QMGreensite_merged

in whichUis a unitary operator with matrix elements in the unprimed
basis given by

Uij¼hijUjji¼hijj^0 i: ½ 2 : 79 Š

The representation of an operator in the two basis sets is then given by
the similarity transformation

A^0 ¼UAU
^1 : ½ 2 : 80 Š

Using these results, the expectation value ofA^0 is

^0

(^)
A^0
^0
¼hj
U
^1 UAU
^1 Uji
¼hj
Aji
, ½ 2 : 81 Š
which justifies earlier assertions that the results of calculating the
expectation value of an operator do not depend on the choice of
basis set.
In order to clarify these ideas, the transformation between a basis set
consisting of the eigenfunctions ofIzand a basis set consisting of the
eigenfunctions ofIxis presented. The eigenfunction equations forIxare
defined as
Ix’ 1
¼
1
2
’ 1
, Ix’ 2
¼

1
2
’ 2
, ½ 2 : 82 Š
in which 1 and’ 2 are the (as yet unspecified) eigenfunctions. An
arbitrary wavefunction can be written as

¼c ji þc
(^) ½ 2 : 83 Š
in the basis functions ofIzand as

^0 ¼c 1 ’ 1
(^)
þc 2 ’ 2
(^)
½ 2 : 84 Š
in the basis functions of Ix. Application of [2.78] yields the matrix
equation

^0 ¼U
,
c 1
c 2

¼
U 11 U 12
U 21 U 22

c
c

:
½ 2 : 85 Š
Using [2.82] and [2.84],
Ix^0
^0 ¼
1
2
c 1 ’ 1
(^)
^1
2
c 2 ’ 2
(^) ¼^1
2
c 1

c 2

, ½ 2 : 86 Š
48 CHAPTER 2 THEORETICALDESCRIPTION OFNMR SPECTROSCOPY