Radial Eigenvalue Equations

With the above ansatz for the spinor we can rewrite the stationary Dirac equation for hydrogen-like atoms with the Hamiltonian of the form of Eq. (6.55) 

We may now either multiply the last equation by ((A K,my(0 simply recognize that 

---

The last equation is formally identical to the radial Schrodinger equation with a non-integer value of the angular momentum quantum number. Its spectrum is bounded from below and the discrete eigenvalues are given by... 

This difference formula propagates axial-velocity u information from the lower boundary (e.g., stagnation surface) up toward the inlet-manifold boundary. At the stagnation surface, a boundary value of the axial velocity is known, u = 0. A dilemma occurs at the upper boundary, however. At the upper boundary, Eq. 6.106 can be evaluated to determine a value for the inlet velocity u j. However, in the finite-gap problem, the inlet velocity is specified as a boundary condition. In general, the velocity evaluated from the discrete continuity equation is not equal to the known boundary condition, which is a temporary contradiction. The dilemma must be resolved through the eigenvalue, which is coupled to the continuity equation through the V velocity and the radial-momentum equation. 

The continuity equation at the inlet boundary can be viewed as a constraint equation. Referring to the difference stencil (Fig. 17.14), it is seen that this first-order equation itself is evaluated at the boundary and no explicit boundary condition is needed. Moreover, since the inlet temperature, pressure, and composition are specified, the density is fixed and thus dp/dt = 0. Therefore, at the boundary, the continuity equation (Eq. 17.15) has no time derivative it is an algebraic constraint. There is no explicit boundary condition for A. At the inlet boundary, the value of A must be determined in such a way that all the other boundary conditions are satisfied. Being an eigenvalue, A s effect is felt through its influence on the V velocity in the radial momentum equation, and subsequently by V s influence on u through the continuity equation. 

For w = 1 or 2 they have the general form of a radial eigenvalue problem arising from some Hamiltonian. In fact, the radial parts of the nonrelativistic hydrogenic Hamiltonian, Klein-Gordon, and second-order iterated Dirac Hamiltonians with 1/r potential can all be expressed in this form for w = 1 and suitable choices of the parameters , rj, x. Similarly, the three-dimensional isotropic harmonic oscillator radial equation has this form for w = 2. 

T. E. Simos, A new Numerov-type method for computing eigenvalues and resonances of the radial Schrodinger equation, Int. J. Mod. Phys. C-Phys. Comput., 1996, 7(1), 33 1. 

Substituting (7.27) into (7.26) and using the angular-momentum eigenvalue equation (6.35), we obtain an ordinary differential equation for the radial function R(r) ... 

The LDA radial Schrodinger equation is solved by matching the outward numerical finite-difference solution to sin inward-going solution (which vanishes at infinity) of the same energy, near the classical turning point. Continuity of P t(r) = rAn/(r) and its derivative determines the eigenvalue /. The second order differential equation is actually solved as a pair of simultaneous first-order equations, so that the nonrelativistic and relativistic (Dirac equation) procedures appear similar. 

After using the numerical solution of the eigenvalues equation, the radial electric field in the direction of polarization is found as ... 

By operating with (114) and (98) on (113) we obtain the radially reduced Dirac eigenvalue equations... 

Le Roy et al. [OSRloy] give an alternative treatment of the frequency data (Direct-potential fit) which is based on the eigenvalues of the radial Schrddinger equation. The potential is expressed in form of a modified Morse function where the exponent is written in form of a power series expansion. The coefficients and those in the non-adiabatic correction terms are determined in a direct fit to the experimental eigenvalue differences, see [05Roy] for details and results. 

For a general potential V(r, r2,r ), invariant under rotations and translations, and symmetric, but not necessarily pairwise, one remarks that the same projection governs the radial equations of 1) and 2). As seen in eqs. (8.30) and (8.31), this is Vj,( ), the hyperscalar projection of the potential. At this approximation (hereafter denoted E ), one has to compare the eigenvalue equations (i = 1,2),... 

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. 

In Equation 1.3, the radial function Rnl (r) is defined by the quantum numbers n and l and the spherical harmonics YJ" depend on the quantum numbers l and W . When the spin of the electron is taken into account, the normalized antisymmetric function is written as a Slater determinant. The corresponding eigenvalues depend only on n and l of each single electron, which determine the electronic configuration of the system. 

In general, the Slater function is not an exact solution of any Schrodinger equation (except the Is- wavefunction, which is the exact solution for the hydrogen-atom problem). Nevertheless, asymptotically, the orbital exponent C is directly related to the energy eigenvalue of that state. Actually, at large distances from the center of the atom, the potential is zero. Schrodinger s equation for the radial function R(r) is... 

In the steady stagnation-flow formulation the thermodymanic pressure may be assumed to be constant and treated as a specified parameter. The small pressure variations in the axial direction, which may be determined from the axial momentum equaiton, become decoupled from the system of governing equations (Section 6.2). The small radial pressure variations associated with the pressure-curvature eigenvalue A are also presumed to be negligible. While this formulation works very well for the steady-state problem, it can lead to significant numerical difficulties in the transient case. A compressible formulation that retains the pressure as a dependent variable (not a fixed parameter) relieves the problem [323],... 

We choose a particular eigenstate (3.65) lVm(r) of and replace L by its eigenvalue, obtaining the radial equation... 

We consider a spherical potential which is zero up to r = a, where it is infinite. The radial boundary condition results in the radial equation (4.10) being an eigenvalue problem with eigenvalues which are positive with respect to the zero of energy. We define a wave number k by... 

---