Spin - Methods of Molecular Quantum Mechanics - page 76

is the spin vector operator (analogous to the orbital angular momentum
vector operator L , but whose observable values can be half-integer
as well). It is assumed that S and its components satisfy the same
commutation rules as those of the angular momentum operators. 2
The two spin states are assumed to be kets normalized to 1 and
orthogonal to each other:

hajai¼hbjbi¼ 1 ; hajbi¼hbjai¼ 0 ð 5 : 3 Þ

It may sometimes be useful to associate with spin the formal variable s
(as distinct from r , the triplet of coordinates specifying the position in
space of the electron) and write Equation 5.3 formally as
ð ds a ð ds b ð ds a ð ds b

ð s Það s Þ¼ ð s Þbð s Þ¼ 1 ; ð s Þbð s Þ¼ ð s Það s Þ¼ 0

ð 5 : 4 Þ

An electron with spin in a uniform magnetic field B (B x ¼ B y ¼ 0 ;

B z ¼ B) acquires a potential energy

b e BS z V ð s Þ¼m S B ¼ g e ð 5 : 5 Þ

where g e 2 is the intrinsic electron g-factor (the so-called anomaly
of spin), b e is the Bohr magneton (the unit of magnetic moment), given by

eh
2mc ¼ 9

274 015 10 24 JT 1 b e ¼ : ð 5 : 6 Þ

and m S , the magnetic moment operator associated with the electron

spin S , is given by

b e S m S ¼ g e ð 5 : 7 Þ

The total one-electron Hamiltonian including spin will be

h ðr; s Þ¼ h 0 ðrÞþ g e

b e BS z

ð 5 : 8 Þ

½ S x ; S y ¼ iS z ; ½ S y ; S z ¼ iS x ; ½ S z ; S x ¼ iS y ; ½ S 2 ; S k ¼ 0

2

; k ¼ x ; y ; z.