QMGreensite_merged

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 U
¼UðÞ¼Ix
 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 Š

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 Š

Satisfying the simultaneous system of equations in [2.85] and [2.88]
requires thatU 11 ¼U 12 andU 21 ¼
U 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 Š

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 Š
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 Š
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