the classical and quantum dynamics of the multispherical nanostructures, 2004, p.400

[...]... from the subsequent chapters 1.1 Basic Equations Maxwell equations are the basis for the theory of electromagnetic field They have been written by English physicist G Maxwell in 1873, and they were the generalization of the experimental facts available then The state of field is described by the vectors of electric E and magnetic H fields accordingly, which can variate both in space r, and in time t The. .. become the Fourier integral The solution for free fields E and B can then be written as the sum of plane waves with arbitrary amplitudes It is also the solution of the Eq (1.8) and the similar equation for B However generally it is necessary to take into account the presence of currents j which can radiate a field, i.e to be the sources of fields In contrast to the vacuum case the problem of calculations of. .. permitivities of the vacuum, connected by a relation ε0 µ0 = c−2 (where c ≈ 3·108 m/sec is the light velocity in vacuum) ρ is the density of the electric charge and j is the vector of density of the electric current From Eqs (1.1)–(1.4) one can derive the important equation binding the electrical charge and current in the form: ∂ρ + ∇ · j = 0 ∂t (1.5) In the simplest case of homogeneous, isotropic and linear... equations have the forms ∇×H = j+ ∇×E = − ∂ D, ∂t ∂ B, ∂t (1.1) (1.2) ∇ · D = ρ, (1.3) ∇ · B = 0, (1.4) 3 4 Classical Dynamics where D = εE, B = µH are the vectors of the electric and magnetic induc¯ ¯ ¯ tion respectively, ε = ε0 ε and µ = µ0 µ are the dielectric and magnetic per¯ mittivities, ε and µ are the relative dielectric and magnetic permittivities, and ε0 and µ0 are the dielectric and the magnetic... , R (1.25) where V is the volume occupied with current j, R = r− r , and r is vector of a point of observations, whereas r is the position of current j (see Fig 1.2) One can see from (1.25) that for a retarded case the value of potential A(r, t) at moment t is determined by the value of the current j in the previous moment t − R/c, where the shift R/c is the time of propagation of a signal from a point... Lorentz–Mie theory and its extensions 2.2.1 Lorentz–Mie theory of elastic scattering 2.2.2 Theory of spontaneous emission 2.2.3 Mie scattering by concentrically stratified spheres Peculiarities of the modes of an open spherical cavity 2.3.1 Indexes and order of a whispering-gallery mode 2.3.2 The problem of normalization of the whispering-gallery modes... important case is the oscillate mode, when ψ1,2 = cos(ωt − kx) = cos ω(t − xk/ω), with ω and k as the arbitrary constants jointed by the ratio ω/k = ±v (1.11) In Eq (1.11) one refers to the quantities ω and k as the frequency and waves number respectively, and v is the phase velocity of wave Similar result can be received, if you take ψ1,2 = sin(ωt − kx) As (1.9) is the linear equation, the sum of its solutions... all other planes parallel to the given one, shifted for a distance of λ = 2π/k Quantity λ is referred to as the wavelength, and instead of angular frequency ω is often used the frequency f = ω/2π Then in (1.16) for the sign + one can write ω/k = f λ = v Note that in materials the √ phase velocity is c/n, where n = ε is the refraction index of this material 1.1.3 Electromagnetic waves Thus, for the solution... Preface The author’s experience shows that the resources and speed of wellknown packages often become unacceptably low for even some mediumlevel problems What can we do? The answer is single: study the necessary material from C++, spend some time creating your own library of classes, and then carry out the engineering research in parallel with numerical simulations on the basis of your model Such efforts and. .. (1.23) the source is the electrical charge ρ whereas in vector potential equation (1.24) the source is the electrical current j Maxwell Equations 9 → r z → R y → r´ x Fig 1.2 Geometry of problem An important circumstance is the relativistic covariant form of the Lorenz gauge [Landau & Lifshits, 1975; Panofsky & Phillips, 1962] The solution of the Eq (1.24) for A, is determined by a current j(r, t) and . written permission from the Publisher. Copyright © 2004 by Imperial College Press THE CLASSICAL AND QUANTUM DYNAMICS OF THE MULTISPHERICAL NANOSTRUCTURES KwangWei _Classical & Quantum. pmd 10/4/2005,. alt="" ~hp Classical and Quantum Dynamics of thp Mu1 t isp her ical Nanos tr uc turps This page intentionally left blank ~hp Classical and Quantum Dynamics of thp Multisp her. of a problem appears rather complicated because of the complex structure of a system, and also due to the fact that it is an open system. For example, to calculate the frequency dependences of