Page 89 - PRINCIPLES OF QUANTUM MECHANICS as Applied to Chemistry and Chemical Physics

80 General principles of quantum theory

In some of the derivations presented in this section, operators need not be
hermitian. However, we are only interested in the properties of hermitian
operators because quantum mechanics requires them. Therefore, we have
implicitly assumed that all the operators are hermitian and we have not
bothered to comment on the parts where hermiticity is not required.

3.6 Hilbert space and Dirac notation
This section introduces the basic mathematics of linear vector spaces as an
alternative conceptual scheme for quantum-mechanical wave functions. The
concept of vector spaces was developed before quantum mechanics, but Dirac
applied it to wave functions and introduced a particularly useful and widely
accepted notation. Much of the literature on quantum mechanics uses Dirac's
ideas and notation.
A set of complete orthonormal functions ø i (x) of a single variable x may be
regarded as the basis vectors of a linear vector space of either ®nite or in®nite
dimensions, depending on whether the complete set contains a ®nite or in®nite
number of members. The situation is analogous to three-dimensional cartesian
space formed by three orthogonal unit vectors. In quantum mechanics we
usually (see Section 7.2 for an exception) encounter complete sets with an
in®nite number of members and, therefore, are usually concerned with linear
vector spaces of in®nite dimensionality. Such a linear vector space is called a
Hilbert space. The functions ø i (x) used as the basis vectors may constitute a
discrete set or a continuous set. While a vector space composed of a discrete
set of basis vectors is easier to visualize (even if the space is of in®nite
dimensionality) than one composed of a continuous set, there is no mathema-
tical reason to exclude continuous basis vectors from the concept of Hilbert
space. In Dirac notation, the basis vectors in Hilbert space are called ket
vectors or just kets and are represented by the symbol jø i i or sometimes
simply by jii. These ket vectors determine a ket space.
When a ket jø i i is multiplied by a constant c, the result c jø i iˆjcø i i is a
ket in the same direction as jø i i; only the magnitude of the ket vector is
^
changed. However, when an operator A acts on a ket jø i i, the result is another
ket jö i i
^ ^
jö i iˆ Ajø i iˆjAø i i
In general, the ket jö i i is not in the same direction as jø i i nor in the same
direction as any other ket jø j i, but rather has projections along several or all
^
basis kets. If an operator A acts on all kets jø i i of the basis set, and the
^
resulting set of kets jö i iˆjAø i i are orthonormal, then the net result of the