Changes

'''Bra-ket 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 \cdot,\cdot \rangle</math>.  In the Dirac Notation, we use the symbol <math>\left|\psi\right\rangle</math> 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 \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:

<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>

In the Dirac Notation, the functional defined by <math>\psi</math> is instead represented by:

In the Dirac notation, the functional defined by <math>\psi</math> is instead represented by:

<math>\left\langle\psi\right| \dot= f_\psi\left(x\right)</math>

<math>\left\langle\psi\right| \dot= f_\psi\left(x\right)</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 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.

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.

<math>\psi\left(x\right)=\left\langle x|\psi\right\rangle</math>

<math>\psi\left(x\right)=\left\langle x|\psi\right\rangle</math>

==References==

==References==

<references/>

<references/>