The quantum numbers are not sufficient to fully characterize the physical state of the electrons in an atom. In 1926, Otto Stern and Walther Gerlach carried out an experiment that could not be explained in terms of the three quantum numbers and showed that there is, in fact, another quantum-mechanical degree of freedom that needs to be included in the theory.

The experiment is illustrated in the figure below:

It is known that a current loop in a nonuniform magnetic field
experiences a net force. This is illustrated below:

where is called the

The fact that the beam splits into 2 beams suggests
that the electrons in the atoms have a degree of freedom capable
of coupling to the magnetic field. That is, an electron
has an *intrinsic magnetic moment* arising from
a degree of freedom that has no classical analog.
The magnetic moment must take on only 2 values according
to the Stern-Gerlach experiment. The intrinsic property that
gives rise to the magnetic moment must have some analog to an
angular momentum and hence, must be a property that, unlike
charge and mass, which are simple numbers, is a
*vector property*.
This property is called the *spin*, ,
of the electron. As the expression above suggests, the
intrinsic magnetic moment of the electron must
be propertional to the spin

In quantum mechanics, spin share numerous features in common
with angular momentum, which is why we represent
it as a vector. In particular, spin is quantized, i.e. we have
certain allowed values of spin. Like angular momentum, the
value of the magnitude squared of spin is fixed, and
one of its components is as well. For an electron, the
allowed values of are

while has just one value , corresponding to a general formula , where . For this reason, the electron is called a

where for electrons. Note that .

For an electron in a *uniform* magnetic field ,
the energy is determined by the spin :

We choose the uniform field to be along the -direction . Since the field lines flow from the north pole to the south pole, this choice of the field means that the north pole lies below the south pole on the -axis. In this case,

Since , we see that the lowest energy configuration uas antiparallel to . Given the two values of , we have two allowed energies or two energy levels corresponding to the two values of :

Using the given expression for , which is negative, we obtain the two energy levels

Unlike position and momentum, which have clear classical analogs, spin does not. But if we think of spin in pseudoclassical terms, we can think of a spinning charged particle, which is similar to a loop of current. Thus, if the particle spins about the -axis, then points along the -axis. Since the spinning charge is negative, the left-hand rule can be applied. When the fingers of the left hand curl in the direction of the spin, the thumb points in the direction of the spin. A spinning charge produces a magnetic field similar to that of a tiny bar magnet. In this case, the spin vector points toward the south pole of the bar magnet. Now if the spinning particle is placed in a magnetic field, it tends to align the spin vector in the opposite to the magnetic field lines, as the figure above suggests.

Now, when the electron is placed in a nonuniform magnetic field, with the field increasing in strength toward the north pole of the field source, the spin-down () electrons have their bar-magnet poles oriented such that the south pole points toward the north pole of the field source, and these electrons will be attracted toward the region of stronger field. The spin-up () electrons have their bar-magnet north poles oriented toward the north pole of the field source and will be repelled to the region of weaker field, thus causing the beam to split as observed in the Stern-Gerlach experiment.

The implication of the Stern-Gerlach experiment is that we need to
include a fourth quantum number, in our description of the
physical state of the electron. That is, in addition to give its
principle, angular, and magnetic quantum numbers, we also need to
say if it is a spin-up electron or a spin-down electron.

Note that we have added spin into the our quantum theory as a kind of
*a posteriori* consideration, which seems a little contrived.
In fact, the existence of the spin degree of freedom can be
derived in a very natural way in the relativistic version of
quantum mechanics, where it simply pops out of the relativistic
analog of the Schrödinger equation, known as the *Dirac equation*.