Page 36 - PRINCIPLES OF QUANTUM MECHANICS as Applied to Chemistry and Chemical Physics

1.7 Stern±Gerlach experiment 27
x
z

y

Magnet
Figure 1.11 A cross-section of the magnet in Figure 1.10.

direction, resulting in a force F x in the x-direction acting on each silver atom.
This force is given by

@V @B
F x ˆÿ ˆ M cos è
@x @x
where M and B are the magnitudes of the vectors M and B and è is the angle
between the direction of the magnetic moment and the positive x-axis. Thus,
the inhomogeneous magnetic ®eld de¯ects the path of a silver atom by an
amount dependent on the orientation angle è of its magnetic moment. If the
angle è is between 08 and 908, then the force is positive and the atom moves in
the positive x-direction. For an angle è between 908 and 1808, the force is
negative and the atom moves in the negative x-direction.
As the silver atoms escape from the oven, their magnetic moments are
randomly oriented so that all possible values of the angle è occur. According to
classical mechanics, we should expect the beam of silver atoms to form, on the
detection plate, a continuous vertical line, corresponding to a gaussian distribu-
tion of impacts with a maximum intensity at the center (x ˆ 0). The outer
limits of this line would correspond to the magnetic moment of a silver atom
parallel (è ˆ 08) and antiparallel (è ˆ 1808) to the magnetic ®eld gradient
(@B=@x). What is actually observed on the detection plate are two spots,
located at each of the outer limits predicted by the classical theory. Thus, the
beam of silver atoms splits into two distinct components, one corresponding to
è ˆ 08, the other to è ˆ 1808. There are no trajectories corresponding to
intermediate values of è. There is nothing unique or special about the vertical
direction. If the magnet is rotated so that the magnetic ®eld gradient is along
the y-axis, then again only two spots are observed on the detection plate, but
are now located on the horizontal axis.
The Stern±Gerlach experiment shows that the magnetic moment of each