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Algebraic Approach to Radial Ladder Operators in the Hydrogen Atom

Unidad Azcapotzalco

Published Jan 1, 2007• 0 citations
OpenAlex

Abstract

We add a phase variable and its corresponding operator to the description of the hydrogen atom. With the help of these additions, we device operators that act as ladder operators for the radial system. The algebra defined by the commutation relations between those operators has a Casimir operator coincident with the radial Hamiltonian of the problem. The algebra happens to be the well-known su(1,1) Lie algebra, hence the phase-dependent eigenfunctions calculated within our scheme belong in a representation of that algebra, a fact that may be useful in certain applications. © 2007 Wiley Periodicals, Inc. Int J Quantum Chem 107: 1608 -1613, 2007

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We add a phase variable and its corresponding operator to the description of the hydrogen atom. With the help of these additions, we device operators that act as ladder operators for the radial system. The algebra defined by the commutation relations between those operators has a Casimir operator coincident with the radial Hamiltonian of the problem. The algebra happens to be the well-known su(1,1) Lie algebra, hence the phase-dependent eigenfunctions calculated within our scheme belong in a representation of that algebra, a fact that may be useful in certain applications. © 2007 Wiley Periodicals, Inc. Int J Quantum Chem 107: 1608 -1613, 2007
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.
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.
The radial form of Hylleraas’ equation for the hydrogen atom, Λl‖El〉=4ℏ4a−2‖El〉 (a=Bohr radius), is considered and it is shown that the operator Λl can be factorized. Hence ladder operators P±l are derived that are linear in the position operator r and are nonlinear functions of the momentum operator p. It is proven that P±l‖El〉 =2ℏ2a−1[1+(l+ (1)/(2) ± (1)/(2) )2 ×(2Mℏ−2a2E)]1/2‖E,l±1〉. In the momentum representation of wave mechanics the solutions to these equations are the radial momentum-space wavefunctions for the hydrogen atom. Thus a simple method of calculating these wavefunctions is obtained. The results complement the familiar operator solution for the hydrogen atom that is based on factorization of the radial Hamiltonian and yields operators that are linear in p and are nonlinear functions of r.
A novel realization of the classical SU(2) algebra is introduced for the Dirac relativistic hydrogen atom defining a set of operators that allow the factorization of the problem. An extra phase is needed as a new variable in order to define the algebra. We take advantage of the operators to solve the Dirac equation using algebraic methods. A similar path to the one used in the angular momentum case is used; hence, the radial eigenfunctions so calculated comprise nonunitary representations of the algebra. One of the interesting properties of such nonunitary representations is that they are not labeled by integer nor by half-integer numbers, as occurs in the usual angular momentum representation.
The quantum mechanics of the hydrogen atom can be solved algebraically by the dynamical symmetry. In this note we introduced firstly Pauli's derivation of the complete relations among the various basic states with the same eigenvalue of the Hamiltonian. Secondly we also derived the same results by using algebraic method and some relations among the associated Legendre functions. As a byproduct, we also obtained a recursive relation of the radial wave functions. By using this recursive relation, we can derive the general radial wave function from the most simple radial wave function which has only an monomial term, much like deriving a general state from the highest weight state by using ladder operators.