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'''Bra-ket notation''', also known as '''Dirac notation''', is essentially the language of quantum mechanics. It was invented by a man named Paul Dirac and originally named after him.<ref>http://www.quantiki.org/wiki/index.php/Bra-ket_notation</ref> 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.

'''Bra-ket notation''', also known as '''Dirac notation''', is essentially the language of quantum mechanics. It was invented by a man named Paul Dirac and originally named after him.<ref>http://www.quantiki.org/wiki/index.php/Bra-ket_notation</ref> 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]] <math>\mathcal{H}</math> which, by definition, has an [[inner product]], typically denoted by <math>\langle , \rangle</math>.  In bra-ket notation, the symbol <math>\left|\psi\right\rangle</math> is used to represent an element of the Hilbert Space in question.  This vector is called a "ket".  However, by [[Reisz' Representation Theorem]], each element <math>\psi</math> 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 non-relativistic quantum mechanics, states are said to reside in a [[Hilbert Space]] <math>\mathcal{H}</math> which, by definition, has an [[inner product]], typically denoted by <math>\langle , \rangle</math>.  In bra-ket notation, the symbol <math>\left|\psi\right\rangle</math> 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 <math>\psi</math> of the Hilbert space also uniquely defines a [[linear functional]] which resides in the [[dual space]] in terms of the inner product, as follows:

<math>f_\psi\left(x\right) = \left\langle x,\psi\right\rangle, x\in\mathcal{H}</math>

<math>f_\psi\left(x\right) = \left\langle x,\psi\right\rangle, x\in\mathcal{H}</math>