VI. i] QUANTIZATION IN SPACE 85
As the electron can move round the orbit in opposite directions
there will be twice as many states as there are orbits.
To illustrate this result, let us consider in particular the simple
cases k = 1, k = 2, k = 3. These are shown in Fig. 13 :
¿ = 1 k = 2 fc ~ 3
Fig. 13.—Spatial Position of Orbital Planes.
When k — 1, as the value a = - is excluded, we are left with
2
only one possibility, namely when cos a is unity, and therefore
a is zero. The orbital plane coincides with the equatorial
plane. But there are two possible senses for the rotation of the
electron, in this plane, and so we get two possible states for
our atom.
When k = 2 there are two possible values of a given by
cos a — I and cos a — f = 1. Consequently a is either 6o° or zero.
This means that besides the equatorial plane there is another
possible inclination of the orbital plane, namely at an angle of
60 0 with the equatorial plane. The orbital plane, inclined at
6o°, can be rotated arbitrarily about the direction of the lines
of force.
When k = 3 there are three possibilities, cos a = cos a — §,
cosa = f = 1. In each of these orientations in space the electron
may describe either circular or elliptic orbits. It should be
mentioned that in this simple treatment both the core and the
electron have been regarded as possessing no magnetic properties.
Sommerfeld remarks : “ Without doubt this spatial quantizing
is one of the most surprising results of the quantum theory.
When we consider the simplicity with which the positions are
derived and how simple is the result, it seems almost like
magic.”
We see, then, that for what may be termed the “ one-
quantum ” atom there are two positions, and two only, for the
vector of the magnetic moment in the magnetic field. These
correspond to the two directions of rotation in the figure, and
may be called the parallel and the anti-parallel positions. It
may be supposed that one half of the atoms assume the
first position, and the other half the second. That may seem
surprising if we think of material magnets, because we are not
