vertical line to a somewhat wider oval. The atoms are dispersed from
left to right along a certain scale of measurement according to their
random value ofLx. The spectrometer is a device for determining
Lx, a continuously varying number.
But that’s all the classical theory. Quantum mechanically,Lxis
quantized, so that only certain very specific values ofFxcan occur.
Although the discussion of precession above is really classical rather
than quantum-mechanical, the result ofFzaveraging out to zero
turns out to be approximately right if the field is strong. Therefore
we expect to see well separated vertical bands on the glass plate
corresponding to the quantized values ofLx. This is approximately
what is seen in figure a, although the field rapidly weakens outside
thex-yplane, so we get the slightly more complicated pattern like
a sideways lipstick kiss. Since we observe two values ofLx(the two
“lips”), we conclude from these results that a silver atom has spin
1 /2, so thatLxtakes on the values−~/2 and +~/2. Although it
took about five years for the experiment to be interpreted completely
correctly, we now understand the Stern-Gerlach experiment to be
not just a confirmation of the quantization of angular momentum
along any given axis but also the first experimental evidence that
the electron has an intrinsic spin of 1/2.
Discussion Questions
A Could the Stern-Gerlach experiment be carried out with a beam of
electrons?
B A few weeks after the Stern-Gerlach experiment’s results became
public, Einstein and Ehrenfest carried out the following reasoning, which
seemed to them to make the results inexplicable. Before a particular sil-
ver atom enters the magnetic field, its magnetic momentmis randomly
oriented. Once it enters the magnetic field, it has an energym·B. Unless
there is a mechanism for the transfer of energy in or out of the atom, this
energy can’t change, and therefore the magnetic moment can only pre-
cess about theBvector, but the angle betweenmandBmust remain the
same. Therefore the atom cannot align itself with the field. (They consid-
ered various mechanisms of energy loss, such as collisions and radiation,
and concluded that all of them were too slow by orders of magnitude to
have an effect during the atom’s time of flight.) It seemed to them that
as soon as the atom left the oven, it was somehow required to have an-
ticipated the direction of the field and picked one of two orientations with
respect to it. How can this paradox be resolved?
C Suppose we send a beam of oxygen molecules, withL=~, through a
Stern-Gerlach spectrometer, throwing away the emerging parts withx= −1 and +1 to make a beam of the purex= 0 state. Now we let this beam
pass through a second spectrometer that is identical but oriented along
thezaxis. Can we produce a beam in which every molecule has both
x= 0 andz= +1? Hint: See the example in fig. d, p. 923.
Section 14.1 The Stern-Gerlach experiment 959