By R Hagedorn
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Post yr word: First released July 1st 1980
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Additional resources for Introduction to field theory and dispersion relations
203) and using Eq. 210), we convert E(r, ω ) [Eq. 205)] and B(r, ω ) [Eq. 211) and ˆ (r, ω ) = (iω )−1 ∇ × Eˆ (r, ω ). c.. 203), ˆ (r, ω ) = (µ0 ω 2 )−1 ∇ × ∇ × Eˆ (r, ω ). 50) (Appendix B). Obviously, the Hamiltonian of the composed system can be given in the form of Hˆ = d3 r ∞ 0 dω h¯ ω fˆ † (r, ω )fˆ(r, ω ). c. c. 46) (Apˆ and −ε 0 Eˆ are respectively the transverse part and pendix B). Note that Π the longitudinal part of a common vector ﬁeld, and Eˆ can be attributed to a ˆ scalar potential V, −∇Vˆ (r) = Eˆ (r).
Schubert and Wilhelmi (1986)] that this approximation may be justiﬁed as long as the electric ﬁeld is weak compared with the intra-atomic electric ﬁeld (Eatom ≈ 1010 Vm−1 ), to which the active charges are subjected, owing to their interaction with the atomic cores. Similarly, the nonlinear multipolar-coupling term in Eq. 150) may be disregarded [see, e. , Loudon (1983)]. 31 In the electric-dipole ˆ (r A ) is retained and Eq. 121) approximation, only the zeroth-order term A simpliﬁes to 2 Qa ˆ (r A ) + ∑ Q a A ˆ 2 (r A ).
4 Dielectric background media The new Hamiltonian Hˆ expressed in terms of the new variables formally looks like the old minimal-coupling Hamiltonian Hˆ expressed in terms of the originally used variables. Expressing in Hˆ the new variables in terms of the original ones, we arrive at a multipolar-coupling Hamiltonian Hˆ = + 1 2 1 ˆ ˆ⊥ dr3 ε− 0 Π ( r) + P A ( r) 1 2 1 ∑ ma pˆ a + Q a a 1 0 2 ˆ ( r) + µ0−1 ∇ × A 2 ˆ [r A + s(rˆ a − rˆ A )]] ds s(rˆ a − rˆ A ) × [∇ × A 2 ˆ Coul . 163) look the same at ﬁrst glance but in fact they are different.