Editing Bra–ket notation (section)

====Bras and kets as row and column vectors====

In the simple case where we consider the vector space <math>\Complex^n</math>, a ket can be identified with a [[column vector]], and a bra as a [[row vector]]. If, moreover, we use the standard Hermitian inner product on <math>\Complex^n</math>, the bra corresponding to a ket, in particular a bra {{math|{{bra|''m''}}}} and a ket {{math|{{ket|''m''}}}} with the same label are [[conjugate transpose]]. Moreover, conventions are set up in such a way that writing bras, kets, and linear operators next to each other simply imply [[matrix multiplication]].<ref name="bra–ket Notation Trivializes Matrix Multiplication">{{cite web |url=http://algassert.com/post/1629 |title=Gidney, Craig (2017). Bra–Ket Notation Trivializes Matrix Multiplication}}</ref> In particular the outer product <math>|\psi\rangle\langle\phi| </math> of a column and a row vector ket and bra can be identified with matrix multiplication (column vector times row vector equals matrix).

For a finite-dimensional vector space, using a fixed [[orthonormal basis]], the inner product can be written as a matrix multiplication of a row vector with a column vector:
<math display="block"> \langle A | B \rangle \doteq A_1^* B_1 + A_2^* B_2 + \cdots + A_N^* B_N =
\begin{pmatrix} A_1^* & A_2^* & \cdots & A_N^* \end{pmatrix}
\begin{pmatrix} B_1 \\ B_2 \\ \vdots \\ B_N \end{pmatrix}</math>
Based on this, the bras and kets can be defined as:
<math display="block">\begin{align}
\langle A | &\doteq \begin{pmatrix} A_1^* & A_2^* & \cdots & A_N^* \end{pmatrix} \\
| B \rangle &\doteq \begin{pmatrix} B_1 \\ B_2 \\ \vdots \\ B_N \end{pmatrix}
\end{align}</math>
and then it is understood that a bra next to a ket implies matrix multiplication.

The [[conjugate transpose]] (also called ''Hermitian conjugate'') of a bra is the corresponding ket and vice versa:
<math display="block">\langle A |^\dagger = |A \rangle, \quad |A \rangle^\dagger = \langle A |</math>
because if one starts with the bra
<math display="block">\begin{pmatrix} A_1^* & A_2^* & \cdots & A_N^* \end{pmatrix} \,,</math>
then performs a [[complex conjugation]], and then a [[matrix transpose]], one ends up with the ket
<math display="block">\begin{pmatrix} A_1 \\ A_2 \\ \vdots \\ A_N \end{pmatrix}</math>

Writing elements of a finite dimensional (or [[mutatis mutandis]], countably infinite) vector space as a column vector of numbers requires picking a [[basis (linear algebra)|basis]]. Picking a basis is not always helpful because quantum mechanics calculations involve frequently switching between different bases (e.g. position basis, momentum basis, energy eigenbasis), and one can write something like "{{math|{{ket|''m''}}}}" without committing to any particular basis. In situations involving two different important basis vectors, the basis vectors can be taken in the notation explicitly and here will be referred simply as "{{math|{{ket|''−''}}}}" and "{{math|{{ket|''+''}}}}".