Simplified ladder operators for the hydrogen atom

Carl W David

Oinam Babynanda Devi, C Amuba Singh

By invoking a standard variable transformation x → y ≡ x 2 that connects the eigenvalue problems of the hydrogen atom system (Coulomb potential) and the Kratzer oscillator system , we obtain a system of ladder operators Ω nl ± ( l = 0,1,2, ... , n − 1) that connect the degenerate energy eigenstates (belonging to different l -values) of the hydrogen atom system. Using such quantum number specific operators as the precursors, we develop a general method of constructing quantum number independent ladder operators ; and then investigate the general conditions for these operators to form a realization of the su (2) algebra. In the process, we demonstrate that, in general, a finite-dimensional linear vector space carries a realization of the su (2) algebra. Applying the method to the case at hand, we show that for the hydrogen atom system the operators and form an su (2) algebra. The Hamiltonian is identified with the Casimir operator of the algebra, thereby revealing a dynamical SU (2) group symmetry of the hydrogen atom system.

Robert Golub, S. K. Lamoreaux

Abstract Modern applications of quantum mechanics are illustrated, and the simplifications offered by operator and algebraic techniques are highlighted. The various "pictures" (Schrödinger, Heisenberg, and Interaction or Dirac) are discussed and their origins are motivated. It is shown that operator techniques simplify the harmonic oscillator and angular momentum through the use of ladder operators. The WKB approximation boundary conditions are developed through the use of complex number algebra. Other topics that benefit from these techniques are discussed, including the squeezing of optical fields and various alternative solutions to the hydrogen atom.

Randell L Mills

Carl W David

An error laden note (Am. J. Phys., 34, 984,(1966)) concerning the ladder operator solution to the hydrogen atom electronic energy levels is corrected.