Page 204 - PRINCIPLES OF QUANTUM MECHANICS as Applied to Chemistry and Chemical Physics

7.1 Electron spin 195

tum in any direction has only the value "=2or ÿ"=2. This spin angular
momentum is in addition to the orbital angular momentum of the electronic
motion about the nucleus. They further assumed that the spin imparts to the
electron a magnetic moment of magnitude e"=2m e , where ÿe and m e are the
electronic charge and mass. The interaction of an electron's magnetic moment
with its orbital motion accounts for the splitting of the spectral lines in the
alkali and alkaline-earth metal atoms. A combination of spin and relativistic
effects is needed to explain the ®ne structure of the hydrogen-atom spectrum.
The concept of spin as introduced by Uhlenbeck and Goudsmit may also be
applied to the Stern±Gerlach experiment, which is described in detail in
Section 1.7. The explanation for the splitting of the beam of silver atoms into
two separate beams by the external inhomogeneous magnetic ®eld requires the
introduction of an additional parameter to describe the behavior of the odd
electron. Thus, the magnetic moment of the silver atom is attributed to the odd
electron possessing an intrinsic angular momentum which can have one of only
two distinct values.
Following the hypothesis of electron spin by Uhlenbeck and Goudsmit, P. A.
M. Dirac (1928) developed a quantum mechanics based on the theory of
relativity rather than on Newtonian mechanics and applied it to the electron.
He found that the spin angular momentum and the spin magnetic moment of
the electron are obtained automatically from the solution of his relativistic
wave equation without any further postulates. Thus, spin angular momentum is
an intrinsic property of an electron (and of other elementary particles as well)
just as are the charge and rest mass.
In classical mechanics, a sphere moving under the in¯uence of a central
force has two types of angular momentum, orbital and spin. Orbital angular
momentum is associated with the motion of the center of mass of the sphere
about the origin of the central force. Spin angular momentum refers to the
motion of the sphere about an axis through its center of mass. It is tempting to
apply the same interpretation to the motion of an electron and regard the spin
as the angular momentum associated with the electron revolving on its axis.
However, as Dirac's relativistic quantum theory shows, the spin angular
momentum is an intrinsic property of the electron, not a property arising from
any kind of motion. The electron is a structureless point particle, incapable of
`spinning' on an axis. In this regard, the term `spin' in quantum mechanics can
be misleading, but its use is well-established and universal.
Prior to Dirac's relativistic quantum theory, W. Pauli (1927) showed how spin
could be incorporated into non-relativistic quantum mechanics. Since the
subject of relativistic quantum mechanics is beyond the scope of this book, we
present in this chapter Pauli's modi®cation of the wave-function description so