Page 92 - PRINCIPLES OF QUANTUM MECHANICS as Applied to Chemistry and Chemical Physics

3.6 Hilbert space and Dirac notation 83
^ y
^ y
(cA) ˆ c A (3:38)
where c is any complex constant, and that
^
^
^
^ y
(A ‡ B) ˆ A ‡ B y (3:39)
y
^^
To obtain the adjoint of the product AB of two operators, we apply equation
^
^
^^
(3.33a), ®rst to AB, then to A, and ®nally to B
^^ y ^^ ^ ^ y ^ ^ y
y
hø j j(AB) jø i iˆhABø j jø i iˆhBø j jA jø i iˆhø j jB A jø i i
Thus, we have the relation
^^
y
^ ^ y
y
(AB) ˆ B A (3:40)
^ ^
^
^^
^
y
If A and B are hermitian (self-adjoint), then we have (AB) ˆ BA and further, if
^^
^
^
A and B commute, then the product AB is hermitian or self-adjoint.
The outer product of a bra höj and a ket jøi is jøihöj and behaves as an
operator. If we let this outer product operate on another ket j÷i, we obtain the
expression jøihöj÷i, which can be regarded in two ways. The scalar product
höj÷i is a complex number multiplying the ket jøi, so that the complete
expression is a ket parallel to jøi. Alternatively, the operator jøihöj acts on the
ket j÷i and transfroms j÷i into a ket proportional to jøi.
To ®nd the adjoint of the outer product j÷ihöj of the ket j÷i and the bra höj,
^
we let A in equation (3.35) be equal to j÷ihöj and obtain
y
hø j j(j÷ihöj) jø i iˆhø i j(j÷ihöj)jø j i ˆhø i j÷i höjø j i
ˆh÷jø i ihø j jöiˆhø j jöih÷jø i iˆhø j j(jöih÷j)jø i i
Setting equal the operators in the left-most and right-most integrals, we ®nd
that
y
(j÷ihöj) ˆjöih÷j (3:41)
Projection operator
^
We de®ne the operator P i as the outer product of jø i i and its corresponding bra
^
P i jø i ihø i j jiihij (3:42)
^
and apply P i to an arbitrary ket jöi
^
P i jöiˆjiihijöi
^
Thus, the result of P i acting on jöi is a ket proportional to jii, the proportion-
^
ality constant being the scalar product hø i jöi. The operator P i , then, projects
jöi onto jø i i and for that reason is known as the projection operator. The
^ 2
operator P is given by
i
^ 2 ^ ^ ^
P ˆ P i P i ˆjiihijiihijˆjiihijˆ P i
i
^ n
where we have noted that the kets jii are normalized. Likewise, the operator P i