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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 complex number . This process is represented mathematically by writing
Let us narrow our focus to those functionals which 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 , satisfies
where and are any two kets in a given ket space.
Consider an -dimensional ket space [i.e., a finite-dimensional, or denumerably infinite dimensional (i.e., ), space]. Let the (where runs from 1 to ) represent independent ket vectors in this space. A general ket vector can be written1
where the are an arbitrary set of complex numbers. The only way the functional can satisfy Eq. (10) for all vectors in the ket space is if
where the are a set of complex numbers relating to the functional.
Let us define basis functionals which satisfy
It follows from the previous three equations that
But, this implies that the set of all possible linear functionals acting on an -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, and , 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 of the ket space, there is a corresponding element, which it is also convenient to label , 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. For a general ket vector , specified by Eq. (11), the corresponding bra vector is written
where the are the complex conjugates of the . is termed the dual vector to . It follows, from the above, that the dual to is , where is a complex number. More generally,
Recall that a bra vector is a functional which acts on a general ket vector, and spits out a complex number. Consider the functional which is dual to the ket vector
acting on the ket vector . This operation is denoted . Note, however, that we can omit the round brackets without causing any ambiguity, so the operation can also be written . This expression can be further simplified to give . According to Eqs. (11), (12), (16), and (18),
Mathematicians term the inner product of a bra and a ket.2 An inner product is (almost) analogous to a scalar product between a covariant and contravariant vector in some curvilinear space. It is easily demonstrated that
Consider the special case where . It follows from Eqs. (12) and (20) that 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 Eq. (11)]. This property of bra and ket vectors is essential for the probabilistic interpretation of quantum mechanics, as will become apparent later.
Two kets and are said to be orthogonal if
which also implies that .
Given a ket which is not the null ket, we can define a normalized ket , where
with the property
Here, is known as the norm or ``length'' of , and is analogous to the length, or magnitude, of a conventional vector. Since and 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-functions become Dirac delta-functions, completeness must be assumed (it cannot be proved), and the normalization convention is somewhat different. More of this later.