Operators radial momentum

With the definition of a radial momentum in Eq. (4.108), we can now write the kinetic energy operator in spherical coordinates as... 

Eq. (6.12) can also be written in terms of a radial momentum operator pr... 

To achieve this separation of variables in the Dirac equation, we must be able to factorize the operator a p. The nuclear potential V is already a function of r only. We follow the procedure in Schiff (1968) and introduce the radial operators for momentum and velocity... 

In Eq. (14), /max is the maximum of the orbital angular momentum quantum numbers of the active electron in either the initial or final states, I nl, n l ) is the radial transition integral, that contains only the radial part of both initial and final wavefunctions of the jumping electron and a transition operator. Two different forms for this have been employed, the standard dipole-length operator, P(r) = r, and another derived from the former in such a way that it accounts explicitly for the polarization induced in the atomic core by the active electron [9],... 

In practice, the transformation of any operator to irreducible form means in atomic spectroscopy that we employ the spherical coordinate system (Fig. 5.1), present all quantities in the form of tensors of corresponding ranks (scalar is a zero rank tensor, vector is a tensor of the first rank, etc.) and further on express them, depending on the particular form of the operator, in terms of various functions of radial variable, the angular momentum operator L(1), spherical functions (2.13), as well as the Clebsch-Gordan and 3n -coefficients. Below we shall illustrate this procedure by the examples of operators (1.16) and (2.1). Formulas (1.15), (1.18)—(1.22) present concrete expressions for each term of Eq. (1.16). It is convenient to divide all operators (1.15), (1.18)—(1.22) into two groups. The first group is composed of one-electron operators (1.18), the first two... 

In this section, we will examine the role of interelectronic repulsion in the perspective of the internal symmetries of the shell. The key observation is that in a d-only approximation — i.e. if the t2g-orbital functions can be written as products of a common radial part and a spherical harmonic angular function of rank two - the interelectronic repulsion operator and the pseudo-angular momentum operators commute [2]. This implies that the dominant part of the... 

The value AP can change in the axial direction in the hollow fiber (AP is the pressure drop in the membrane matrix due to the momentum transfer, the velocity through the membrane is u0 , where e is the membrane porosity). Kelsey etal. [11] have solved the equation system in all three cases, namely for closed-shell operation, partial ultrafiltration and complete ultrafiltration and have plotted the dimensionless axial and radial velocities as well as the flow streamlines. Typical axial and radial velocity profiles are shown in the hollow-fiber membrane bioreactor at several axial positions in Figure 14.8 plotted by Kelsey etal. [ 11]. This figure illustrates clearly the change of the relative values of both the axial and the radial velocity [V=vL/(u0Ro), U=u/u0 where uc is the inlet centerline axial velocity]. 

In order to apply the representation theory of so(2,1) to physical problems we need to obtain realizations of the so(2, 1) generators in either coordinate or momentum space. For our purposes the realizations in three-dimensional coordinate space are more suitable so we shall only consider them (for N-dimensional realizations, see Cizek and Paldus, 1977, and references therein). First we shall show how to build realizations in terms of the radial distance and momentum operators, r, pr. These realizations are sufficiently general to express the radial parts of the Hamiltonians we shall consider linearly in the so(2,1) generators. Then we shall obtain the corresponding realizations of the so(2,1) unirreps which are bounded from below. The basis functions of the representation space are simply related to associated Laguerre polynomials. For finding the eigenvalue spectra it is not essential to obtain these explicit realizations of the basis functions, since all matrix elements can... 

The radial distance and momentum operators satisfy the simple commutation relation4... 

It is clear from Eq. (22) that a different REP arises for each pseudospinor. The complete REP is conveniently expressed in terms of products of radial functions and angular momentum projection operators, as has been done for the nonrelativistic Hartrce Fock case (23). Atomic orbitals having different total angular momentum j but the same orbital angular momentum / are not degenerate in j-j coupling. Therefore the REP is expressed as... 

In theory, an infinite number of calculations for highly excited states is required to complete the expansion of the EP given by Eq. (24), since there are only a few occupied valence orbitals in neutral atoms. This difficulty also exists in the nonrelativistic case and is resolved by using the closure property of the projection operator with the assumption that radial parts of EPs are the same for all orbitals having higher angular momentum quantum numbers than are present in the core. The same approximation is applicable in the present... 

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