Arbitrary kets and bras, expressed as a linear combination of the
eigenkets or eigenbras, have the representations
ji ¼c ji þc(^)
¼c
1
0
þc
0
1
¼
c
c
,
hj¼c hj þc
(^)
¼c 10
þc 01
¼ c c
:
½ 2 : 75
Thus, the matrix representation of a ket is the column vector whose
elements are the coefficients from the expansion in terms of basis kets.
The results of operator manipulations can be expressed using matrix
algebra. For example,
Ixji ¼^12
01
10
1
0
¼^12
0
1
¼^12
; Ix
¼^12
01
10
0
1
¼^12
1
0
¼^12 ji ;
Iyji ¼^1
2
0 i
i 0
1
0
¼^1
2
0
i
¼i
2
(^)
; Iy
(^)
¼^1
2
0 i
i 0
0
1
¼^1
2
i
0
¼i
2
ji ;
Izji ¼
1
2
10
0 1
1
0
¼
1
2
1
0
¼
1
2 ji^ ; Iz^
¼
1
2
10
0 1
0
1
¼
1
2
0
1
¼
1
2
;
½ 2 : 76
express the results of the Cartesian spin operators acting on thej iand
j ikets. These results should be compared with [2.31]. Similarly, the
orthogonality relations are obtained as
hi j ¼ 10
1
0
¼1,
j
¼ 10
0
1
¼0,
j
¼ 01
1
0
¼0,
j
¼ 01
0
1
¼ 1 :
½ 2 : 77
The matrix representations of operators and wavefunctions depend
upon the particular basis set employed. Matrix representations using
different basis sets can be interconverted using unitary transformations.
If^0 is the representation of a wavefunction in one (primed) basis set
andis the representation in another (unprimed basis), then
j^0 i¼Uji, ½ 2 : 78
2.2 THEDENSITYMATRIX 47