Vector spherical wave functions harmonics

The vector spherical wave functions can be expressed in terms of vector spherical harmonics as follows ... 

In the above analysis, and Y are expressed in the principal coordinate system, but in general, it is necessary to transform these vector functions from the principal coordinate system to the particle coordinate system through a rotation. The vector quasi-spherical wave functions can also be defined for biaxial media (e 7 y z) by considering the expansion of the tangential vector function T>c (3,a)Va + T>is j3, a)vjj in terms of vector spherical harmonics. 

The vector spherical harmonics YjtM form an orthogonal system. The state of the photon with definite values of j and M is described by a wave function which in general is a linear combination of three vector spherical harmonics... 

The boundary conditions (4.39), the orthogonality of the vector harmonics, and the form of the expansion of the incident field dictate the form of the expansions for the scattered field and the field inside the sphere the coefficients in these expansions vanish for all m = = 1. Finiteness at the origin requires that we take y (kjr), where kj is the wave number in the sphere, as the appropriate spherical Bessel functions in the generating functions for the vector harmonics inside the sphere. Thus, the expansion of the field (Ej,H,) is... 

Consider now the field scattered by an isotropic, optically active sphere of radius a, which is embedded in a nonactive medium with wave number k and illuminated by an x-polarized wave. Most of the groundwork for the solution to this problem has been laid in Chapter 4, where the expansions (4.37) and (4.38) of the incident electric and magnetic fields are given. Equation (8.11) requires that the expansion functions for Q be of the form M N therefore, the vector spherical harmonics expansions of the fields inside the sphere are... 

The nuclear quadrupole moment is an expectation value with respect to the nuclear wave function. For the nuclear ground state the nuclear wave function depends upon a radial parameter a, the nuclear spin quantum number / and its projection Mj so that the corresponding ket-vector is denoted as a, I, Mj). The properties of the nuclear spin (in general, an angular momentum) are well known and they can be fully exploited in expressing such an expectation value. For this purpose let us rewrite the electrostatic interaction energy, making use of the expansion in terms of the spherical harmonic functions... 

The concept of GMM and the details of computation were presented by Yu-Lin Xu in a series of papers (e.g. Xu 1995,1997,1998a). The main idea is to express the scattering field of the particle j as an infinite series of spherical vector wave functions (SVWF) (i.e. spherical harmonics for vector fields) that are defined for... 

The polarization unit vector of a linearly polarized vector plane wave is given by (1.18). If the vector plane wave propagates along the z-axis we have / = a = 0 and for / = 0, the spherical vector harmonics are zero unless m = 1. Using the special values of the angular functions and when (3 = 0,... 

The regular and radiating spherical vector wave functions can be expressed as integrals over vector spherical harmonics [26]... 

The delta function corresponds to Einstein s equation, which says that the kinetic energy of the emitted electron Ef equals the difference of the photon energy h(a and the energy level of the initial state of the sample, The final state is a plane wave with wave vector k, which represents the electrons emitted in the direction of k. Apparently, the dependence of the matrix element 1 j) on the direction of the exit electron, k, contains information about the angular distribution of the initial state on the sample. For semiconductors and d band metals, the surface states are linear combinations of atomic orbitals. By expressing the atomic orbital in terms of spherical harmonics (Appendix A),... 

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