The Runge-Lenz vector, which accounts for the accidental degeneracy of the non-relativistic Kepler problem, has been the subject matter of many studies, both in quantum mechanics and in classical mechanics. Much less attention has been paid to the Johnson-Lippmann operator which accounts for the accidental degeneracy of the relativistic Kepler problem in Dirac`s quantum-mechanical description. In the present communication we discuss the properties of the Johnson-Lippmann operator. We show its relation to the non-relativistic Runge-Lenz vector and draw a connection to Sommerfield`s early discussion of the relativistic Kepler problem. This enables us, inter alia, to give an explanation of the apparent coincidence of the energy expressions of the two theories.
We study the quantum mechanical harmonic oscillator in two and three dimensions, with particular attention to the solutions as represents of their respective symmetry groups: O(2), O(3), and O(2,1). Solving the Schrodinger equation by separating variables in polar coordinates, we obtain wavefunctions characterized by a principal quantum number, the group Casimir eigenvalue, and one observable component of orbital angular momentum, with eigenvalue
m+s, for integer
m and real constant parameter
s. In each symmetry group,
s splits the solutions into two inequivalent representations, one associated with
s=0, which recovers the familiar description of the oscillator as a product of one-dimensional solutions, and the other with
s>0 (in three dimensions,
s=0,1/2) whose solutions are non-separable in Cartesian coordinates, and are hence overlooked by the standard Fock space approach. In two dimensions, a single set of creation and annihilation operators forms a ladder representation for the allowed oscillator states for any
s, and the degeneracy of energy states is always finite. However, in three dimensions, the integer and half-integer eigenstates are qualitatively different: the former can be expressed as finite dimensional irreducible tensors under O(3) or O(2,1), and a ladder representation can be constructed via irreducible tensor products of the vector creation operator multiplet, while the latter exhibit infinite degeneracy and the finite-dimensional ladder representation fails for these states. These results are closely connected to the breaking of a unitary symmetry of the harmonic oscillator Hamiltonian recently discussed by Bars.
Symmetry is a cornerstone of modern physics, offering profound insights into the fundamental laws governing the universe. In quantum mechanics, symmetries lead to conserved quantities and simplify the resolution of complex systems. This paper explores the universal symmetries inherent in three paradigmatic quantum mechanical systems: the one-dimensional quantum harmonic oscillator, the hydrogen atom, and the angular momentum operator. While renowned for their exact solvability, their underlying symmetry structures extend beyond explicit Hamiltonian invariances. We examine both geometric and dynamical (hidden) symmetries, which arise from non-trivial conserved quantities and often explain ``accidental'' degeneracies in energy spectra. For the quantum harmonic oscillator, we delve into the SU(1,1) or SU(2) dynamical symmetry associated with its creation and annihilation operators. The hydrogen atom's unique SO(4) symmetry, stemming from the conservation of the Runge-Lenz vector in addition to angular momentum, is thoroughly investigated. The Lie algebra of angular momentum operators forms the basis of SO(3) or SU(2) symmetry, governing rotational invariance. By employing Lie groups and Lie algebras, we reveal the deep mathematical unity underpinning these seemingly disparate systems. This comparative analysis demonstrates how these universal symmetry principles not only explain solvability and degeneracy patterns but also offer a general framework for understanding quantum phenomena. The paper hypothesizes that the recurring algebraic structures of su(2) and its extensions across these systems signify a deeper, universal blueprint for quantum system behavior, serving as a foundational guide for understanding both exactly solvable and inherently non-solvable quantum phenomena through the lens of symmetry breaking or approximation. This exploration provides a robust foundation for advanced topics in quantum field theory and particle physics.