Laplace-Runge-Lenz vector

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. 

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... 

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