[S 1z , H ] = [S 1z , S 1+ ]S 2− + S 1+ [S 1z , S 2− ] + [S 1z , S 1− ]S 2+ + S 1− [S 1z , S 2+ ] + + [S 1z , S 2+ ]S 3− + S 2+ [S 1z , S 3− ] + [S 1z , S 2− ]S 3+ + S 2− [S 1z , S 3+ ]
S j± = S jx
y j
± iS ,
j = 1, 2, 3.
S jz , S i± = 0,
i = 1, 2, 3.
y j
[S jz , S j± ] = [S jz , S jx
± iS ] = ± S ±
[S 1z , H ] = S 1+ S 2− [S 2z , H ] [S 2z , H ] = [S 3z , H ] =
j
+ 2
− S − S 1
[S 3z , H ] + 1 2
1
+ 2
+ 2 3
2
+ 3
− S S − + S − S − S S − + S − S
+ S 2+ S 3−
+ 3
− S − S 2
[S z , H ] = 0, z
S z S 1z , S 2z
S 3z
S z
1/2
s = 1/2, m =
±1/2
s = 3/2, m =
−3/2, −1/2, 1/2, 3/2.
