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Canadian Journal of Physics

Constructing physical systems with dynamical symmetry

Yan Li, Fulin Zhang, Rui-Juan Gu, Jing‐Ling Chen, L. C. Kwek

Published Nov 9, 2013• 0 citations
Publisher

Abstract

An approach to constructing quantum systems with dynamical symmetry is proposed. As examples, we construct generalized systems of the hydrogen atom and harmonic oscillator, which can be regarded as the systems with position-dependent mass. They have symmetries that are similar to the corresponding ones, and can be solved by using the algebraic method. We also exhibit an example of the method applied to the noncentral field.

DOI
10.1139/cjp-2013-0212

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An approach to constructing quantum systems with dynamical symmetry is proposed. As examples, we construct generalized systems of the hydrogen atom and harmonic oscillator, which can be regarded as the systems with position-dependent mass. They have symmetries that are similar to the corresponding ones, and can be solved by using the algebraic method. We also exhibit an example of the method applied to the noncentral field.
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.
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.
About 20 years ago almost simultaneously a number of authors had introduced the notion of a dynamic group or dynamic symmetry for quantum systems [ 1 ] (also known as spectrum generating algebra [ 2 ] and non-invariance group [ 3 ]. Introducing this notion required a clearer understanding what is to be called the symmetry of an equation describing a physical system [ 4 ] and to investigate concrete examples of quantum systems. In [ 4–7 ] it had been shown that the dynamic symmetry of hydrogen atom is 0(4,2), as a dynamic group describing Landau levels of an electron in magnetic field can serve either non-homogeneous symplectic group ISp(4,R) or U(2,1) [ 8 , 9 ].
Structured as a dialogue between a mathematician and a physicist, Symmetry and Quantum Mechanics unites the mathematical topics of this field into a compelling and physically-motivated narrative that focuses on the central role of symmetry. Aimed at advanced undergraduate and beginning graduate students in mathematics with only a minimal background in physics, this title is also useful to physicists seeking a mathematical introduction to the subject. Part I focuses on spin, and covers such topics as Lie groups and algebras, while part II offers an account of position and momentum in the context of the representation theory of the Heisenberg group, along the way providing an informal discussion of fundamental concepts from analysis such as self-adjoint operators on Hilbert space and the Stone-von Neumann Theorem. Mathematical theory is applied to physical examples such as spin-precession in a magnetic field, the harmonic oscillator, the infinite spherical well, and the hydrogen atom.