Bra-ket notation

Dirac Notation is essentially the language of quantum mechanics. Although observable quantities are associated with linear operators, and states are represented by vectors, the required computations can be greatly simplified through the use of the Dirac Bracket Notation.

In non-relativistic quantum mechanics, states are said to reside in a Hilbert Space which, by definition, has an inner product, typically denoted by . In the Dirac Notation, we use the symbol to represent an element of the Hilbert Space in question. This vector is called a "ket". However, by Reisz' Representation Theorem, each element of the Hilbert space also uniquely defines a linear functional which resides in the dual space in terms of the inner product, as follows:

In the Dirac Notation, the functional defined by is instead denoted by:

And in this case, is called a "bra". When the bra is written next to a ket , we understand the pair to form a "bracket" giving us an inner product:

The notation is deceptively simple. The elegant nature of the Dirac Bracket Notation allows physicists to treat linear functionals represented as bras in a very intuitive fashion (they preserve nearly all of the familiar algebraic properties of numbers except commutativity). Computations of inner products are naturally suggested, and the problem is not bogged down in excessive notation -- it is essentially distilled down to its algebraic content alone.

Othogonality of Bras and Kets

Because the bracket represents an inner product, certain concepts from linear algebra will continue to play a role. One is that of orthogonality. By using the Gram-Schmidt Process, and set of linearly independent kets can be orthogonalized, and we may often times assume that such a procedure has been carried out. In addition, because of the probabilistic interpretation of wave mechanics, we may actually take the kets to be normalized. In the Dirac Notation, the normalization condition reads:

While, by orthogonality, if :

So for any two kets from a countable orthonormal set indexed by integers, we may write:

Where is the Kronecker Delta function. As far as discrete sets of states are concerned, the Dirac Notation has an important role to play in the representation of linear operators by matrices. Suppose we have a complete orthonormal set of kets:

And a linear operator . Then the entries of the matrix representation of in the basis defined by the kets is simply:

Likewise, if the kets are known, but a set of eigenvalues of the form:

is also known, then a linear operator possessing those eigenvalues can be constructed. The matrix representation for a linear operator with eigenvalues from :

The kets may also represent a continuous set of states. In this, Dirac also found it necessary to develop what is known as the Dirac Delta Function. Then, for a continuous set of complete kets indexed by the continuous variables and :