Radial solutions

In summary, separation of variables has been used to solve the full r,0,( ) Schrodinger equation for one electron moving about a nucleus of charge Z. The 0 and (j) solutions are the spherical harmonics YL,m (0,(1>)- The bound-state radial solutions... 

All orbitals with = 0 are called s orbitals, and the subscripts 1,0 associated with R(r) refer to this function as the first radial solution to the complete set of s orbitals. The next one (similarly with = 0) will be R2 0(r), and so on. The total wave function is the product of the radial and the angular function and we have ... 

In fact we know from (4.50) that the radial solution is a function of kr and from (4.52) that it is independent of M. Using the transformation (4.9) to simplify the differential operator we find that the radial equation... 

The radial solutions uA E,r) to the one-electron equation inside the spheres are of the form... 

Koeiiing D D and Arbman G O 1975 Use of energy derivative of the radial solution in an augmented plane wave method application to copper J. Phys. F Met. Phys. 5 2041... 

Novakowski K. S. P. A. Lapcevic. 1994. Field measurement of radial solute transport in fractured rock. Water resources research. 30 (1) pp. 37-43... 

Different basis equations are applied in these regions. Inside the spheres, one uses the radial solutions of equations. A linear combination of radial functions ui r)Yim r) and their derivatives with respect to the linearization parameters / is utilized. These functions and derivatives are matched on the sphere boimdaries. 

DERIVATION SUMMARY The Radial Solution. Again, we started from a reasonable guess, this time for the radial wavefunction with several free parameters, and operated on it to see what had to happen in order to satisfy the eigenvalue equation. The eigenvalue equation in this case was the complete Schrodinger equation, and we obtained (i) the same energies as the Bohr model, and (ii) radial wavefunctions that required / to be a positive integer less than n. 

It is easy to show that the radial solution can also be written as... 

This sum clearly truncates after a finite number of terms if a is a negative integer. The two functions Fi and F2 appearing in the radial solution of the point nucleus Dirac equation are... 

Inserting this approximate value for Fi and the exact value for Fz into the expression for the radial solutions, we get for 2pi/2... 

Here we derive exact, closed-form, analytical solutions for lineal and radial flows where the growth of the mudcake and the progress of the invasion front are strongly coupled. The first solution was given in Chin et al. (1986), but the radial solution available at the time did not model spurt, and also required numerical analysis. The full solution presented here appears for the first time. 

Simultaneous mudcake buildup and filtrate invasion in a radial geometry (liquid flows). Here, we will reconsider the simultaneous mudcake buildup and filtrate invasion problem just discussed, but we will use realistic radial coordinates. Note that the exact//neor flow solution in Chin et al. (1986) includes the all-important effect of mud spurt. But while that paper alluded to progress towards a radial solution, the work at that time could not account for any spurt at all because of mathematical complexities and, furthermore, turned to numerical solution as a last resort. Thus, a useful solution was not available, and any applications to time lapse analysis would await further progress. Since then, the result of some significant efforts have led to a closed-form solution. The resulting solution and derivation are described in detail here. This availability, together with the simple recipe for mudcake properties alluded to, brings time lapse analysis closer to reality. 

The Laguerre radial solutions form an orthonormal set of functions and the orthogonality of the various members of the set is accomplished using a system of (n i l) radial nodes (zero crossings in the r-coordinate). The 2s orbital has a small part like the Is in it... 

Because of spherical symmetry, the radial and angular parts can again be separated. The radial solutions turn out to be of the form r e ) for... 

The radial solutions are here distinguished by two quantum numbers - recognizing that, for each / > 0, we may solve (6.3.11) and obtain a complete set of orthonormal radial functions rR i(r) in L2(R+)... 

---