By Cox R.T., Hubbard J.C.
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Extra resources for A Statistical Quantum Theory of Regular Reflection and Refraction
121). 143) can also be constructed. 174)]. Hence, the corresponding dynamical variables of the medium must be included in the quantization scheme, which implies extension of the Hilbert space. For the sake of transparency let us ﬁrst restrict our attention to the case when the dielectric medium is the only matter that is present. 170) for ρ = j = 0. Let us again restrict our attention to 23) Note that with respect to the integration measure ε (r)d3 r in the deﬁnition of scalar products, the differential operator ε−1 (r)∇ × ∇× is Hermitian.
123). 4) as ˙ ( r) . 126), the terms P˙ A and ∇ × M A are also called polarization and magnetization currents respectively. 15) for the charge and current densities, respectively, and using, e. 1 The multipolar-coupling Lagrangian The total derivative with respect to time of a function of the generalized coordinates can of course be added to the Lagrangian L of a system to obtain a new Lagrangian L , which yields the same equations of motion and is therefore fully equivalent to the old Lagrangian.
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). 55). Clearly, the associated mode operators do not evolve freely, because of the interaction with the medium. 219) [together with Eq. 2 The minimal-coupling Hamiltonian When additional charged particles are present, then the interaction of the particles with the medium-assisted electromagnetic ﬁeld can be described by the 25) Note that G ⊥( ) (r, r , ω ) = d3 s δ ⊥( ) (r − s) G (s, r , ω ).