Central field problem

The method of superposition of configurations is essentially based on the assumption that the basic orbitals form a complete set. The most popular basis used so far in the literature is certainly formed by the hydrogen-like functions, which set contains a discrete and a continuous part. The discrete subset corresponds physically to the bound states of an electron around a proton, whereas the continuous part corresponds to a free electron scattered by a proton, or classically to the elliptic and hyperbolic orbits, respectively, in a central-field problem. 

The selection rules (3.76)—(3.78) were derived without reference to the form of the radial functions hence they are valid for any one-particle central-field problem. For example, they hold well for the sodium valence electron, which moves outside a closed-shell structure. 

Two coupled first order differential equations derived for the atomic central field problem within the relativistic framework are transformed to integral equations through the use of approximate Wentzel-Kramers-Brillouin solutions. It is shown that a finite charge density can be derived for a relativistic form of the Fermi-Thomas atomic model by appropriate attention to the boundary conditions. A numerical solution for the effective nuclear charge in the Xenon atom is calculated and fitted to a rational expression. 

For D = 3, and putting zq = z in Eq. (24) to obtain the Slater sum S, use of the explicit form of V in Eq. (22) readily allows one to verify that the diagonal form of Eq. (24) is indeed an exact solution of Eq. (25). Later, Amovilli and March [20] made similar progress on central field problems. It remains of interest to treat atoms in intense electric fields by direct use of the Slater sum rather than by use of the off-diagonal canonical density matrix. 

As to future directions, the problem of the canonical density matrix, or equivalently the Feynman propagator, for hydrogen-like atoms in intense external fields remain an unsolved problem of major interest. Not unrelated, differential equations for the diagonal element of the canonical density matrix, the important Slater sum, are going to be worthy of further research, some progress having already been made in (a) intense electric fields and (b) in central field problems. Finally, further analytical work on semiclassical time-dependent theory seems of considerable interest for the future. 

The equations for the two radial wave functions g and f in the Dirac central field problem are [9.9,10]... 

The hydrogen atom is a typical case of the central-field problem. As was shown in Fig. 19.5, the proton is at the center with a charge + e while the electron is at a distance r with a charge —e. The coulombic force acts along the line of centers and corresponds to a potential energy, V(r) = —e /4ncQr. 

When Dirac proposed his equation for the electron [99], he already considered the case of a central scalar field and studied the relation to Schrodinger s wave equation and its extensions including spin by Pauli and Darwin. The solution of the Dirac central-field problem, however, was due to Darwin [103] and Gordon [119] who in turn recovered Sommerfeld s energy expression but, of course, as an energy eigenvalue. 

The selection rules for the different types of polarization are, of course, significant only when there is a unique z direction, due, for example, to the presence of a uniform magnetic field. This subject will be discussed more fully in the following chapter, where the Zeeman effect is considered. It is apparent from the derivation of the above selection rules that they are not limited to the hydrogen atom but are valid for any central field problem where the angular portion of the wave function is identical with that of the hydrogen atom. 

The central field problem distinguishes celestial mechanics from other areas of classical dynamics. This deals with the motion of a test particle, whose mass is negligible with respect to the central body, in the gravitational field of a point mass. The extended version of this problem is to allow the central mass to have a finite spatial extent, to depart from spherical symmetry, and perhaps to rotate. The basic Newtonian problem is the foUowing. 

Here the solution to the central field problem makes specific use of the gravitational force. The substitution of... 

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