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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 the English physicist Paul Dirac and is 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 \cdot,\cdot \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>

<math>\left\langle\varphi|\psi\right\rangle = \left\langle\psi,\varphi\right\rangle</math>

<math>\left\langle\varphi|\psi\right\rangle = \left\langle\psi,\varphi\right\rangle</math>

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.

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.

== Othogonality of Bras and Kets ==

== 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:

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 oftentimes 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:

<math>\left\langle\psi|\psi\right\rangle = 1</math>

<math>\left\langle\psi|\psi\right\rangle = 1</math>

<math>Q=\sum_{n}\lambda_n\left|n\right\rangle\left\langle n\right|</math>

<math>Q=\sum_{n}\lambda_n\left|n\right\rangle\left\langle n\right|</math>

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 <math>x^\prime</math> and <math>x^{\prime\prime}</math>:

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|Kronecker delta]] function.  For a continuous set of complete kets indexed by the continuous variables <math>x^\prime</math> and <math>x^{\prime\prime}</math>:

<math>\left\langle x^\prime|x^{\prime\prime}\right\rangle = \delta\left(x^\prime-x^{\prime\prime}\right)</math>

<math>\left\langle x^\prime|x^{\prime\prime}\right\rangle = \delta\left(x^\prime-x^{\prime\prime}\right)</math>

==References==

==References==

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[[category:physics]]

 

[[Category:Quantum Mechanics]]