Adiabatic quantum transport. by Avron

By Avron

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130) 1 Lint = − ∑ Q a 0 a ˙ [r A + s(ra − r A )] ds (ra − r A )A − [s(ra − r A ) × r˙ a ][∇ × A[r A + s(ra − r A )]] . 136) It is worth noting that the transverse electromagnetic field appears in the new ˙ In parLagrangian in terms of the field strengths B = ∇ × A and E⊥ = −A. ticular, the charges interact with the transverse field via the coupling of the magnetization to the induction field and the coupling of the polarization to the transverse electric field, d3 r [P A (r)E⊥ (r) + M A (r)B(r)] Lint = = ∑ Qa a 1 0 ds (ra − r A )E⊥ [r A + s(ra − r A )] + [s(ra − r A ) × r˙ a ]B[r A + s(ra − r A )] .

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. 24) Note that the photons defined in this way do not only refer to the transverse part of the electromagnetic field. 4 Dielectric background media nonmagnetic matter, so that Eq. 171) holds. 203) which obey the continuity equation ∇jN (r, ω ) = iωρN (r, ω ). 207) together with the boundary condition at infinity, G (r, r , ω ) → 0 if |r − r | → 0 (for the properties of the Green tensor, see Appendix A).

83), with increasing V the modes become more and more dense in the k domain. 90) we see that in the limit as V → ∞ (i. 91) ˆ (r) accordingly. The commutation relations for the operators aˆ σ (k) and Π ˆ and a†σ (k) may be found from the original commutation relations as given 10) Nevertheless, the set of modes obtained from the boundary conditions that are realized by perfectly reflecting walls can be used to describe the electromagnetic field inside the cavity correctly. Note that the number of modes which contribute to the field can drastically increase with decreasing distance of the point of observation from the cavity walls, because of the wrong boundary conditions.

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