Up: Fundamental Concepts
Previous: Ket Space
A snack machine inputs coins plus some code entered on a key pad, and
(hopefully) outputs a snack. It also does so in a deterministic manner: i.e.,
the same money plus the same code produces the same snack
(or the same error message) time after time.
Note that the input and output of the machine have completely different natures.
We can imagine building a rather abstract snack machine which inputs ket
vectors and outputs complex numbers in a deterministic fashion. Mathematicians
call such a machine a functional. Imagine a general functional, labeled
, acting on a general ket vector, labeled
, and spitting out a general
. This process is represented mathematically by writing
Let us narrow our focus to those functionals that preserve the linear dependencies
of the ket vectors upon which they operate. Not surprisingly, such functionals
are termed linear functionals. A general linear functional, labeled
are any two kets in a given ket space.
-dimensional ket space [i.e., a finite-dimensional, or
denumerably infinite dimensional (i.e.,
runs from 1 to
independent ket vectors in this space.
A general ket vector can be written
are an arbitrary set of complex numbers.
The only way that the functional
can satisfy Equation (10) for all vectors in the ket
space is if
are a set of complex numbers relating to the functional.
Let us define
Here, the Kronecker delta symbol is defined such that
It follows from the previous three equations that
But, this implies that the set of all possible linear functionals acting
-dimensional ket space is itself an
-dimensional vector space.
This type of vector
space is called a bra
space (after Dirac), and its constituent vectors
(which are actually functionals of the ket space) are called bra vectors.
Note that bra vectors are
quite different in nature to ket vectors (hence, these
vectors are written in mirror image notation,
, so that they can never be confused).
Bra space is an example of what mathematicians call a dual
vector space (i.e., it is dual to the original ket space). There is
a one to one correspondence between the elements of the ket space and those
of the related bra space. So, for every element
ket space, there is a corresponding element, which it is also convenient to
, in the bra space. That is,
where DC stands for dual correspondence.
There are an infinite number of ways of setting up the correspondence between
vectors in a ket space and those in the related bra space. However,
only one of these has any physical significance. (See Section 1.11.) For a general ket vector
, specified by Equation (11), the corresponding bra vector is written
are the complex conjugates of the
is termed the dual vector to
. It follows, from the
above, that the dual to
a complex number. More generally,
Recall that a bra vector is a functional that acts on a general ket vector,
and spits out a complex number. Consider the functional which is dual to the
acting on the ket vector
operation is denoted
. Note, however, that
we can omit the round brackets without causing any ambiguity, so the
operation can also be written
expression can be further simplified
. According to Equations (11), (13), (16),
the inner product of a bra and a ket. An inner product is (almost) analogous to a
scalar product between covariant and contravariant vectors in
It is easily demonstrated that
Consider the special case where
follows from Equations (19) and (20)
is a real number, and that
The equality sign only holds if
is the null ket [i.e., if
all of the
are zero in Equation (11)]. This property of bra and ket
vectors is essential for the probabilistic interpretation of quantum mechanics,
as will become apparent in Section 1.11.
are said to be orthogonal
which also implies that
Given a ket
, which is not the null ket,
we can define a normalized ket
with the property
is known as the norm
or ``length'' of
is analogous to the length, or magnitude, of a conventional vector. Because
represent the same physical state, it makes sense
to require that all kets corresponding to physical states have unit norms.
It is possible to define a dual bra space for a ket space of nondenumerably
infinite dimensions in much the same manner as that described above. The
main differences are that summations over discrete labels become integrations
over continuous labels, Kronecker delta symbols become Dirac delta functions,
completeness must be assumed (it cannot be proved), and the normalization convention
is somewhat different. (See Section 1.15.)
Up: Fundamental Concepts
Previous: Ket Space