Wave function radial factor

Here, P and Q are the radial large and small components of the wave function. The factor of i has been introduced to make the radial components real. The angular functions are two-component spinors, that is, a product of angular and spin functions the spin variable r has been explicitly shown. 

The radial factor in the total wave function must therefore be an eigenstate of... 

The radial factors of the hydrogen-like atom total wave functions ip r, 0, tp) are related to the functions Sni(p) by equation (6.23). Thus, we have... 

As a function of r the general expression for the radial factors of the wave function becomes... 

These manipulations have brought us to a familiar equation we recognize (4.23) as the Schrodinger equation (1.132) for a one-dimensional harmonic oscillator with force constant ke. Before we can conclude that (4.23) and (1.132) have the same solutions, we must verify that the boundary conditions are the same. For quadratic integrability, we require that S(q) vanish for q = oo. Also, since the radial factor F(R) in the nuclear wave function is... 

The wave function F for a single electron, in a hydrogen atom for example, may be written as a product of four factors. These are the radial function R(r), which is dependent only on the radial distance rfrom the nucleus two angular functions 0(0) and ( ), which depend only on the angles 0 and (j>... 

An interesting improvement of the SRC model has been discussed by Wang and Bulou (1995). They considered simplified expansion factors used in the Hartree-Fock radial wave-functions for the f-electrons. With these factors it was possible to introduce a k dependence for the pressure-induced change of different Slater parameters. This procedure would remove the weak point of the model which could not account for the observed -dependence of the parameters under pressure. 

The term (r2) is the expectation value of the radial wavefunction. Since A and (r2) are functions of radial wave functions, usually these parameters are empirically evaluated. The matrix elements and factors are dependent only on the angular parts of the wave functions and can be exactly evaluated. 

In order to test such an application we have calculated the spin and charge structure factors from a theoretical wave function of the iron(III)hexaaquo ion by Newton and coworkers ( ). This wave function is of double zeta quality and assumes a frozen core. Since the distribution of the a and the B electrons over the components of the split basis set is different, the calculation goes beyond the RHF approximation. A crystal was simulated by placing the complex ion in a lOxIOxlOA cubic unit cell. Atomic scattering factors appropriate for the radial dependence of the Gaussian basis set were calculated and used in the analysis. 

S-wave scattering is the only practical outcome since P-wave final neutron states are not accessible to thermal neutrons, because these wave functions have negligible amplitude at the small radial values that are typical of atomic nuclei. It is convenient to rewrite the equation as a dynamical structure factor (or Scattering Law), which emphasises the dynamics of the sample. 

From Figure 5.6, the R21 wave function has no radial nodes, and the R31 function has one radial node the R41 (not shown) function has two radial nodes. The R e wave functions have n- -l radial nodes. Because the angular part of the np wave function always has a nodal plane, the total wave function has n - I nodes (n - 1 radial and 1 angular), which is the same number as an s orbital with the same principal quantum number. The R2i(r) function (that is. Rip) in Table 5.2 contains the factor a, which is proportional to r (u = Zrjao) and causes it to vanish at the nucleus. This is true of all the radial wave functions except the ns functions, and it means that the probability is zero for the electron to be at the nucleus for all wave functions with > 0 (p, d, f,...). Physically, electrons with angular momentum are moving around the nucleus, not toward it, and cannot penetrate toward the nucleus. 

For Eq. (11) S is the Bragg vector S = 2ttH, IT is the row vector (htk,l) and the scalar S - S = 4ir sin 0/A. The index / covers the N atoms in the unit cell. The atomic scattering factor f (S) is the Fourier-Bessel transform of the electronic, radial density function of the isolated atom. This density function is usually derived from a spin-restricted Hartree-Fock wave function for the atom in its ground state. The structure fac-... 

The eigenfunctions of the free electron confined in the same prolate spheroids are expressed as products of regular radial and angular spheroidal wave functions [16] Chapter 21, in the respective coordinates u and v, and the eigenfunctions of Equations (34) and (35). The radial functions are expressed as infinite series of spherical Bessel functions of order m + s and argument kfu. Its eigenvalues are determined by the boundary condition on the radial factor,... 

The radial wave functions i j(r) for = 1, 2, and 3 and 1 = 0 and 1 are shown plotted in Figure 21-2. The abscissas represent values of p hence the horizontal scale should be increased by the factor n in order to show R r) as functions of the electron-nucleus distance r. It will be noticed that only for s states (with 1 = 0) is the wave function different from zero at r = 0. The wave function crosses the p axis n — l — 1 times in the region between p = 0 and p = oo. 

These functions satisfy the normalization condition (20). It should be noted that the ratio of the scale factors in (48) and (49) is - (1 — e /c )/ I + Cmc/c ) 0iZl2n. Thus, Qnit r) is several orders of magnitude smaller than Pn ( ) for Z = 1. For this reason, P K and QnK are referred to as the large and small components of the radial Dirac wave function, respectively. 

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