Physical Chemistry , 1st ed.

12.2 Spin

Not long before quantum mechanics was developed, an important experimen-
tal observation was made. In 1922, Otto Stern and W. Gerlach attempted to
measure the magnetic moment of the silver atom. They passed vaporized sil-
ver atoms through a magnetic field and recorded the pattern that the beam of
atoms made after it passed through the magnetic field. Surprisingly, the beam
split into two parts. The experiment is illustrated in Figure 12.1.
Attempts to explain this in terms of the Bohr theory and quantized angular
momentum of electrons in their orbits failed. Finally, in 1925, George
Uhlenbeck and Samuel Goudsmit proposed that this result could be explained
if it was assumed that the electron had its ownangular momentum. This an-
gular momentum was an intrinsic property of the electron itself and not a con-
sequence of any motion of the electron. In order to explain the experimental
results, Uhlenbeck and Goudsmit proposed that components of the intrinsic
angular momentum, called spin angular momentum,had quantized values of
either ^12 or ^12 . (Recall that hhas units of angular momentum.)
Since that proposal, it has become understood that all electrons have an in-
trinsic angular momentum called spin.Although commonly compared to the
spinning of a top, the spin angular momentum of an electron is not due to any
rotation about the axis of the particle. Indeed, it would be impossible for us to
determine that an electron is actually spinning. Spin is a property of a parti-
cle’s very existence. This property behaves as if it were an angular momentum,
so for all intents and purposes it is considered an angular momentum.
Like the angular momentum of an electron in its orbit, there are two mea-
surables for spin that can be observed simultaneously: the square of the total
spin and the zcomponent of the spin. Because spin is an angular momentum,
there are eigenvalue equations for the spin observables that are the same as for
Lˆ^2 and Lˆz, except we use the operators Sˆ^2 and ˆSzto indicate the spin observ-
ables. We also introduce the quantum numbers sand msto represent the quan-
tized values of the spin of the particles. (Do not confuse s, the symbol for the

12.2 Spin 371

S

Beam of

Ag atoms

N

Oven

Beam split

into two

Magnet

Glass

plate

Figure 12.1 A diagram of the Stern-Gerlach experiment. A beam of silver atoms passed
through a magnetic field splits into two separate beams. This finding was used to propose the ex-
istence of spin on the electron.