# Bra-ket notation

Bra-ket notation, also known as Dirac notation, is essentially the language of quantum mechanics. It was invented by the Enlish physicist Paul Dirac and is named after him.[1] 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 $\mathcal{H}$ which, by definition, has an inner product, typically denoted by $\langle , \rangle$. In bra-ket notation, the symbol $\left|\psi\right\rangle$ is used to represent an element of the Hilbert Space in question. This vector is called a "ket". However, by Riesz Representation Theorem, each element ψ of the Hilbert space also uniquely defines a linear functional which resides in the dual space in terms of the inner product, as follows:

$f_\psi\left(x\right) = \left\langle x,\psi\right\rangle, x\in\mathcal{H}$

In the Dirac notation, the functional defined by ψ is instead represented by:

$\left\langle\psi\right| \dot= f_\psi\left(x\right)$

And in this case, $\left\langle\psi\right|$ is called a "bra". When the bra is written next to a ket $\left|\varphi\right\rangle$, we understand the pair to form a "bracket" giving us an inner product:

$\left\langle\varphi|\psi\right\rangle = \left\langle\psi,\varphi\right\rangle$

The notation is deceptively simple. The elegant nature of the Dirac Bra-ket 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.

## Orthogonality 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:

$\left\langle\psi|\psi\right\rangle = 1$

While, by orthogonality, if $\psi\neq\phi$:

$\left\langle\psi|\phi\right\rangle = 0$

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

$\left\langle i|j\right\rangle = \delta_{i,j}$

Where δi,j 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:

$\beta = \left\lbrace\left|1\right\rangle, \left|2\right\rangle,\ldots,\left|n\right\rangle,\ldots\right\rbrace$

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

$Q_{i,j} = \left\langle i\right|Q\left|j\right\rangle$

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

$S=\left\lbrace\lambda_1,\lambda_2,\ldots,\lambda_n,\ldots\right\rbrace$

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

$Q=\sum_{n}\lambda_n\left|n\right\rangle\left\langle n\right|$

The kets may also represent a continuous set of states. In such circumstances (which would include, for example, a free particle), Dirac also found it necessary to develop what is known as the Dirac delta function as an analogue to the Kronecker delta function. For a continuous set of complete kets indexed by the continuous variables $x^\prime$ and $x^{\prime\prime}$:

$\left\langle x^\prime|x^{\prime\prime}\right\rangle = \delta\left(x^\prime-x^{\prime\prime}\right)$

This relation should be thought of as a strictly canonical for use in integration, since writing the Dirac Delta Function in the absence of an integral is somewhat dubious. However, this does allow us to consider wavefunctions in terms of the Dirac Notation. Referring to the article on the Dirac Delta Function, we see that:

$\left\langle x|\psi\right\rangle = \int\delta\left(x^\prime-x\right)\psi\left(x^\prime\right)dx^\prime$

And thus that the wavefunction in position space for a state $\left|\psi\right\rangle$ can be written as:

$\psi\left(x\right)=\left\langle x|\psi\right\rangle$

## References

1. http://www.quantiki.org/wiki/index.php/Bra-ket_notation