Editing Bra–ket notation (section)

==Quantum mechanics==
In [[quantum mechanics]] and [[quantum computing]], bra–ket notation is used ubiquitously to denote [[quantum state]]s. The notation uses [[angle bracket]]s, {{char|<math>\langle</math>}} and {{char|<math>\rangle</math>}}, and a [[vertical bar]] {{char|<math>|</math>}}, to construct "bras" and "kets".

A '''ket''' is of the form <math>|v \rangle</math>. Mathematically it denotes a [[vector space|vector]], <math>\boldsymbol v</math>, in an abstract (complex) [[vector space]] <math>V</math>, and physically it represents a state of some quantum system.

A '''bra''' is of the form <math>\langle f|</math>. Mathematically it denotes a [[linear form]] <math>f:V \to \Complex</math>, i.e. a [[linear map]] that maps each vector in <math>V</math> to a number in the complex plane <math>\Complex</math>. Letting the linear functional <math> \langle f|</math> act on a vector <math>|v\rangle</math> is written as <math>\langle f | v\rangle \in \Complex</math>.

Assume that on <math>V</math> there exists an inner product <math>(\cdot,\cdot)</math> with [[Antilinear map|antilinear]] first argument, which makes <math>V</math> an [[inner product space]]. Then with this inner product each vector <math>\boldsymbol \phi \equiv |\phi\rangle</math> can be identified with a corresponding linear form, by placing the vector in the anti-linear first slot of the inner product: <math>(\boldsymbol\phi,\cdot) \equiv \langle\phi|</math>. The correspondence between these notations is then <math>(\boldsymbol\phi, \boldsymbol\psi) \equiv \langle\phi|\psi\rangle</math>. The [[linear form]] <math>\langle\phi|</math> is a [[covector]] to <math>|\phi\rangle</math>, and the set of all covectors forms a subspace of the [[dual vector space]] <math> V^\vee</math>, to the initial vector space <math>V</math>. The purpose of this linear form <math>\langle\phi|</math> can now be understood in terms of making projections onto the state <math>\boldsymbol \phi,</math> to find how linearly dependent two states are, etc.

For the vector space <math>\Complex^n</math>, kets can be identified with column vectors, and bras with row vectors. Combinations of bras, kets, and linear operators are interpreted using [[matrix multiplication]]. If <math>\Complex^n</math> has the standard Hermitian inner product <math>(\boldsymbol v, \boldsymbol w) = v^\dagger w</math>, under this identification, the identification of kets and bras and vice versa provided by the inner product is taking the [[Hermitian conjugate]] (denoted <math> \dagger</math>).

It is common to suppress the vector or linear form from the bra–ket notation and only use a label inside the typography for the bra or ket. For example, the spin operator <math>\hat \sigma_z</math> on a two-dimensional space <math>\Delta</math> of [[spinor]]s has [[eigenvalue]]s <math display="inline">\pm \frac{1}{2}</math> with eigenspinors <math>\boldsymbol \psi_+,\boldsymbol \psi_- \in \Delta</math>. In bra–ket notation, this is typically denoted as <math>\boldsymbol \psi_+ = |+\rangle</math>, and <math>\boldsymbol \psi_- = |-\rangle</math>. As above, kets and bras with the same label are interpreted as kets and bras corresponding to each other using the inner product. In particular, when also identified with row and column vectors, kets and bras with the same label are identified with [[Hermitian conjugate]] column and row vectors.

Bra–ket notation was effectively established in 1939 by [[Paul Dirac]];<ref name=Dirac /><ref>{{harvnb|Shankar|1994|loc=Chapter 1}}</ref> it is thus also known as Dirac notation, despite the notation having a precursor in [[Hermann Grassmann]]'s use of <math>[\phi{\mid}\psi]</math> for inner products nearly 100 years earlier.<ref name="Grassmann">{{harvnb|Grassmann|1862}}</ref><ref>[https://www.youtube.com/watch?v=VtBRKw1Ab7E&t=2561 Lecture 2 | Quantum Entanglements, Part 1 (Stanford)], Leonard Susskind on complex numbers, complex conjugate, bra, ket. 2006-10-02.</ref>