Scalar Wave Functions

The spherical wave functions form a set of characteristic solutions to the Helmholtz equation and are given by [215] 

The spherical wave functions can be expressed as integrals over plane waves [26,70,215] 

The spherical wave expansion of the plane wave exp(jfe r) is of significant importance in electromagnetic scattering and is given by [215] 

The quasi-plane waves have been introduced by Devaney [46], and for z 0, they are defined as 

As in (B.3), the expansion of quasi-plane waves in terms of radiating spherical wave functions is given by 

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The approximation of Fresnel is scalar approximation. Let u(, r],0-0) be the scalar wave function of the laser beam falling onto the optical element, and u( X,y,Cl) will the be scalar wave function in the plane Z = Cl. Then [3,4]... 

The spin of elementary particles is not described by the square-root Klein-Gordon equation. The solutions of the square-root Klein-Gordon equation are scalar wave functions. Real electrons have spin and should be described by a matrix-wave equation. 

An analysis of the transformation properties of the Fock-Klein-Gordon equation and of the Dirac equation leads to the conclusion that satisfaction of the first of these equations requires the usual (i.e., scalar) wave function, whereas the second equation requires a bispinor character of the wave function. Scalar functions describe spinless particles (because they cannot be associated with the Pauli matrices), while bispinors in the Dirac equation are associated with the Pauli matrices, and describe a particle of 1/2 spin. 

In an Abelian theory [for which I (r, R) in Eq. (90) is a scalar rather than a vector function, Al=l], the introduction of a gauge field g(R) means premultiplication of the wave function x(R) by exp(igR), where g(R) is a scalar. This allows the definition of a gauge -vector potential, in natural units... 

This part of our chapter has shown that the use of the two variables, moduli and phases, leads in a direct way to the derivation of the continuity and Hamilton-Jacobi equations for both scalar and spinor wave functions. For the latter case, we show that the differential equations for each spinor component are (in the nearly nomelativistic limit) approximately decoupled. Because of this decoupling (mutual independence) it appears that the reciprocal relations between phases and moduli derived in Section III hold to a good approximation for each spinor component separately, too. For velocities and electromagnetic field strengths that ate nomrally below the relativistic scale, the Berry phase obtained from the Schrddinger equation (for scalar fields) will not be altered by consideration of the Dirac equation. 

We use s, p, and d partial waves, 16 energy points on a semi circular contour, 135 special k-points in the l/12th section of the 2D Brillouin zone and 13 plane waves for the inter-layer scattering. The atomic wave functions were determined from the scalar relativistic Schrodinger equation, as described by D. D. Koelling and B. N. Harmon in J. Phys. C 10, 3107 (1977). 

When is a one component scalar function, one can take the square root of Eq. (9-237) and one thus obtains the relativistic equation describing a spin 0 particle discussed in Section 9.4. This procedure, however, does not work for a spin particle since we know that in the present situation the amplitude must be a multicomponent object, because in the nonrelativistic limit the amplitude must go over into the 2-component nonrelativistic wave function describing a spin particle. Dirac, therefore, argued that the square root operator in the present case must involve something operating on these components. 

The scalar product of two Dirac wave functions is defined as... 

Sales response to advertising, 265 Salpeter, E. E641 Sample, adequacy of, 319 Sampling theorem, 245 Scalar product of two wave functions, 549,553... 

Since we have a compact region (all the momenta are now discrete) it is more convenient to employ the energy spectrum formulation to obtain the Casimir energy. The wave function for the massless scalar field in the cavity is... 

A wave is described by a wave function y(f, /), either scalar (as pressure p) or vector (as u or v) at position r and time t. The wave function is the solution of a wave equation that describes the response of the medium to an external stress (see below). 

In this chapter, we also discussed several schemes that allow for the computation of scalar observables without explicit construction and storage of the eigenvectors. This is important not only numerically for minimizing the core memory requirement but also conceptually because such a strategy is reminiscent of the experimental measurement, which almost never measures the wave function explicitly. Both the Lanczos and the Chebyshev recursion-based methods for this purpose have been developed and applied to both bound-state and scattering problems by various groups. 

The scalar relativistic contribution is computed as the first-order Darwin and mass-velocity corrections from the ACPF/MTsmall wave function, including inner-shell correlation. 

The wave function, V /(r), is a function of the vector position variable r. To determine it at every point in space it is convenient to take advantage of the fact that the potential V(r) depends only on the scalar interatomic distance r. In spherical coordinates (Figure 1.2), the Laplacian operator V2 has the form... 

The solution of the vector wave equation can be written in terms of the generating function ij/, which is a solution of the scalar wave equation... 

Abstract. The elements of the second-order reduced density matrix are pointed out to be written exactly as scalar products of specially defined vectors. Our considerations work in an arbitrarily large, but finite orthonormal basis, and the underlying wave function is a full-CI type wave function. Using basic rules of vector operations, inequalities are formulated without the use of wave function, including only elements of density matrix. 

Therefore, M and N have all the required properties of an electromagnetic field they satisfy the vector wave equation, they are divergence-free, the curl of M is proportional to N, and the curl of N is proportional to M. Thus, the problem of finding solutions to the field equations reduces to the comparatively simpler problem of finding solutions to the scalar wave equation. We shall call the scalar function ip a generating function for the vector harmonics M and N the vector c is sometimes called the guiding or pilot vector. 

We have now done enough work to construct generating functions that satisfy the scalar wave equation in spherical polar coordinates ... 

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