262 
THE QUANTUM 
This means that not only the angular momentum, but also the eccen 
tricity can have only certain prescribed values. 
The expression for the energy, W, may now be written in terms 
of and n r 
™ _ 2Tr 2 mg 4 i __ 2i t 
w ~ A 2 (n* + n r )* “ T 2 “ V* ' • • 
so that the important fact emerges from this result that the energy 
depends only on the sum of the two integers of and n r . These 
integers, for convenience, may be referred to as the “ azimuthal 
quantum number,” and the “ radial quantum number,” respectively. 
The energy depends on the “ total quantum number,” n, which is the 
sum of these partial quantum numbers and n T . 
The frequency of the radiation emitted is given by 
(20) 
We thus obtain the Balmer series, and other associated series, for 
hydrogen, just as before ; but the series will now be composed of 
sharply defined lines. Since the total quantum number can be made 
up in various ways, each line can be obtained by the passage of an 
electron from several different orbits, which can be elliptic or circular, 
to some inner orbit. 
It is easy to show that the semi-axes a and b of the elliptic orbit 
are given by 
foZ h 2 
a= ^W^ + n ^’ h = + ■ • (21) 
When n r = o, we get the circles of Bohr’s theory. We may notice 
that all orbits having the same value for the energy have equal major 
axes, in fact W = <? 2 /2a. Again, the frequency of the electron in its 
orbit is given by 
v — ft — __ 2roo 
2nabm h^{n^-{-n r ) z (n^ + n r ) z ' * ' 
where rods the fundamental Rydberg frequency. 
From equations (19) and (22) we see that W may be expressed 
in the simple form 
W = 1(«^ + (23)