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Radial Ladder Operators in 3D Coulomb Interaction

H. N. Núñez‐Yépez, J. Lopez-Bonilla, A. L. Salas-Brito

Published Sep 7, 2015• 0 citations
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Abstract

We show that to obtain the ladder operators for the non-relativistic quantum 3D Coulomb problem it is enough to solve a simple linear system of first-order differential equations. We first assume the most general form for the operators compatible with problem and after, by applying the basic commutation relations they must comply with, we first deduce and then solve a differential equations system; in this way we find the usual form of the Coulomb radial ladder operators. The computations are explicitly done only for the creation operator since for the annihilation operator they are almost the same.

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We show that to obtain the ladder operators for the non-relativistic quantum 3D Coulomb problem it is enough to solve a simple linear system of first-order differential equations. We first assume the most general form for the operators compatible with problem and after, by applying the basic commutation relations they must comply with, we first deduce and then solve a differential equations system; in this way we find the usual form of the Coulomb radial ladder operators. The computations are explicitly done only for the creation operator since for the annihilation operator they are almost the same.
We show that the supersymmetric radial ladder operators of the three-dimensional isotropic harmonic oscillator are contained in the spherical components of the creation and annihilation operators of the system. Also, we show that the constants of motion of the problem, written in terms of these spherical components, lead us to second-order radial operators. Further, we show that these operators change the orbital angular momentum quantum number by two units and are equal to those obtained by the Infeld–Hull factorization method.
A procedure to obtain the operational solutions of second order differential equations related with Sturm–Liouville problems is presented. The method is based on the commutation relation between the ladder operators themselves, with a certain structure, and the position and momentum operators. Even though the creation and annihilation operators, derived by the proposed approach, factorize as expected the corresponding differential equation, the method does not use, as original premise factoring, the differential relation under consideration. That is, the displayed procedure is quite different, simple, and direct when compared with other procedures such as the factorization method of Infeld and Hull. To illustrate the above, the usefulness of the proposed procedure is shown by finding the ladder operators associated to the quantum numbers n and I for various potential wave functions. © 1992 John Wiley & Sons, Inc.
The Morse oscillator, radial Coulomb and radial harmonic oscillator problems can be solved exactly using a variety of algebraic methods. These problems correspond to different realizations of the so(2,1) algebra and a comparison of the generators of the algebra may be used to identify mappings between each pair of systems. The resultant transition operators act as ladder, or energy changing, operators in the cases of the Coulomb and harmonic oscillator potentials, whereas they act as shift operators, acting at constant energy, in the case of the Morse potential. This is a consequence of the so(2,1) dynamical symmetry, whereby the Morse Hamiltonian is expressible solely in terms of the Casimir operator of the algebra. An alternative algebraic approach, the use of the method of supersymmetric quantum mechanics, or factorization, produces in each case a set of shift operators. Relations between the various ladder and shift operators may be identified by means of the appropriate mappings, and these results can be generalized so as to relate the one dimensional Morse oscillator to the radial Coulomb and radial harmonic oscillator potentials involving an arbitrary number of angular dimensions.
With the use of a simple similarity transformation which brings the radial wave equations of the Dirac-Coulomb problem into a form nearly identical to those of the Schr\"odinger and Klein-Gordon equations, we derive simplified solutions to the Dirac-Coulomb equation for both the bound and continuum states following the familiar standard procedure adopted in the derivation of the conventional solutions. We show that to obtain the desired form of the second-order radial equations we can still work with a first-order partial differential equation rather than with the second-order Dirac equation widely employed in the derivation of the simplified solutions, and thus we can avoid the task of reducing the solutions of the second-order equations to those of the original Dirac equation. The transformed Dirac-Coulomb radial equations are so simple that one can apply the WKB method to them in the same way that one applies the WKB approximation to the Schr\"odinger radial equation, without making the further approximations commonly invoked, and the Sommerfeld-Dirac discrete spectrum follows immediately. For small Z\ensuremath{\alpha}, we also present approximate expressions for both the transformed and the original Dirac-Coulomb wave functions, which are valid for all energies and certainly valid in the quasirelativistic approximation which takes account of relativistic effects. The structure of these approximate expressions also suggests one of the methods to obtain relativistic approximate wave functions from the Pauli-spinor wave function for more general central potentials arising in atoms, molecules, and solids.