Runge-Lenz vectors

For an introduction to the Runge-Lenz vectors in the classical context, see [Mi]. 

In quantum mechanical context this invariant of the Kepler problem is often referred to as a Runge-Lenz-Pauli vector. In classical mechanics context it is also sometimes referred to as a Laplace-Runge-Lenz vector. 

This is just the so called vector rule. In general special properties of V give (j J V j) = 0 (for example, in the hydro-genic realization considered in Section VII, V is the Runge-Lenz vector and J V = 0). 

Thus, we obtain only finite dimensional unirreps for so(4) and only infinite dimensional ones for so(3,1). For our applications to perturbation theory we shall only need the so called hydrogenic case (cf. Section VII) where V is the Laplace-Runge-Lenz vector. For the realization of the generators in this manner we shall show that j0 — 0 and q is the principal quantum number. The unirreps of so(3,1) may be of interest in scattering problems which deal with the continuum states of the hydrogen atom. 

In order to obtain a realization of so(4), the factor — 2H must be removed from Eq. (162c). There are two possibilities H can be replaced by one of its continuum or bound-state energy eigenvalues. The former choice leads to a realization of so 3, 1) and the latter to a realization of, so(4). Thus, if we replace H by the bound-state energy E , and define the modified Laplace-Runge-Lenz vector... 

Note that here X defines a different quantity than the Runge-Lenz vector, Eq. (159). 

The same is true for two more special cases, namely the relativistic particle with spin (yielding the correct Dirac energy levels), and a relativistic particle without fine structure. In these cases, the Runge-Lenz vector is no longer a constant of motion, and the O4 symmetry of the nonrelativistic problem is broken. It is, however, not seriously broken, and an analogue of the Runge-Lenz vector, the so-called Johnson-Lippman operator [10]... 

Analytically the gobbling is represented by oscillations of the Runge-Lenz vector A which points along the orbit s major axis. The components of A parallel and transverse to the field B, which we label A and A respectively in a precessing frame, vary together with a transverse component of the orbital momentum, Ly. 

The system exhibits competing symmetries - spherical symmetry due to the Coulomb potential and axial symmetry due to the Stark potential (applied electric field). Parity and angular momentiun, i, are no longer conserved only the magnetic quantiun niunber, m, is a good quantum number in this system. However, the z-component of the Runge-Lenz vector (besides Lz) is a constant of the motion in this system. 

---