a/Bottom: A schematic di-
agram of the Stern-Gerlach
experiment. The z direction
is out of the page. The entire
apparatus is about 10 cm long.
Top: A portion of Gerlach’s cel-
ebratory 1922 postcard to Niels
Bohr, with a photo showing the
results. A coordinate system is
superimposed. The orientation is
flipped downward by 90 degrees
compared to the schematic. The
photo was taken through a micro-
scope, and Gerlach drew the 1.0
mm scale on after the magnified
photo had been printed.
The entire apparatus was sealed inside a vacuum chamber with
the best vacuum obtainable at the time. A sample of silver was
heated to 1000◦C, evaporating it. The atoms leaving the oven en-
countered two narrow slits, so that what emerged was a beam with
a width of only 0.03 mm, or about a third of the width of a human
hair. The atoms then encountered a magnetic field. Because the
atoms were electrically neutral, we would normally expect them to
be unaffected by a magnetic field. But in the planetary model of
the atom, we imagine the electrons as orbiting in circles like lit-
tle current loops, which would give the atom a magnetic dipole
momentm. Even if we are sophisticated enough about quantum
mechanics not to believe in the circular orbits, it is reasonable to
imagine that such a dipole moment would exist. When a dipole
encounters anonuniform field, it experiences a force (example 7,
p. 589). In this example, the forces in thexandzdirections would
beFx=m·(∂B/∂x) andFz=m·(∂B/∂z). (Because of Gauss’s
law for magnetism, these two derivatives are not independent — we
have∂Bx/∂x+∂Bz/∂z= 0.) The rapidly varying magnetic field for
this experiment was provided by a pair of specially shaped magnet
poles, described in example 27, p. 743.
Because electrons have charge, we expect the motion of an elec-
tron to give it a magnetic dipole momentm. But they also have
mass, so for exactly the same reasons, we expect there to be some
angular momentumLas well. The analogy is in fact mathematically
exact, as discussed in sec. 11.2.4, p. 695. Therefore this experiment
with dipoles and magnetic fields is actually a probe of the behavior
of angular momentum at the atomic level.
Luckily for Stern and Gerlach, who had no modern knowledge of
atomic structure, the silver atoms that they chose to use do happen
to have nonzero totalL, and therefore nonzerom. The atoms come
out of the oven with random orientations.
Classically, we would expect the following. Each atom has an
energym·Bdue to its interaction with the magnetic field, and
this energy is conserved, so that the componentmxstays constant.
However, there is a torquem×B, and this causes the direction of
the atom’s angular momentum to precess, i.e., wobble like a top,
with its angular momentum forming a cone centered on thexaxis
(example 25, p. 289). This precession is extremely fast, carrying out
about 10^10 wobbles per second, so that the atom precesses about 10^6
times while traveling the 3.5 cm length of the spectrometer. So even
though the forcesFxandFzare typically about the same size, the
rapid precession causesFzto average out to nearly zero, and only
a deflection in thexdirection is expected. Because the orientations
of the atoms are random as they enter the magnetic field, they will
have every possible value of Lx ranging from−|L|to +|L|, and
therefore we expect that when the magnetic field is turned on, the
effect should be to smear out the image on the glass plate from a958 Chapter 14 Additional Topics in Quantum Physics