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)