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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 Š

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 Š

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 Š
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 Š
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