On the Problem of Degeneracy in Quantum Mechanics

Abstract

DEGENERACY I N QUANTUM MECHANICS 64iwhere D' is defined as in Eq. ( 19) and A4 and 84 are arbitrary constants. This solution represents a state of the rotating space charge in which "spokes" of increased density maintain a fixed position relative to a coordinate system rotating with angular velocity Qp. Fig. 2 is intended to represent this state of affairs. The a.c. at any point is then due only to changes in charge density and not at all to changes in electron velocity. DIscUssIoNWe have assumed throughout that the boundaries are perfectly conducting and that electrons travel in circles around the axis so that no current Rows from the inner to the outer conductor. Thus no d.c. power is being fed into the system, and we cannot expect to make any estimates of a.c. power output. This is also evident from the fact that currents and voltages calculated from our solutions are always ~/2 out of phase. For the same reason it is impossible to estimate the resistance which a tube will present to an a.c. signal. This situation could be remedied by introducing a small d.c. radial velocity, but this increases the complexity of the a.c. equations so that they are quite unmanageable by our present technique.It is a pleasure to acknowledge our indebtedness to Mr. W. C. Hahn for many stimulating discussions and suggestions.

Superintegrability with third order integrals of motion, cubic algebras, and supersymmetric quantum mechanics. I. Rational function potentials

“…where ␣ 0 , ... , 3 are constants. The Casimir operator of a polynomial algebra is an operator that commutes with all elements of the algebra:…”

“…The most well known examples of ͑maximally͒ superintegrable systems are the Kepler-Coulomb 1,2 system V͑x ជ͒ = ␣ / r and the harmonic oscillator V͑x ជ͒ = ␣r 2 . 3,4 A systematic search for superintegrable systems in two-dimensional Euclidean space E 2 was started some time ago. 5,6 In 1935 Drach 7 published two articles on two-dimensional Hamiltonian systems with third order integrals of motion and found ten such integrable classical potentials in complex Euclidean space E 2 ͑C͒.…”

Superintegrability of the caged anisotropic oscillator

“…If l : m : n = 2 : 1 : 1, then the Hamiltonian also separates in the rotational parabolic and elliptic cylindrical coordinate systems [9], giving rise to additional conserved quantities and accidental degeneracy. However, if l+m+n > 4, then the Hamiltonian is still superintegrable [10,11,12], even though it now only separates in rectangular Cartesians. Further examples of systems which are superintegrable but not separable in more than one coordinate system include the Calogero-Moser problem [13,14] and the generalized Coulomb problem [15].…”

Superintegrability of the Fock–Darwin system

“…This connection was studied long time ago for a Hamiltonian describing a particle in a central potential [10,11,12,13,14]. In general terms, a quantum system of n degrees of freedom is called integrable if it has n algebraically independent symmetry operators, including the Hamiltonian, commuting with each other.…”