the lifetimes of states. 

Absorption and emission spectroscopies provide experimental values for the quantized energies of atomic electrons. The theory of quantum mechanics provides a mathematical explanation that links quantized energies to the wave characteristics of electrons. These wave properties of atomic electrons are described by the Schrddinger equation, a complicated mathematical equation with numerous terms describing the kinetic and potential energies of the atom. 

The quantum mechanical perturbation theory that allows us to calculate the probability of absorption and stimulated emission shows (see Section 5.3) that both... 

There are several reasons for starting this account with a discussion of electromagnetic radiation. Historically, it was in this area that the quantum theory first developed. It is easier here to understand the evidence for the theory, and to appreciate some of its paradoxical consequences, than it is in the quantum theory of matter. The applications of the light-quantum hypothesis, as it was first called, also provide key pieces of evidence for the quantization of energy in atoms and molecules. Studies of the absorption and emission of radiation—the field of spectroscopy—and of the effect of light on chemical reactions—photochemistry—are very important areas of modem chemistry, in which the quantum nature of radiation is crucial. 

What has been presented here is a semiclassical theory of TJ 1) quantum electrodynamics. Here the electromagnetic field is treated in a purely classical manner, but where the electromagnetic potential has been normalized to include one photon per some unit volume. Here the absorption and emission of a photon is treated in a purely perturbative manner. Further, the field normalization is done so that each unit volume contains the equivalent of n photons and that the energy is computed accordingly. However, this is not a complete theory, for it is known that the transition probability is proportional to n + 1. So the semiclassical theory is only appropriate when the number of photons is comparatively large. 

The EM theory of Metal Enhanced Fluorescence (MEF) was studied and developed extensively in the 70-80 s [2,3,4,5,6], All the EM mechanims involved in MEF can be understood within classical EM theory [3,4,5,6] as confirmed in the simplest cases by quantum studies [2,12,17,18], In most of these models, the emitter is depicted as a simple two- (or three-) level system, i.e. only one emission wavelength is considered. This is appropriate in general to understand modifications of absorption or emission rates, but it entirely ignores the spectral profile of the fluorescence emission. We will first review... 

Max Planck in 1900 derived the correct form of the blackbody radiation law by introducing a bold postulate. He proposed that energies involved in absorption and emission of electromagnetic radiation did not belong to a continuum, as implied by Maxwell s theory, but were actually made up of discrete bundles—which he called quanta. Planck s idea is traditionally regarded as the birth of quantum theory. A quantum associated with radiation of frequency v has the energy... 

All the considerations that follow are only valid for radiation that is stimulated thermally. Radiation is released from all bodies and is dependent on their material properties and temperature. This is known as heat or thermal radiation. Two theories are available for the description of the emission, transfer and absorption of radiative energy the classical theory of electromagnetic waves and the quantum theory of photons. These theories are not exclusive of each other but instead supplement each other by the fact that each describes individual aspects of thermal radiation very well. 

F. London, Zur Theorie und Systematik der Molekularkrafte. Zeits. fur Phys. 63 (1930) 245. P.A.M. Dirac, The Quantum Theory of the Emission and Absorption of Radiation. Proc. Roy. Soc. Lond. A114 (1927) 243. 

Many linear polyenes exhibit anomalous fluorescence behaviour in the sense that the fluorescence rate constant kf calculated from the absorption spectrum (Equation 2.11) is much smaller than that determined by lifetime and quantum yield measurements (Equation 3.33). In 1972, Hudson and Kohler reported high-resolution absorption and emission spectra of all-trart.v-l, 8-diphenylocta-l, 3,5,7-tetraene at low temperature, which proved that the lowest excited singlet state Si was not that reached by the strongly allowed 7t,Jt transition (/ 1.5) that is predicted by MO theory and observed at 410nm.3" Rather, very weak (f 0.06), structured absorption that had been hidden under the tail of the jt,jt -absorption in solution spectra was detected at slightly longer wavelengths. 

The selection rules of the new quantum theory allow electric dipole transitions between levels of the same n, but since the probability of spontaneous transition depends on the cube of the frequency, such transitions in hydrogen are exceedingly improbable. Stimulated transitions, on the other hand, may take place under quite small alternating electric fields of the appropriate frequency. Absorption and emission of energy by the atom are equally probable, so that a change in an assembly of atoms may only be detected if the two states between which transitions are taking place are unequally populated at the outset. 

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