Editing Bra–ket notation (section)

==Linear operators==
{{see also|Linear operator}}

===Linear operators acting on kets===

A linear operator is a map that inputs a ket and outputs a ket. (In order to be called "linear", it is required to have [[linear operator|certain properties]].) In other words, if <math>\hat A</math> is a linear operator and <math>|\psi\rangle</math> is a ket-vector, then <math>\hat A |\psi\rangle</math> is another ket-vector.

In an <math>N</math>-dimensional Hilbert space, we can impose a basis on the space and represent <math>|\psi\rangle</math> in terms of its coordinates as a <math>N \times 1</math> [[column vector]]. Using the same basis for <math>\hat A</math>, it is represented by an <math>N \times N</math> complex matrix. The ket-vector <math>\hat A |\psi\rangle</math> can now be computed by matrix multiplication.

Linear operators are ubiquitous in the theory of quantum mechanics. For example, observable physical quantities are represented by [[self-adjoint operator]]s, such as [[energy]] or [[momentum]], whereas transformative processes are represented by [[unitary operator|unitary]] linear operators such as rotation or the progression of time.

===Linear operators acting on bras===

Operators can also be viewed as acting on bras ''from the right hand side''. Specifically, if {{math|'''''A'''''}} is a linear operator and {{math|{{bra|''φ''}}}} is a bra, then {{math|{{bra|''φ''}}'''''A'''''}} is another bra defined by the rule
<math display="block">\bigl(\langle\phi|\boldsymbol{A}\bigr) |\psi\rangle = \langle\phi| \bigl(\boldsymbol{A}|\psi\rangle\bigr) \,,</math>
(in other words, a [[function composition]]). This expression is commonly written as (cf. [[energy inner product]])
<math display="block">\langle\phi| \boldsymbol{A} |\psi\rangle \,.</math>

In an {{math|''N''}}-dimensional Hilbert space, {{math|{{bra|''φ''}}}} can be written as a {{math|1 × ''N''}} [[row vector]], and {{math|'''''A'''''}} (as in the previous section) is an {{math|''N'' × ''N''}} matrix. Then the bra {{math|{{bra|''φ''}}'''''A'''''}} can be computed by normal matrix multiplication.

If the same state vector appears on both bra and ket side,
<math display="block">\langle\psi|\boldsymbol{A}|\psi\rangle \,,</math>
then this expression gives the [[expectation value (quantum mechanics)|expectation value]], or mean or average value, of the observable represented by operator {{math|'''''A'''''}} for the physical system in the state {{math|{{ket|''ψ''}}}}.

===Outer products===

A convenient way to define linear operators on a Hilbert space {{math|{{mathcal|H}}}} is given by the [[outer product]]: if {{math|{{bra|''ϕ''}}}} is a bra and {{math|{{ket|''ψ''}}}} is a ket, the outer product
<math display="block"> |\phi\rang \, \lang \psi| </math>
denotes the [[finite-rank operator|rank-one operator]] with the rule 
<math display="block"> \bigl(|\phi\rang \lang \psi|\bigr)(x) = \lang \psi | x \rang |\phi \rang.</math>

For a finite-dimensional vector space, the outer product can be understood as simple matrix multiplication:
<math display="block"> |\phi \rangle \, \langle \psi | \doteq 
\begin{pmatrix} \phi_1 \\ \phi_2 \\ \vdots \\ \phi_N \end{pmatrix}
\begin{pmatrix} \psi_1^* & \psi_2^* & \cdots & \psi_N^* \end{pmatrix}
= \begin{pmatrix}
\phi_1 \psi_1^* & \phi_1 \psi_2^* & \cdots & \phi_1 \psi_N^* \\
\phi_2 \psi_1^* & \phi_2 \psi_2^* & \cdots & \phi_2 \psi_N^* \\
\vdots & \vdots & \ddots & \vdots \\
\phi_N \psi_1^* & \phi_N \psi_2^* & \cdots & \phi_N \psi_N^* \end{pmatrix}
</math>
The outer product is an {{math|''N'' × ''N''}} matrix, as expected for a linear operator.

One of the uses of the outer product is to construct [[projection operator]]s. Given a ket {{math|{{ket|''ψ''}}}} of norm 1, the orthogonal projection onto the [[Linear subspace|subspace]] spanned by {{math|{{ket|''ψ''}}}} is
<math display="block">|\psi\rangle \, \langle\psi| \,.</math>
This is an [[idempotent]] in the algebra of observables that acts on the Hilbert space.

===Hermitian conjugate operator===
{{main|Hermitian conjugate}}
Just as kets and bras can be transformed into each other (making {{math|{{ket|''ψ''}}}} into {{math|{{bra|''ψ''}}}}), the element from the dual space corresponding to {{math|''A''{{ket|''ψ''}}}} is {{math|{{bra|''ψ''}}''A''<sup>†</sup>}}, where {{math|''A''<sup>†</sup>}} denotes the Hermitian conjugate (or adjoint) of the operator {{math|''A''}}. In other words,
<math display="block"> |\phi\rangle = A |\psi\rangle \quad \text{if and only if} \quad \langle\phi| = \langle \psi | A^\dagger \,.</math>

If {{math|''A''}} is expressed as an {{math|''N'' × ''N''}} matrix, then {{math|''A''<sup>†</sup>}} is its conjugate transpose.