Wavefunction quantum light theory

The uncertainty principle shows that the classical trajectory of a particle, with a precisely determined position and momentum, is really an illusion. It is a very good approximation, however, for macroscopic bodies. Consider a particle with mass I Xg, and position known to an accuracy of 1 pm. Equation 2.41 shows that the uncertainty in momentum is at least 5 x 10 29 kg m s-1, corresponding to a velocity of 5 x 10 JO m s l. This is totally negligible for any practical purpose, and it illustrates that in the macroscopic world, even with very light objects, the uncertainty principle is irrelevant. If we wanted to, we could describe these objects by wave packets and use the quantum theory, but classical mechanics gives essentially the same answer, and is much easier. At the atomic and molecular level, however, especially with electrons, which are very light, we must abandon the idea of a classical trajectory. The statistical predictions provided by Bom s interpretation of the wavefunction are the best that can be obtained. 

Quantum-mechanical expressions for the polarizability and other higher-order molecular response tensors are obtained by taking expectation values of the operator equivalent of the electric dipole moment (2.5) using molecular wavefunctions perturbed by the light wave (2.4). This particular semi-classical approach avoids the complications of formal time-dependent perturbation theory it has a respectable pedigree, being found in Placzek s famous treatise on the Raman effect [9], and also in the books by Born and Huang [lO] and Davydov [ll]. Further details of the particular version outlined here can be found in my own book [12]. 

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. 

Before going any further, it will be helpful to understand the physical significance of a wavefunction. The interpretation most widely used is based on a suggestion made by the German physicist Max Born. He made use of an analogy with the wave theory of light, in which the square of the amplitude of an electromagnetic wave is interpreted as its intensity and therefore (in quantum terms) as the number of photons present. The Born interpretation asserts ... 

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