148 THE QUANTUM [ X<5
that structure within the atom which may be regarded as the
origin of the quantum magnetic tubes.
Consider the case of an atom consisting of a positive core
and a negative electron. There are two possibilities presenting
themselves. The core of the atom may be a magneton, or the
negative electron may be a magneton.
Without entering into a discussion of these alternatives, it
will be sufficient for our present purpose to assume that the
magneton is closely associated with an integral number of quan
tum tubes of frequency v T . As in Bohr’s theory, the electron is
supposed to describe an orbit about the nucleus, the size of the
orbit and the frequency being determined by the usual quantum
relations. For simplicity we shall consider the case of a massive
nucleus, which may be regarded as at rest.
Let n denote the number of quantum tubes linked with the
orbit and also passing through the magneton, and n T the num
ber of quantum tubes associated with the magneton which are
not linked with the orbit.
Then, in accordance with the principles discussed in the
earlier portion of this chapter, the electrokinetic energy of the
system in its initial state is given by
T = \njiv r + \nh{y -f v r ) . . . io : 46
Denoting the corresponding quantities in the final state by
accented letters, we have, similarly,
T # = \n r 'hv/ + \n'h{y' + v T ') . . . 10 :47
To proceed further we have to make some specific assumptions
with regard to the frequencies. In the first place, we assume
that the frequency v r remains unchanged in passing from the
initial to the final state, and we identify this frequency with that
of the emitted radiation.
In systems which emit hydrogen-like spectra the kinetic
energy is numerically the same as Bohr’s W. In his theory the
kinetic energy in the new orbit is greater than that in the old,
but the amount of potential energy set free is exactly twice the
change in the kinetic energy. Consequently, an amount of
energy is available for radiation which is equal to the change
in the kinetic energy. This is
T' — T = §[(w/ + n') — (n r + n)]hv r + \n'hv' — \nhv 10 : 48
We now assume that in the change from one state to the other
one quantum tube has been liberated from the magneton, and
put
\hv T — T' — T 10 : 49
Hence
\hv T — l[{n T ' + n') — (n T + n) ]hv r -f- \n'hv' — \nhv
10 :50