Editing Bra–ket notation (section)

===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''}}}}.