Page 90 - PRINCIPLES OF QUANTUM MECHANICS as Applied to Chemistry and Chemical Physics

3.6 Hilbert space and Dirac notation 81

operation is a rotation of the basis set jø i i about the origin to a new basis set
^
jö i i. In the situation where A acting on jø i i gives a constant times jø i i (cf.
equation (3.5))
^ ^
Ajø i iˆ jAø i iˆ á i jø i i
^
the ket jAø i i is along the direction of jø i i and the kets jø i i are said to be
^
eigenkets of the operator A.
^
^
Although the expressions Ajø i i and jAø i i are completely equivalent, there
^
is a subtle distinction between them. The ®rst, Ajø i i, indicates the operator A ^
^
being applied to the ket jø i i. The quantity jAø i i is the ket which results from
that application.

Bra vectors
The functions ø i (x) are, in general, complex functions. As a consequence, ket
space is a complex vector space, making it mathematically necessary to
introduce a corresponding set of vectors which are the adjoints of the ket
vectors. The adjoint (sometimes also called the complex conjugate transpose)
of a complex vector is the generalization of the complex conjugate of a
complex number. In Dirac notation these adjoint vectors are called bra vectors
y
or bras and are denoted by hø i j or hij. Thus, the bra hø i j is the adjoint jø i i of
y
the ket jø i i and, conversely, the ket jø i i is the adjoint hø i j of the bra hø i j
y
jø i i ˆhø i j
y
hø i j ˆjø i i
These bra vectors determine a bra space, just as the kets determine ket space.
The scalar product or inner product of a bra höj and a ket jøi is written in
Dirac notation as höjøi and is de®ned as

höjøiˆ ö (x)ø(x)dx

The bracket (bra-c-ket)in höjøi provides the names for the component
vectors. This notation was introduced in Section 3.2 as a shorthand for the
scalar product integral. The scalar product of a ket jøi with its corresponding
bra høj gives a real, positive number and is the analog of multiplying a
complex number by its complex conjugate. The scalar product of a bra hø j j
^
^
^
and the ket jAø i i is expressed in Dirac notation as hø j jAjø i i or as hjjAjii.
^
These scalar products are also known as the matrix elements of A and are
sometimes denoted by A ij .
To every ket in ket space, there corresponds a bra in bra space. For the ket