Page 206 - PRINCIPLES OF QUANTUM MECHANICS as Applied to Chemistry and Chemical Physics

7.2 Spin angular momentum 197
^ ^ ^ ^
S ˆ iS x ‡ jS y ‡ kS z
^ 2 ^ 2 x ^ 2 y ^ 2 z
S ˆ S ‡ S ‡ S
These components obey the commutation relations
^
^
^
^ ^
^
^
^
^
[S x , S y ] ˆ i"S z , [S y , S z ] ˆ i"S x , [S z , S x ] ˆ i"S y (7:2)
or, equivalently
^ ^ ^ (7:3)
S 3 S ˆ i"S
Thus, the quantum-mechanical treatment of generalized angular momentum
presented in Section 5.2 may be applied to spin angular momentum. The spin
^
^
^
^
^
operator S is identi®ed with the operator J and its components S x , S y , S z with
^ ^ ^
J x , J y , J z . Equations (5.26) when applied to spin angular momentum are
1
3
^ 2 2 s ˆ 0, ,1, ,2, ... (7:4)
S jsm s iˆ s(s ‡ 1)" jsm s i,
2 2
^ m s ˆÿs ÿ s ‡ 1, ... , s ÿ 1, s (7:5)
S z jsm s iˆ m s "jsm s i,
where the quantum numbers j and m are now denoted by s and m s . The
^
^ 2
simultaneous eigenfunctions jsm s i of the hermitian operators S and S z are
orthonormal
(7:6)
hs9m9 s jsm s iˆ ä ss9 ä m s m9 s
The raising and lowering operators for spin angular momentum as de®ned by
equations (5.18) are
^ ^ ^
S ‡ S x ‡ iS y (7:7a)
^ ^ ^ (7:7b)
S ÿ S x ÿ iS y
and equations (5.27) take the form
^ p  (7:8a)
(s ÿ m s )(s ‡ m s ‡ 1) "js, m s ‡ 1i
S ‡ jsm s iˆ
^ p  (7:8b)
(s ‡ m s )(s ÿ m s ‡ 1) "js, m s ÿ 1i
S ÿ jsm s iˆ
In general, the spin quantum numbers s and m s can have integer and half-
integer values. Although the corresponding orbital angular-momentum quan-
tum numbers l and m are restricted to integer values, there is no reason for
such a restriction on s and m s .
Every type of particle has a speci®c unique value of s, which is called the
spin of that particle. The particle may be elementary, such as an electron, or
composite but behaving as an elementary particle, such as an atomic nucleus.
4
All He nuclei, for example, have spin 0; all electrons, protons, and neutrons
1
2
have spin ; all photons and deuterons ( H nuclei) have spin 1; etc. Particles
2
3
1
with spins 0, 1, 2, ... are called bosons and those with spins , , ... are
2 2
fermions. A many particle system of bosons behaves differently from a many