BioPHYSICAL chemistry

Spin

From Schrödinger’s equation for the hydrogen atom
we obtained three indices, n,l, and ml, that represent
the quantization of energy, angular momentum, and
the zprojection of angular momentum respectively.
Based upon these three indices, there are, at most, two
electrons per orbital. At the time that the hydrogen atom
was originally being formulated, it was realized that
there were, at most, two electrons per orbital; however,
there was no reason for the pairing of electrons in
each orbital. As you already know, there is a fourth
quantum number, spin. Electrons are paired in orbitals
with one electron as a spin up and the other as a spin
down. Thus, each electron is defined uniquely by the
four quantum numbers, n,l, and ml, and the spin
projection ms.
The term spin derives from its contribution to the total
angular momentum. Spin is observed experimentally
in the Stern–Gerlach experiment in which a beam
of silver atoms was shot through a heterogeneous
magnetic field (Figure 12.11). If the atoms had a
continuous range of angular momentum, as would be
allowed classically, a broad, continuous band would
be observed. However, only two bands are observed
and this quantization must arise from two states of
angular momentum. This must arise from another con-
tribution to angular momentum that is quantized. For these experiments,
silver atoms were used with a total of 47 electrons: 46 would be paired
together, leaving one unpaired electron. The quantization must be from the
unpaired electron. This was assigned as arising from a specific property of
the electron, termed spin. Spin is a fundamental property of the electron,
having a fixed value, like charge, that allows the electron to interact with
a magnetic field.
The expressions that define the spin operators are written in a fashion
similar to those for orbital angular momentum. In this case, both the total
squared momentum and the zprojection are quantized by the quantum
numbers land ml:

L^2 ψ=l(l+1)Z^2 ψ (12.39)

Lzψ=mlZψ

For the spin, the corresponding quantum numbers are the spin quantum
number, s, and the z projection of the spin,ms. These two quantum numbers

CHAPTER 12 THE HYDROGEN ATOM 257

Ions

In

Ions

In

B Classical prediction

continuous

Observed discrete

bands

B

Figure 12.11A schematic

representation of the experimental

arrangement for the Stern–Gerlach

experiment showing the observed

experimental outcome of two bands.