Harold V. McIntosh

Symmetry and Degeneracy

Throughout this article we shall be describing wave equations, both Schrodingers and Diracs for a wide variety of potentials. The notation for the parameters appearing in these equations, their eigenvalues and eigenfunctions is now well standardized and nearly universal, and we shall frequently refer to them by name, without further ceremony; for example, the magnetic quantum number m. Furthermore, we shall take ℏ = c = 1, as well as taking 1 for a particles mass, except that we will retain an explicit m in relativistic formulas.

On Accidental Degeneracy in Classical and Quantum Mechanics

The theory of accidental degeneracy is surveyed, particular attention being paid to the connection between the accidental degeneracy of the two-dimensional isotropic harmonic oscillator and the theory of angular momentum.

On the Degeneracy of the Two-Dimensional Harmonic Oscillator

A further study of the degeneracy of the two dimensional harmonic oscillator is made, both in the isotropic and anisotropic cases. By regarding the Hamiltonian as a linear operator acting through the Poisson bracket on functions of the coordinates and momenta, a method applicable generally to bilinear Hamiltonians, it is shown how all possible rational constants of the motion may be generated. They are formed from eigenfunctions of the Hamiltonian which are linear combinations of the coordinates and momenta, and which belong to negative pairs of eigenvalues. Canonical coordinates, which may be visualized geometrically for the isotropic oscillator in terms of the Hopf mapping, place the symmetry group responsible for the accidental degeneracy clearly in evidence. Surprisingly, one finds that the unitary unimodular group SU2, is the symmetry group in all cases, even including that of an anisotropic oscillator with incommensurable frequencies. The lack of a quantum-mechanical analogy in the latter case is du...

Search for a Universal Symmetry Group in Two 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 Schrodingers 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 s...

Symmetry‐Adapted Functions Belonging to the Symmetric Groups

Youngs factorization of idempotents belonging to the symmetric groups is given a necessary and sufficient characterization, by means of a lemma due to Burrow. The use of these idempotents is contrasted with Yamanouchis representation, and finally the equivalence of Lowdins path diagram method to the group‐theoretical treatment of the angular momentum states arising from the coupling of an assemblage of spin ½ particles is demonstrated.

Quantization as an Eigenvalue Problem

Publisher Summary The fundamental and productive version of quantum mechanics has been the one introduced by Erwin Schrodinger. The symbolic and operational techniques have been indispensable in providing a vocabulary for the teaching, discussion, and application of quantum mechanics. When the moment arrives that matrix elements have to be calculated and results obtained, it is the Schrodinger equation that is introduced and has to be solved. The term eigenvalue problem refers to the acquisition of suitable solutions of the boundary value problem of the differential equation and not to the diagonalization of a matrix. The specification of the boundary conditions in a way that would be adequate for axiomatic considerations was a concern of some delicacy. Schrodinger imposed the requirements of continuity, single-valuedness, and finiteness.

Quantization and a Green’s Function for Systems of Linear Ordinary Differential Equations

One of the traditional puzzles for students of quantum mechanics is the reconciliation of the quantification principle that wave functions must be square integrable with the reality that continuum wave functions do not respect this requirement. By protesting that neither are they quantized the problem can be sidestepped, although some ingenuity may still be required to find suitable boundary conditions. Further difficulties await later on when particular systems are studied in more detail. Sometimes all the solutions, and not just some of them, are square integrable. Then it is necessary to resort to another principle, such as continuity or finiteness of the wave function, to achieve quantization. Examples where such steps have to be taken can be found both in the Schroedinger equation and the Dirac equation. The ground state of the hydrogen atom, the hydrogen atom in Minkowski space, the theta component of angular momentum, all pose problems for the Schroedinger equation. Finiteness of wavefunction alone is not a reliable principle because it fails in the radial equation of the Dirac hydrogen atom. Thus the quantizing conditions which have been invoked for one potential or another seem to be quite varied.