Page 91 - PRINCIPLES OF QUANTUM MECHANICS as Applied to Chemistry and Chemical Physics

82 General principles of quantum theory

cjø i i, the corresponding bra is c hø i j. We can also write cjø i i as jcø i i,in
which case the corresponding bra is hcø i j, so that

hcø i jˆ c hø i j
^
^
For every linear operator A that transforms jø i i in ket space into jö i iˆjAø i i,
^ y
there is a corresponding linear operator A in bra space which transforms hø i j
^
^
^ y
into hö i jˆhAø i j. This operator A is called the adjoint of A. In bra space the
transformation is expressed as
^
hAø i jˆhø i jA ^ y
Thus, for bras the operator acts on the vector to its left, whereas for kets the
operator acts on the vector to its right.
^
To ®nd the relationship between A and its adjoint A , we take the scalar
^ y
^
product of hAø j j and jø i i
^ ^ y
hAø j jø i iˆhø j jA jø i i (3:33a)
or in integral notation

^

^ y
(Aø j ) ø i dx ˆ ø A ø i dx (3:33b)
j
^
A comparison with equation (3.8) shows that if A is hermitian, then we have
^
^
A ˆ A and A is said to be self-adjoint. The two terms, hermitian and self-
^ y
adjoint, are synonymous. To ®nd the adjoint of a non-hermitian operator, we
apply equations (3.33). For example, we see from equation (3.10) that the
adjoint of the operator d=dx is ÿd=dx.

Since the scalar product højöi is equal to höjøi , we see that
^ ^
hAø j jø i iˆhø i jAjø j i (3:34)
Combining equations (3.33a) and (3.34) gives
^ y ^
hø j jA jø i iˆhø i jAjø j i (3:35)
^
^ y
If we replace A in equation (3.35) by the operator A , we obtain
^ y y ^ y
hø j j(A ) jø i iˆhø i jA jø j i (3:36)
^ y
^ y y
where (A ) is the adjoint of the operator A . Equation (3.35) may be rewritten
as
^ y ^
hø i jA jø j i ˆhø j jAjø i i
and when compared with (3.36), we see that
^ y y ^
hø j j(A ) jø i iˆhø j jAjø i i
We conclude that
^ y y
(A ) ˆ A ^ (3:37)
From equation (3.35) we can also show that