686 BELL SYSTEM TECHNICAL JOURNAL 



in equation (135) ; the permissible mode, or rather modes, of vibration 

 involve nodal planes and double-cones, which the reader may work 

 out for himself with the aid of (135). 



Second Excited State, w = 3 (the state from which the atom departs 

 when it emits the line //-alpha). Three Eigenfunktionen X3, 0, ^3, 1 

 and A'3, 2- The first corresponds to a vibration with two nodal 

 spheres, and perfect spherical symmetry. The second and third corre- 

 spond to vibrations with one nodal sphere, and with a steady diminu- 

 tion of amplitude from the centre outward, respectively; but being 

 multiplied with the spherical harmonics Yi and F2, they describe 

 modes which are not endowed with spherical symmetry, and involve 

 nodal double-cones and nodal planes. 



Generally: the state distinguished by the numeral n enjoys n 

 distinct Eigenfunktionen, describing vibrations having respectively 

 0, 1, 2, 3 • • • (w — 1) nodal spheres; to the Eigenfunktion with the 

 maximum number of nodal spheres corresponds a single mode of 

 vibration which is spherically symmetric, to the others various modes 

 with varying members of nodal double-cones and planes. 



If this is destined to be the "language of the future" for describing 

 the data of experiment, it will be necessary to have dictionaries for 

 translating it out of (or into) the "language of the present," the 

 vocabulary of the Bohr-Sommerfeld atom-model in which Stationary 

 States are represented by electron-orbits. They will contain defini- 

 tions such as these: the numeral n is the total-quantum-number of 

 the electron-orbits — the numeral / is one unit smaller than the 

 azimuthal quantum-number k of the electron-orbit — the numeral 

 (w — / — 1), to which the number of nodal spheres is equal, is the 

 radial quantum-number of the electron-orbit. To elucidate these 

 "definitions" of the future dictionary, I recall that the Bohr-Sommer- 

 feld atom-model provided, for the hydrogen atom in its state of energy- 

 values En, a family of n distinct electron-orbits, of which one is circular 

 while the other {n — 1) are ellipses of varying degrees of eccen- 

 tricity.^^ These ellipses were selected by laying down the conditions, 

 that the integral Sp,pdip of the angular momentum p^ around the 

 orbit shall be equal to the product of // by some integer k equal to or 

 less than the prescribed n\ and the integral fprdr of the radial 

 momentum f p,dr shall be equal to the product of // by the integer 

 [n — Iz); so that the sum of the integrals fp^dip and fprdr shall be 

 equal to the product of h by n. The quantities n, k and n — k were 

 given the names total, azimuthal, radial quantum-number. "Defini- 



'"The introduction some twenty months ago of the "spinning electron" caused a 

 modification of this picture; for those who accept the modification, it is the "language 

 of antiquity" which is comjjared in this jxTragraph with the "language of the future." 



