Editing Bra–ket notation (section)

===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>