Editing Bra–ket notation (section)

==Properties==
Bra–ket notation was designed to facilitate the formal manipulation of linear-algebraic expressions. Some of the properties that allow this manipulation are listed herein. In what follows, {{math|''c''<sub>1</sub>}} and {{math|''c''<sub>2</sub>}} denote arbitrary [[complex number]]s, {{math|''c''*}} denotes the [[complex conjugate]] of {{math|''c''}}, {{math|''A''}} and {{math|''B''}} denote arbitrary linear operators, and these properties are to hold for any choice of bras and kets.

===Linearity===
* Since bras are linear functionals, <math display="block">\langle\phi| \bigl( c_1|\psi_1\rangle + c_2|\psi_2\rangle \bigr) = c_1\langle\phi|\psi_1\rangle + c_2\langle\phi|\psi_2\rangle \,. </math>
* By the definition of addition and scalar multiplication of linear functionals in the dual space,<ref>[http://bohr.physics.berkeley.edu/classes/221/1112/notes/hilbert.pdf Lecture notes by Robert Littlejohn] {{Webarchive|url=https://web.archive.org/web/20120617144946/http://bohr.physics.berkeley.edu/classes/221/1112/notes/hilbert.pdf |date=2012-06-17 }}, eqns 12 and 13</ref> <math display="block">\bigl(c_1 \langle\phi_1| + c_2 \langle\phi_2|\bigr) |\psi\rangle = c_1 \langle\phi_1|\psi\rangle + c_2 \langle\phi_2|\psi\rangle \,. </math>

===Associativity===
Given any expression involving complex numbers, bras, kets, inner products, outer products, and/or linear operators (but not addition), written in bra–ket notation, the parenthetical groupings do not matter (i.e., the [[associative property]] holds). For example:
:<math>\begin{align}
\lang \psi| \bigl(A |\phi\rang\bigr) = \bigl(\lang \psi|A\bigr)|\phi\rang \, &\stackrel{\text{def}}{=} \, \lang \psi | A | \phi \rang \\
\bigl(A|\psi\rang\bigr)\lang \phi| = A\bigl(|\psi\rang \lang \phi|\bigr) \, &\stackrel{\text{def}}{=} \, A | \psi \rang \lang \phi |
\end{align}</math>
and so forth. The expressions on the right (with no parentheses whatsoever) are allowed to be written unambiguously ''because'' of the equalities on the left. Note that the associative property does ''not'' hold for expressions that include nonlinear operators, such as the [[antilinear]] [[T-symmetry|time reversal operator]] in physics.

===Hermitian conjugation===
Bra–ket notation makes it particularly easy to compute the Hermitian conjugate (also called ''dagger'', and denoted {{math|†}}) of expressions. The formal rules are:
* The Hermitian conjugate of a bra is the corresponding ket, and vice versa.
* The Hermitian conjugate of a complex number is its complex conjugate.
* The Hermitian conjugate of the Hermitian conjugate of anything (linear operators, bras, kets, numbers) is itself—i.e., <math display="block">\left(x^\dagger\right)^\dagger=x \,.</math>
* Given any combination of complex numbers, bras, kets, inner products, outer products, and/or linear operators, written in bra–ket notation, its Hermitian conjugate can be computed by reversing the order of the components, and taking the Hermitian conjugate of each.

These rules are sufficient to formally write the Hermitian conjugate of any such expression; some examples are as follows:
* Kets: <math display="block">
\bigl(c_1|\psi_1\rangle + c_2|\psi_2\rangle\bigr)^\dagger = c_1^* \langle\psi_1| + c_2^* \langle\psi_2| \,.
</math>
* Inner products: <math display="block">\langle \phi | \psi \rangle^* = \langle \psi|\phi\rangle \,.</math> Note that {{math|{{bra-ket|''φ''|''ψ''}}}} is a scalar, so the Hermitian conjugate is just the complex conjugate, i.e., <math display="block">\bigl(\langle \phi | \psi \rangle\bigr)^\dagger = \langle \phi | \psi \rangle^*</math>
* Matrix elements: <math display="block">\begin{align}
\langle \phi| A | \psi \rangle^\dagger &= \left\langle \psi \left| A^\dagger \right|\phi \right\rangle \\
\left\langle \phi\left| A^\dagger B^\dagger \right| \psi \right\rangle^\dagger &= \langle \psi | BA |\phi \rangle \,.
\end{align}</math>
* Outer products: <math display="block">\Big(\bigl(c_1|\phi_1\rangle\langle \psi_1|\bigr) + \bigl(c_2|\phi_2\rangle\langle\psi_2|\bigr)\Big)^\dagger = \bigl(c_1^* |\psi_1\rangle\langle \phi_1|\bigr) + \bigl(c_2^*|\psi_2\rangle\langle\phi_2|\bigr) \,.</math>