Abstract
Starting from Maxwell and Newton equations, we develop here the theory of emission, dispersion, absorption, and scattering of light by matter based on the oscillator model of the matter. This is the standard material that can be found in many places, see, e.g., Feynman et al. (Feynman Lectures on Physics. Basic Books, New York, 2011). Schwinger et al. (Classical Electrodynamics. Perseus Books, New York, 1998). For our purposes, the most important is the final Sect. 2.10 where we introduce the notion of oscillator strengths and Thomas–Kuhn–Reiche sum rule.
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Notes
- 1.
This identity can be proven either by the direct substitution of Eqs. (1.15) and (1.16) into Eq. (2.31) or from symmetry considerations: on the right, we need quantity with two indices, and the only available is Kronecker delta. The numerical factor follows from setting \(l=k\) and using Einstein summation convention; then \(n_k n_k=1\), \(\delta _{kk}=3\) and \(\int \mathrm {d} \varOmega = 4\pi \).
- 2.
As we show later, see Eqs. (4.18) and (4.55), the oscillator strengths equal in modern notation
$$\displaystyle \begin{aligned} f_k = \frac{2}{3} m \omega_{k0} |\vec{q}_{0k}|{}^2 \,, \end{aligned}$$where the circular frequency \(\omega _{k0}\) is the difference between the energies of the k-th excited and ground stationary states, see Eq. (4.4), and \(\vec {q}_{0k}\) is the matrix element of the coordinate between the ground and k-th excited states, see Eq. (5.7).
References
R.P. Feynman, R.B. Leighton, M. Sands, Feynman Lectures on Physics (Basic Books, New York, 2011)
J. Schwinger, L. de Raad Jr, K.A. Milton, W. Tsai, Classical Electrodynamics (Perseus Books, New York, 1998)
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Zamastil, J. (2023). Classical Electrodynamics. In: Understanding the Path from Classical to Quantum Mechanics. SpringerBriefs in Physics. Springer, Cham. https://doi.org/10.1007/978-3-031-37373-2_2
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