Page 205 - PRINCIPLES OF QUANTUM MECHANICS as Applied to Chemistry and Chemical Physics

196 Spin

as to include spin. His treatment is equivalent to Dirac's relativistic theory in
the limit of small electron velocities (v=c ! 0).

7.2 Spin angular momentum
The postulates of quantum mechanics discussed in Section 3.7 are incomplete.
In order to explain certain experimental observations, Uhlenbeck and Goudsmit
introduced the concept of spin angular momentum for the electron. This
concept is not contained in our previous set of postulates; an additional
postulate is needed. Further, there is no reason why the property of spin should
be con®ned to the electron. As it turns out, other particles possess an intrinsic
angular momentum as well. Accordingly, we now add a sixth postulate to the
previous list of quantum principles.

6. A particle possesses an intrinsic angular momentum S and an associated magnetic
^
moment M s . This spin angular momentum is represented by a hermitian operator S
^
^
^
which obeys the relation S 3 S ˆ i"S. Each type of particle has a ®xed spin
3
1
quantum number or spin s from the set of values s ˆ 0, ,1, ,2, ... The spin s for
2 2
1
the electron, the proton, or the neutron has a value . The spin magnetic moment for
2
the electron is given by M s ˆÿeS=m e .
As noted in the previous section, spin is a purely quantum-mechanical concept;
there is no classical-mechanical analog.
The spin magnetic moment M s of an electron is proportional to the spin
angular momentum S,
g s e g s ì B
M s ˆÿ S ˆÿ S (7:1)
2m e "
where g s is the electron spin gyromagnetic ratio and the Bohr magneton ì B is
de®ned in equation (5.82). The experimental value of g s is 2.002 319 304 and
the value predicted by Dirac's relativistic quantum theory is exactly 2. The
discrepancy is removed when the theory of quantum electrodynamics is
applied. We adopt the value g s ˆ 2 here. A comparison of equations (5.81) and
(7.1) shows that the proportionality constant between magnetic moment and
angular momentum is twice as large in the case of spin. Thus, the spin
gyromagnetic ratio for the electron is twice the orbital gyromagnetic ratio. The
spin gyromagnetic ratios for the proton and the neutron differ from that of the
electron.
^
The hermitian spin operator S associated with the spin angular momentum S
^
^ ^
has components S x , S y , S z , so that