Symmetry and Degeneracy¹

HAROLD V. McINTOSH

ESCUELA SUPERIOR DE FISICA Y MATEMATICAS
INSTITUTO POLITECNICO NACIONAL,
MEXICO

I.	Introduction	$\qquad \qquad$	75	$\quad$	22
II.	Symmetry of the Hydrogen Atom	$\qquad \qquad$	80	$\quad$	5
III.	Symmetry of the Harmonic Oscillator	$\qquad \qquad$	84	$\quad$	8
IV.	Symmetry of Tops and Rotators	$\qquad \qquad$	87	$\quad$	10
V.	Bertrand's Theorem	$\qquad \qquad$	91	$\quad$	13
VI.	Non-Bertrandian Systems	$\qquad \qquad$	95	$\quad$	16
VII.	Cyclotron Motion	$\qquad \qquad$	98	$\quad$	19
VIII.	The Magnetic Monopole	$\qquad \qquad$	101	$\quad$	21
IX.	Two Coulomb Centers	$\qquad \qquad$	105	$\quad$	24
X.	Relativistic Systems	$\qquad \qquad$	109	$\quad$	27
XI.	Zitterbewegung	$\qquad \qquad$	115	$\quad$	32
XII.	Dirac Equation for the Hydrogen Atom	$\qquad \qquad$	120	$\quad$	35
XIII.	Other Possible Systems and Symmetries	$\qquad \qquad$	125	$\quad$	39
XIV.	Universal Symmetry Groups	$\qquad \qquad$	129	$\quad$	42
XV.	Summary	$\qquad \qquad$	134	$\quad$	47
	References	$\qquad \qquad$	137	$\quad$	49

Throughout this article we shall be describing wave equations, both Schrödinger's and Dirac's 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 $\hbar = c = 1$, as well as taking 1 for a particle's mass, except that we will retain an explicit $m$ in relativistic formulas.