in which the prime has been added toIxto emphasize that the eigenbase
ofIxis being utilized. Using [2.75] and [2.78],
I^0 x^0 ¼UIxU^1 U¼UðÞ¼Ix U
1
2
c
þ
1
2
c ji
¼
1
2
U 11 U 12
U 21 U 22
c
c
, ½ 2 : 87
in which the results in [2.76] have been used. Equating [2.86] and [2.87]
yields
c 1
c 2
¼ UU^11 U^12
21 U 22
c
c
: ½ 2 : 88
Satisfying the simultaneous system of equations in [2.85] and [2.88]
requires thatU 11 ¼U 12 andU 21 ¼U 22. The columns ofUmust be
normalized and orthogonal, becauseUis unitary. Thus,U^211 þU^222 ¼ 1
andU^211 U^222 ¼0. Finally, the determinant ofUmust equalþ1, so that
Urepresents a proper rotation. Thus, 2U 11 U 22 ¼1. These additional
constraints give
U¼
1
ffiffiffi
2
p^11
11
, ½ 2 : 89
from which the explicit relationships are obtained:
’ 1
(^)
¼
1
ffiffiffi
2
p ji þ
^
,
’ 2
(^)
¼
1
ffiffiffi
2
p ji
^
:
½ 2 : 90
Using [2.80], the operator,Izfor example, has a matrix representation in
the basis set of theIxeigenfunctions of
I^0 z¼UIzU^1 ¼
1
4
11
11
10
0 1
1 1
11
¼
1
4
11
11
1 1
1 1
¼
1
2
01
10
:
½ 2 : 91
In a particularly important application of these ideas, the matrix
representation of the Hamiltonian operator,H, is calculated in some
convenient basis. The matrixUis then determined such that the new
2.2 THEDENSITYMATRIX 49