2.5.1 DIRECTPRODUCTSPACES
The wavefunctions in the product basis are given by the direct
productsof the wavefunctions for individual spins:
k¼jim 1 jim 2 jimN
N
i¼ 1
jimi jim 1 ,m 2 ,...,mN, ½ 2 : 138
in whichmitakes on all possible values, yielding 2Nwavefunctions for
spin-1/2 nuclei. The total magnetic quantum number associated with a
wavefunction in the product basis is
Mk¼
XN
i¼ 1
mi: ½ 2 : 139
The direct product of two matrices is given by (illustrated for two 2 2
matrices)
A B¼
A 11 A 12
A 21 A 22
B 11 B 12
B 21 B 22
¼
A 11 B A 12 B
A 21 B A 22 B
¼
A 11 B 11 A 11 B 12 A 12 B 11 A 12 B 12
A 11 B 21 A 11 B 22 A 12 B 21 A 12 B 22
A 21 B 11 A 21 B 12 A 22 B 11 A 22 B 12
A 21 B 21 A 21 B 22 A 22 B 21 A 22 B 22
2
(^66)
(^64)
3
(^77)
(^75)
½ 2 : 140
Thus, for example, the four wavefunctions in the product basis of a
two-spin system are
1 ¼ji ¼
1
0
1
0
¼
1
0
0
0
2
(^66)
(^64)
3
(^77)
75 ; 2 ¼^
¼
1
0
0
1
¼
0
1
0
0
2
(^66)
(^64)
3
(^77)
75 ;
3 ¼
¼
0
1
1
0
¼
0
0
1
0
2
(^66)
6
4
3
(^77)
7
5
; 4 ¼
¼
0
1
0
1
¼
0
0
0
1
2
(^66)
6
4
3
(^77)
7
5
:
½ 2 : 141
Next, consider the operator corresponding to the sum of the
components Iz and Sz in a two-spin system. Clearly, the matrix
2.5 QUANTUMMECHANICS OFMULTISPINSYSTEMS 59