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Physica Scripta

Ladder operators and a dynamicalSU(2) group symmetry of the hydrogen atom system

Oinam Babynanda Devi, C Amuba Singh

Published Jan 29, 2014• 3 citations
Publisher

Abstract

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.

DOI
10.1088/0031-8949/89/02/025202

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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.
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
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.
Recently several authors have proposed a universal symmetry group and demonstrated the classical validity of the concept. Supposedly, an SU(n) symmetry group could be constructed for whatsoever system of n degrees of freedom. The claim is assuredly valid for all classically degenerate systems, but is in contradiction with most of the well-known and widely accepted solutions of Schrödinger's equation. We examine the reasons for this discrepancy on the quantum-mechanical level. The construction of the universal symmetry group requires ladder operators, which in most cases are the ladder operators of Infeld and Hill. Complications which owe their origin entirely to numerical relationships imposed by quantization prevent these operators from forming a von Neumann algebra and, in turn, an SU(n) group of constants of the motion. Two important effects are those imposed by anisotropy, wherein not all quanta have the same size, and by non-Cartesian coordinates, wherein quanta in some dimensions are restricted in size by those in other dimensions. These effects can be seen quite clearly in two-dimensional systems: anisotropy in the Cartesian anisotropic harmonic oscillator, and conflict between dimensions in the polar form of the isotropic oscillator. Further complications arise when the two effects are combined, as in the harmonic oscillator or hydrogen atom with ``excess'' angular momentum. Enough residue of the universal symmetry concept remains that many similarities between the hydrogen atom and harmonic oscillator may be understood, including the fact that some levels of the hydrogen atom, which ordinarily transform according to an orthogonal group, may form irreducible representations of the unitary group.
It is shown that the Dirac equation with the Coulomb potential can be solved using the algebra of the three spinor invariants of the Dirac equation without the involvement of the methods of supersymmetric quantum mechanics. The Dirac Hamiltonian is invariant with respect to the rotation transformation, which indicates the dynamical (hidden) symmetry SU(2) of the Dirac equation. The total symmetry of the Dirac equation is the symmetry SO(3)⊗SU(2). The generator of the SO(3) symmetry group is given by the total momentum operator, and the generator of SU(2) group is given by the rotation of the vector-states in the spinor space, determined by the Dirac, Johnson–Lippmann, and the new spinor invariants. It is shown that using algebraic approach to the Dirac problem allows one to calculate the eigenstates and eigenenergies of the relativistic hydrogen atom and reveals the fundamental role of the principal quantum number as an independent number, even though it is represented as the combination of other quantum numbers.
The angular wave functions for a hydrogen atom are well known to be spherical harmonics, and are obtained as the solutions of a partial differential equation. However, the differential operator is given by the Casimir operator of the SU(2)SU(2)SU(2) algebra and its eigenvalue l(l+1)ℏ2l(l+1) \hbar^2l(l+1)ℏ2, where lll is non-negative integer, is easily obtained by an algebraic method. Therefore the shape of the wave function may also be obtained by extending the algebraic method. In this paper, we describe the method and show that wave functions with different quantum numbers are connected by a rotational group in the cases of l=0l=0l=0, 1 and 2.
Abstract Taking a close look at the Infeld–Hull ladder operators for the Kratzer oscillator system, V(x) = [x2 + β(β − 1)x−2]/2, we deduce and explicitly construct energy‐raising and ‐lowering operators for the generalized Morse potential system V(z) = (Ae−4αz − Be−2αz)/2, through a canonical transformation that exists between the two systems. For the Morse potential system, we obtain a system of raising and lowering operators P±(n) (n = 0, 1, 2, 3, … , nmax) with the specific property that P±(n)Φn = c±(n)Φn±1, where Φn denotes the nth energy eigenfunction. While P−(0) annihilates the ground‐state Φ0, the operator P+(nmax), instead of annihilating the highest bound‐state Φ , actually knocks it out of the L2 space spanned by the discrete bound states and becomes inadmissible. Yet, raising and lowering operators P^\hat{P}P^ ± with proper end‐of‐spectrum behavior (i.e., P^\hat{P}P^ −|0〉 = 0 and P^\hat{P}P^ +|nmax〉 = 0) can be constructed in a straightforward way in the energy representation. We show that the operators P^\hat{P}P^ +, P^\hat{P}P^ −, and P^\hat{P}P^ 0 (where P^\hat{P}P^ 0 ≡ (1/2)[P^\hat{P}P^ +, P^\hat{P}P^ −]) form a su(2) algebra only if we restrict them to the (N − 1)‐dimensional subspace spanned by the lowest (N − 1) basis vectors, but not in the full (N + 1)‐dimensional space spanned by the discrete bound states [N ≡ nmax ≡ integral part of (1/2)(B/(2αA\sqrt{A}A​ ) − 1)]. Realization of this su(2) algebra in the position representation (when restricted to the (N − 1)‐dimensional subspace) is also given. © 2005 Wiley Periodicals, Inc. Int J Quantum Chem, 2006