Editing Bra–ket notation (section)

==Vector spaces==

===Vectors vs kets===

In mathematics, the term "vector" is used for an element of any vector space. In physics, however, the term "vector" tends to refer almost exclusively to quantities like [[displacement (vector)|displacement]] or [[velocity]], which have components that relate directly to the three dimensions of [[space]], or relativistically, to the four of [[spacetime]]. Such vectors are typically denoted with over arrows (<math>\vec r</math>), boldface (<math>\mathbf{p}</math>) or indices (<math>v^\mu</math>).

In quantum mechanics, a quantum state is typically represented as an element of a complex [[Hilbert space]], for example, the infinite-dimensional vector space of all possible [[wavefunction]]s (square integrable functions mapping each point of 3D space to a complex number) or some more abstract Hilbert space constructed more algebraically. To distinguish this type of vector from those described above, it is common and useful in physics to denote an element <math>\phi</math> of an abstract complex vector space as a ket <math>|\phi\rangle</math>, to refer to it as a "ket" rather than as a vector, and to pronounce it "ket-<math>\phi</math>" or "ket-A" for {{math|{{ket|''A''}}}}.

Symbols, letters, numbers, or even words—whatever serves as a convenient label—can be used as the label inside a ket, with the <math>|\ \rangle</math> making clear that the label indicates a vector in vector space. In other words, the symbol "{{math|{{ket|''A''}}}}" has a recognizable mathematical meaning as to the kind of variable being represented, while just the "{{math|''A''}}" by itself does not. For example, {{math|{{ket|1}} + {{ket|2}}}} is not necessarily equal to {{math|{{ket|3}}}}. Nevertheless, for convenience, there is usually some logical scheme behind the labels inside kets, such as the common practice of labeling [[stationary state|energy eigenkets]] in quantum mechanics through a listing of their [[quantum number]]s. At its simplest, the label inside the ket is the eigenvalue of a physical operator, such as <math>\hat x</math>, <math>\hat p</math>, <math>\hat L_z</math>, etc.

===Notation===

Since kets are just vectors in a Hermitian vector space, they can be manipulated using the usual rules of linear algebra. For example:

:<math>\begin{align}
|A \rangle &= |B\rangle + |C\rangle \\
|C \rangle &= (-1+2i)|D \rangle \\
|D \rangle &= \int_{-\infty}^{\infty} e^{-x^2} |x\rangle \, \mathrm{d}x \,.
\end{align}</math>

Note how the last line above involves infinitely many different kets, one for each real number {{math|''x''}}.

Since the ket is an element of a vector space, a '''bra''' <math>\langle A|</math> is an element of its [[dual space]], i.e. a bra is a linear functional which is a linear map from the vector space to the complex numbers. Thus, it is useful to think of kets and bras as being elements of different vector spaces (see below however) with both being different useful concepts.

A bra <math>\langle\phi|</math> and a ket <math> |\psi\rangle</math> (i.e. a functional and a vector), can be combined to an operator <math>|\psi\rangle\langle\phi|</math> of rank one with [[outer product]]

:<math>|\psi\rangle\langle\phi| \colon |\xi\rangle \mapsto |\psi\rangle\langle\phi|\xi\rangle ~.</math>

===Inner product and bra–ket identification on Hilbert space===
{{main|Inner product}}

The bra–ket notation is particularly useful in Hilbert spaces which have an inner product<ref>[https://www.youtube.com/watch?v=VtBRKw1Ab7E&t=2834 Lecture 2 | Quantum Entanglements, Part 1 (Stanford)], Leonard Susskind on inner product, 2006-10-02.</ref> that allows [[Hermitian conjugation]] and identifying a vector with a continuous linear functional, i.e. a ket with a bra, and vice versa (see [[Riesz representation theorem]]). The inner product on Hilbert space <math>(\ , \ )</math> (with the first argument anti linear as preferred by physicists) is fully equivalent to an (anti-linear) identification between the space of kets and that of bras in the bra–ket notation: for a vector ket <math>\psi = |\psi\rangle </math> define a functional (i.e. bra) <math>f_\phi = \langle\phi|</math> by

:<math>(\phi,\psi) = (|\phi\rangle, |\psi\rangle) =: f_\phi(\psi) = \langle\phi| \, \bigl(|\psi\rangle\bigr) =: \langle\phi{\mid}\psi\rangle </math>

====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|''+''}}}}".

=== Non-normalizable states and non-Hilbert spaces ===

Bra–ket notation can be used even if the vector space is not a [[Hilbert space]].

In quantum mechanics, it is common practice to write down kets which have infinite [[norm (mathematics)|norm]], i.e. non-[[normalizable wavefunction]]s. Examples include states whose wavefunctions are [[Dirac delta function]]s or infinite [[plane wave]]s. These do not, technically, belong to the [[Hilbert space]] itself. However, the definition of "Hilbert space" can be broadened to accommodate these states (see the [[Gelfand–Naimark–Segal construction]] or [[rigged Hilbert space]]s). The bra–ket notation continues to work in an analogous way in this more general context.

[[Banach spaces]] are a different generalization of Hilbert spaces. In a Banach space {{math|{{mathcal|B}}}}, the vectors may be notated by kets and the continuous [[linear functionals]] by bras. Over any vector space without a given [[topology]], we may still notate the vectors by kets and the linear functionals by bras. In these more general contexts, the bracket does not have the meaning of an inner product, because the [[Riesz representation theorem]] does not apply.