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arXiv:1211.1532

Prepotential approach to systems with dynamical symmetries

Yan Li, Fulin Zhang, Jing‐Ling Chen, L. C. Kwek

Published Nov 7, 2012• 0 citations
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Abstract

A prepotential approach to constructing the quantum systems with dynamical symmetry is proposed. As applications, we derive generalizations of the hydrogen atom and harmonic oscillator, which can be regarded as the systems with position-dependent mass. They have the symmetries which are similar to the corresponding ones, and can be solved by using the algebraic method.

DOI
10.48550/arxiv.1211.1532

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A prepotential approach to constructing the quantum systems with dynamical symmetry is proposed. As applications, we derive generalizations of the hydrogen atom and harmonic oscillator, which can be regarded as the systems with position-dependent mass. They have the symmetries which are similar to the corresponding ones, and can be solved by using the algebraic method.
Following work of Malkin, Man’ko and Lipkin, Dothan proposed a general definition for the dynamical group generators associated to a given Hamiltonian, based on the symmetry group of the corresponding time‐dependent quantum mechanical equation of motion for the system. In this paper we establish a procedure to derive the dynamical symmetry algebra of a quantum system, which arises naturally from Noether’s theorem by allowing time dependent symmetry transformations. The method is applicable to non‐stationary systems, as illustrated for the varying‐mass harmonic oscillator.
During recent years, exact solutions of position-dependent mass Schrödinger equations have inspired intense research activities, based on the use of point canonical transformations, Lie algebraic methods or supersymmetric quantum mechanical techniques. Here we highlight the interest of another approach to such problems, relying on quadratic algebras. We illustrate this point by constructing spectrum generating algebras for a class of d -dimensional radial harmonic oscillators with d ≥ 2 (including the one-dimensional oscillator on the line via some minor changes) and a specific mass choice. This provides us with a counterpart of the well-known su(1, 1) Lie algebraic approach to the constant-mass oscillators.
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 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.