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{{short description|Notation for quantum states}}
{{Cleanup reorganize|date=February 2023}}
{{Quantum mechanics|cTopic=Background}}

'''Bra–ket notation''', also called '''Dirac notation''', is a notation for [[linear algebra]] and [[linear operator]]s on [[complex vector space]]s together with their [[dual space]] both in the finite-dimensional and infinite-dimensional case. It is specifically designed to ease the types of calculations that frequently come up in [[quantum mechanics]]. Its use in quantum mechanics is quite widespread.

Bra–ket notation was created by [[Paul Dirac]] in his 1939 publication ''A New Notation for Quantum Mechanics''. The notation was introduced as an easier way to write quantum mechanical expressions.<ref name=Dirac>{{harvnb|Dirac|1939}}</ref> The name comes from the English word "bracket".

==Quantum mechanics==
In [[quantum mechanics]] and [[quantum computing]], bra–ket notation is used ubiquitously to denote [[quantum state]]s. The notation uses [[angle bracket]]s, {{char|<math>\langle</math>}} and {{char|<math>\rangle</math>}}, and a [[vertical bar]] {{char|<math>|</math>}}, to construct "bras" and "kets".

A '''ket''' is of the form <math>|v \rangle</math>. Mathematically it denotes a [[vector space|vector]], <math>\boldsymbol v</math>, in an abstract (complex) [[vector space]] <math>V</math>, and physically it represents a state of some quantum system.

A '''bra''' is of the form <math>\langle f|</math>. Mathematically it denotes a [[linear form]] <math>f:V \to \Complex</math>, i.e. a [[linear map]] that maps each vector in <math>V</math> to a number in the complex plane <math>\Complex</math>. Letting the linear functional <math> \langle f|</math> act on a vector <math>|v\rangle</math> is written as <math>\langle f | v\rangle \in \Complex</math>.

Assume that on <math>V</math> there exists an inner product <math>(\cdot,\cdot)</math> with [[Antilinear map|antilinear]] first argument, which makes <math>V</math> an [[inner product space]]. Then with this inner product each vector <math>\boldsymbol \phi \equiv |\phi\rangle</math> can be identified with a corresponding linear form, by placing the vector in the anti-linear first slot of the inner product: <math>(\boldsymbol\phi,\cdot) \equiv \langle\phi|</math>. The correspondence between these notations is then <math>(\boldsymbol\phi, \boldsymbol\psi) \equiv \langle\phi|\psi\rangle</math>. The [[linear form]] <math>\langle\phi|</math> is a [[covector]] to <math>|\phi\rangle</math>, and the set of all covectors forms a subspace of the [[dual vector space]] <math> V^\vee</math>, to the initial vector space <math>V</math>. The purpose of this linear form <math>\langle\phi|</math> can now be understood in terms of making projections onto the state <math>\boldsymbol \phi,</math> to find how linearly dependent two states are, etc.

For the vector space <math>\Complex^n</math>, kets can be identified with column vectors, and bras with row vectors. Combinations of bras, kets, and linear operators are interpreted using [[matrix multiplication]]. If <math>\Complex^n</math> has the standard Hermitian inner product <math>(\boldsymbol v, \boldsymbol w) = v^\dagger w</math>, under this identification, the identification of kets and bras and vice versa provided by the inner product is taking the [[Hermitian conjugate]] (denoted <math> \dagger</math>).

It is common to suppress the vector or linear form from the bra–ket notation and only use a label inside the typography for the bra or ket. For example, the spin operator <math>\hat \sigma_z</math> on a two-dimensional space <math>\Delta</math> of [[spinor]]s has [[eigenvalue]]s <math display="inline">\pm \frac{1}{2}</math> with eigenspinors <math>\boldsymbol \psi_+,\boldsymbol \psi_- \in \Delta</math>. In bra–ket notation, this is typically denoted as <math>\boldsymbol \psi_+ = |+\rangle</math>, and <math>\boldsymbol \psi_- = |-\rangle</math>. As above, kets and bras with the same label are interpreted as kets and bras corresponding to each other using the inner product. In particular, when also identified with row and column vectors, kets and bras with the same label are identified with [[Hermitian conjugate]] column and row vectors.

Bra–ket notation was effectively established in 1939 by [[Paul Dirac]];<ref name=Dirac /><ref>{{harvnb|Shankar|1994|loc=Chapter 1}}</ref> it is thus also known as Dirac notation, despite the notation having a precursor in [[Hermann Grassmann]]'s use of <math>[\phi{\mid}\psi]</math> for inner products nearly 100 years earlier.<ref name="Grassmann">{{harvnb|Grassmann|1862}}</ref><ref>[https://www.youtube.com/watch?v=VtBRKw1Ab7E&t=2561 Lecture 2 | Quantum Entanglements, Part 1 (Stanford)], Leonard Susskind on complex numbers, complex conjugate, bra, ket. 2006-10-02.</ref>

==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.

==Usage in quantum mechanics==

The mathematical structure of quantum mechanics is based in large part on [[linear algebra]]:
*[[Wave function]]s and other quantum states can be represented as vectors in a [[Separable space|separable]] [[complex number|complex]] [[Hilbert space]]. (The exact structure of this Hilbert space depends on the situation.) In bra–ket notation, for example, an electron might be in the "state" {{math|{{ket|''ψ''}}}}. (Technically, the quantum states are ''[[Ray (quantum theory)|ray]]s'' of vectors in the Hilbert space, as {{math|''c''{{ket|''ψ''}}}} corresponds to the same state for any nonzero complex number {{math|''c''}}.)
*[[Quantum superposition]]s can be described as vector sums of the constituent states. For example, an electron in the state {{math|{{sfrac|1|√2}}{{ket|1}} + {{sfrac|''i''|√2}}{{ket|2}}}} is in a quantum superposition of the states {{math|{{ket|1}}}} and {{math|{{ket|2}}}}.
*[[Measurement in quantum mechanics|Measurements]] are associated with linear operators (called [[observable]]s) on the Hilbert space of quantum states.
*Dynamics are also described by linear operators on the Hilbert space. For example, in the [[Schrödinger picture]], there is a linear [[time evolution]] operator {{math|''U''}} with the property that if an electron is in state {{math|{{ket|''ψ''}}}} right now, at a later time it will be in the state {{math|''U''{{ket|''ψ''}}}}, the same {{math|''U''}} for every possible {{math|{{ket|''ψ''}}}}.
*[[Normalizable wave function|Wave function normalization]] is scaling a wave function so that its [[norm (mathematics)|norm]] is 1.

Since virtually every calculation in quantum mechanics involves vectors and linear operators, it can involve, and often does involve, bra–ket notation. A few examples follow:

===Spinless position–space wave function===

<div class="skin-invert-image">
{{multiple image
 | left
 | footer    = Components of complex vectors plotted against index number; discrete {{math|''k''}} and continuous {{math|''x''}}. Two particular components out of infinitely many are highlighted.
 | width1    = 225
 | image1    = Discrete complex vector components.svg 
 | caption1  = Discrete components {{math|''A''<sub>''k''</sub>}} of a complex vector {{math|1={{ket|''A''}} = Σ<sub>''k''</sub> ''A''<sub>''k''</sub> {{ket|''e<sub>k</sub>''}}}}.
 | width2    = 230
 | image2    = Continuous complex vector components.svg
 | caption2  = Continuous components {{math|''ψ''(''x'')}} of a complex vector {{math|1={{ket|''ψ''}} = ∫ d''x'' ''ψ''(''x''){{ket|''x''}}}}.
}}
</div>

The Hilbert space of a [[Spin (physics)|spin]]-0 point particle  can be represented in terms of a "position [[basis (linear algebra)|basis]]" {{math|{ {{ket|'''r'''}} }<nowiki/>}}, where the label {{math|'''r'''}} extends over the set of all points in [[position space]]. These states satisfy the eigenvalue equation for the [[position operator]]:
<math display="block"> \hat{\mathbf{r}}|\mathbf{r}\rangle = \mathbf{r}|\mathbf{r}\rangle.</math>
The position states are "[[Dirac_delta_function#Quantum_mechanics|generalized eigenvectors]]", not elements of the Hilbert space itself, and do not form a countable orthonormal basis. However, as the Hilbert space is separable, it does admit a countable dense subset within the [[domain of definition]] of its wavefunctions. That is, starting from any ket {{math|{{ket|Ψ}}}} in this Hilbert space, one may ''define'' a complex scalar function of {{math|'''r'''}}, known as a [[wavefunction]],
<math display="block">\Psi(\mathbf{r}) \ \stackrel{\text{def}}{=}\ \lang \mathbf{r}|\Psi\rang \,.</math>

On the left-hand side, {{math|Ψ('''r''')}} is a function mapping any point in space to a complex number; on the right-hand side,
<math display="block">\left|\Psi\right\rangle = \int d^3\mathbf{r} \, \Psi(\mathbf{r}) \left|\mathbf{r}\right\rangle</math> 
is a ket consisting of a superposition of kets with relative coefficients specified by that function.

It is then customary to define linear operators acting on wavefunctions in terms of linear operators acting on kets, by
<math display="block">\hat A(\mathbf{r}) ~ \Psi(\mathbf{r}) \ \stackrel{\text{def}}{=}\ \lang \mathbf{r}|\hat A|\Psi\rang \,.</math>

For instance, the [[momentum]] operator <math>\hat \mathbf {p}</math> has the following coordinate representation,
<math display="block">\hat{\mathbf{p} } (\mathbf{r}) ~ \Psi(\mathbf{r}) \ \stackrel{\text{def}}{=}\ \lang \mathbf{r} |\hat \mathbf{p}|\Psi\rang = - i \hbar \nabla \Psi(\mathbf{r}) \,.</math>

One occasionally even encounters an expression such as <math >\nabla |\Psi\rang </math>, though this is something of an [[abuse of notation]]. The differential operator must be understood to be an abstract operator, acting on kets, that has the effect of differentiating wavefunctions once the expression is projected onto the position basis, <math>\nabla \lang\mathbf{r}|\Psi\rang \,,</math>
even though, in the momentum basis, this operator amounts to a mere multiplication operator (by {{math|''iħ'''''p'''}}). That is, to say,
<math display="block"> \langle \mathbf{r} |\hat \mathbf{p} = - i \hbar \nabla \langle \mathbf{r}| ~,</math>
or
<math display="block"> \hat \mathbf{p} = \int d^3 \mathbf{r} ~| \mathbf{r}\rangle ( - i \hbar \nabla) \langle \mathbf{r}| ~.</math>

===Overlap of states===
In quantum mechanics the expression {{math|{{bra-ket|''φ''|''ψ''}}}} is typically interpreted as the [[probability amplitude]] for the state {{math|''ψ''}} to [[wavefunction collapse|collapse]] into the state {{math|''φ''}}. Mathematically, this means the coefficient for the projection of {{math|''ψ''}} onto {{math|''φ''}}. It is also described as the projection of state {{math|''ψ''}} onto state {{math|''φ''}}.

===Changing basis for a spin-1/2 particle===
A stationary [[spin-1/2|spin-{{1/2}}]] particle has a two-dimensional Hilbert space. One [[orthonormal basis]] is:
<math display="block">|{\uparrow}_z \rangle \,, \; |{\downarrow}_z \rangle</math>
where {{math|{{ket|↑<sub>''z''</sub>}}}} is the state with a definite value of the [[angular momentum operator|spin operator {{math|''S<sub>z</sub>''}}]] equal to +{{1/2}} and {{math|{{ket|↓<sub>''z''</sub>}}}} is the state with a definite value of the [[angular momentum operator|spin operator {{math|''S<sub>z</sub>''}}]] equal to −{{1/2}}.

Since these are a basis, ''any'' quantum state of the particle can be expressed as a [[linear combination]] (i.e., [[quantum superposition]]) of these two states:
<math display="block">|\psi \rangle = a_{\psi} |{\uparrow}_z \rangle + b_{\psi} |{\downarrow}_z \rangle</math>
where {{math|''a<sub>ψ</sub>''}} and {{math|''b<sub>ψ</sub>''}} are complex numbers.

A ''different'' basis for the same Hilbert space is:
<math display="block">|{\uparrow}_x \rangle \,, \; |{\downarrow}_x \rangle</math>
defined in terms of {{math|''S<sub>x</sub>''}} rather than {{math|''S<sub>z</sub>''}}.

Again, ''any'' state of the particle can be expressed as a linear combination of these two:
<math display="block">|\psi \rangle = c_{\psi} |{\uparrow}_x \rangle + d_{\psi} |{\downarrow}_x \rangle</math>

In vector form, you might write
<math display="block">|\psi\rangle \doteq \begin{pmatrix} a_\psi \\ b_\psi \end{pmatrix} \quad \text{or} \quad |\psi\rangle \doteq \begin{pmatrix} c_\psi \\ d_\psi \end{pmatrix} </math>
depending on which basis you are using. In other words, the "coordinates" of a vector depend on the basis used.

There is a mathematical relationship between <math>a_\psi</math>, <math>b_\psi</math>, <math>c_\psi</math> and <math>d_\psi</math>; see [[change of basis]].

==Pitfalls and ambiguous uses==

There are some conventions and uses of notation that may be confusing or ambiguous for the non-initiated or early student.

===Separation of inner product and vectors===
A cause for confusion is that the notation does not separate the inner-product operation from the notation for a (bra) vector. If a (dual space) bra-vector is constructed as a linear combination of other bra-vectors (for instance when expressing it in some basis) the notation creates some ambiguity and hides mathematical details. We can compare bra–ket notation to using bold for vectors, such as <math>\boldsymbol \psi</math>, and <math>(\cdot,\cdot)</math> for the inner product. Consider the following dual space bra-vector in the basis <math>\{|e_n\rangle\}</math>, where <math>\{\psi_n\}</math> are the complex number coefficients of <math>\langle \psi | </math>:
<math display="block">\langle\psi| = \sum_n \langle e_n| \psi_n</math>

It has to be determined by convention if the complex numbers <math>\{\psi_n\}</math> are inside or outside of the inner product, and each convention gives different results.

<math display="block">\langle\psi| \equiv (\boldsymbol\psi, \cdot ) = \sum_n (\boldsymbol e_n, \cdot ) \, \psi_n</math>
<math display="block">\langle\psi| \equiv (\boldsymbol\psi, \cdot ) = \sum_n (\boldsymbol e_n \psi_n, \cdot ) = \sum_n (\boldsymbol e_n, \cdot ) \, \psi_n^*</math>

===Reuse of symbols===
It is common to use the same symbol for ''labels'' and ''constants''. For example, <math>\hat \alpha |\alpha\rangle = \alpha |\alpha \rangle</math>, where the symbol <math>\alpha</math> is used simultaneously as the ''name of the operator'' <math>\hat \alpha</math>, its ''eigenvector'' <math>|\alpha\rangle</math> and the associated ''eigenvalue'' <math>\alpha</math>. Sometimes the ''hat'' is also dropped for operators, and one can see notation such as <math>A |a\rangle = a |a \rangle</math>.<ref>{{harvnb|Sakurai|Napolitano|2021}} Sec 1.2, 1.3</ref>

===Hermitian conjugate of kets===
It is common to see the usage <math>|\psi\rangle^\dagger = \langle\psi|</math>, where the dagger (<math>\dagger</math>) corresponds to the Hermitian conjugate. This is however not correct in a technical sense, since the ket, <math>|\psi\rangle</math>, represents a [[vector (mathematics and physics)|vector]] in a complex Hilbert-space <math>\mathcal{H}</math>, and the bra, <math>\langle\psi|</math>, is a [[linear functional]] on vectors in <math>\mathcal{H}</math>. In other words, <math>|\psi\rangle</math> is just a vector, while <math>\langle\psi|</math> is the combination of a vector and an inner product.

===Operations inside bras and kets===
This is done for a fast notation of scaling vectors. For instance, if the vector <math>|\alpha \rangle</math> is scaled by <math>1/\sqrt{2}</math>, it may be denoted <math>|\alpha/\sqrt{2} \rangle</math>. This can be ambiguous since <math>\alpha</math> is simply a label for a state, and not a mathematical object on which operations can be performed. This usage is more common when denoting vectors as tensor products, where part of the labels are moved '''outside''' the designed slot, e.g. <math>|\alpha \rangle = |\alpha/\sqrt{2} \rangle_1 \otimes |\alpha/\sqrt{2} \rangle_2</math>.

==Linear operators==
{{see also|Linear operator}}

===Linear operators acting on kets===

A linear operator is a map that inputs a ket and outputs a ket. (In order to be called "linear", it is required to have [[linear operator|certain properties]].) In other words, if <math>\hat A</math> is a linear operator and <math>|\psi\rangle</math> is a ket-vector, then <math>\hat A |\psi\rangle</math> is another ket-vector.

In an <math>N</math>-dimensional Hilbert space, we can impose a basis on the space and represent <math>|\psi\rangle</math> in terms of its coordinates as a <math>N \times 1</math> [[column vector]]. Using the same basis for <math>\hat A</math>, it is represented by an <math>N \times N</math> complex matrix. The ket-vector <math>\hat A |\psi\rangle</math> can now be computed by matrix multiplication.

Linear operators are ubiquitous in the theory of quantum mechanics. For example, observable physical quantities are represented by [[self-adjoint operator]]s, such as [[energy]] or [[momentum]], whereas transformative processes are represented by [[unitary operator|unitary]] linear operators such as rotation or the progression of time.

===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''}}}}.

===Outer products===

A convenient way to define linear operators on a Hilbert space {{math|{{mathcal|H}}}} is given by the [[outer product]]: if {{math|{{bra|''ϕ''}}}} is a bra and {{math|{{ket|''ψ''}}}} is a ket, the outer product
<math display="block"> |\phi\rang \, \lang \psi| </math>
denotes the [[finite-rank operator|rank-one operator]] with the rule 
<math display="block"> \bigl(|\phi\rang \lang \psi|\bigr)(x) = \lang \psi | x \rang |\phi \rang.</math>

For a finite-dimensional vector space, the outer product can be understood as simple matrix multiplication:
<math display="block"> |\phi \rangle \, \langle \psi | \doteq 
\begin{pmatrix} \phi_1 \\ \phi_2 \\ \vdots \\ \phi_N \end{pmatrix}
\begin{pmatrix} \psi_1^* & \psi_2^* & \cdots & \psi_N^* \end{pmatrix}
= \begin{pmatrix}
\phi_1 \psi_1^* & \phi_1 \psi_2^* & \cdots & \phi_1 \psi_N^* \\
\phi_2 \psi_1^* & \phi_2 \psi_2^* & \cdots & \phi_2 \psi_N^* \\
\vdots & \vdots & \ddots & \vdots \\
\phi_N \psi_1^* & \phi_N \psi_2^* & \cdots & \phi_N \psi_N^* \end{pmatrix}
</math>
The outer product is an {{math|''N'' × ''N''}} matrix, as expected for a linear operator.

One of the uses of the outer product is to construct [[projection operator]]s. Given a ket {{math|{{ket|''ψ''}}}} of norm 1, the orthogonal projection onto the [[Linear subspace|subspace]] spanned by {{math|{{ket|''ψ''}}}} is
<math display="block">|\psi\rangle \, \langle\psi| \,.</math>
This is an [[idempotent]] in the algebra of observables that acts on the Hilbert space.

===Hermitian conjugate operator===
{{main|Hermitian conjugate}}
Just as kets and bras can be transformed into each other (making {{math|{{ket|''ψ''}}}} into {{math|{{bra|''ψ''}}}}), the element from the dual space corresponding to {{math|''A''{{ket|''ψ''}}}} is {{math|{{bra|''ψ''}}''A''<sup>†</sup>}}, where {{math|''A''<sup>†</sup>}} denotes the Hermitian conjugate (or adjoint) of the operator {{math|''A''}}. In other words,
<math display="block"> |\phi\rangle = A |\psi\rangle \quad \text{if and only if} \quad \langle\phi| = \langle \psi | A^\dagger \,.</math>

If {{math|''A''}} is expressed as an {{math|''N'' × ''N''}} matrix, then {{math|''A''<sup>†</sup>}} is its conjugate transpose.

==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>

==Composite bras and kets==
Two Hilbert spaces {{math|''V''}} and {{math|''W''}} may form a third space {{math|''V'' ⊗ ''W''}} by a [[tensor product]]. In quantum mechanics, this is used for describing composite systems. If a system is composed of two subsystems described in {{math|''V''}} and {{math|''W''}} respectively, then the Hilbert space of the entire system is the tensor product of the two spaces. (The exception to this is if the subsystems are actually [[identical particles]]. In that case, the situation is a little more complicated.){{Citation needed|date=January 2025}}

If {{math|{{ket|''ψ''}}}} is a ket in {{math|''V''}} and {{math|{{ket|''φ''}}}} is a ket in {{math|''W''}}, the tensor product of the two kets is a ket in {{math|''V'' ⊗ ''W''}}. This is written in various notations:
:<math>|\psi\rangle|\phi\rangle \,,\quad |\psi\rangle \otimes |\phi\rangle\,,\quad|\psi \phi\rangle\,,\quad|\psi ,\phi\rangle\,.</math>

See [[quantum entanglement#Quantum mechanical framework|quantum entanglement]] and the [[EPR paradox#Mathematical formulation|EPR paradox]] for applications of this product.

==The unit operator==
Consider a complete [[orthonormal]] system (''basis''),
<math display="block">\{ e_i \ | \ i \in \mathbb{N} \} \,,</math>
for a Hilbert space {{math|''H''}}, with respect to the norm from an inner product {{math|{{angbr|·,·}}}}.

From basic [[functional analysis]], it is known that any ket <math>|\psi\rangle </math> can also be written as
<math display="block">|\psi\rangle = \sum_{i \in \mathbb{N}} \langle e_i | \psi \rangle | e_i \rangle,</math>
with {{math|{{bra-ket|·|·}}}} the inner product on the Hilbert space.

From the commutativity of kets with (complex) scalars, it follows that
<math display="block">\sum_{i \in \mathbb{N}} | e_i \rangle \langle e_i | = \mathbb{I}</math>
must be the ''identity operator'', which sends each vector to itself.

This, then, can be inserted in any expression without affecting its value; for example
<math display="block">\begin{align}
\langle v | w \rangle &= \langle v | \left( \sum_{i \in \mathbb{N}} | e_i \rangle \langle e_i| \right) | w \rangle \\
                      &= \langle v | \left( \sum_{i \in \mathbb{N}} | e_i \rangle \langle e_i| \right) \left( \sum_{j \in \mathbb{N}} | e_j \rangle \langle e_j |\right)| w \rangle \\
&= \langle v | e_i \rangle \langle e_i | e_j \rangle \langle e_j | w \rangle \,, 
\end{align}</math>
where, in the last line, the [[Einstein summation convention]] has been used to avoid clutter.

In quantum mechanics, it often occurs that little or no information about the inner product {{math|{{bra-ket|''ψ''|''φ''}}}} of two arbitrary (state) kets is present, while it is still possible to say something about the expansion coefficients {{math|1={{bra-ket|''ψ''|''e<sub>i</sub>''}} = {{bra-ket|''e<sub>i</sub>''|''ψ''}}*}} and {{math|{{bra-ket|''e<sub>i</sub>''|''φ''}}}} of those vectors with respect to a specific (orthonormalized) basis. In this case, it is particularly useful to insert the unit operator into the bracket one time or more.

For more information, see [[Resolution of the identity]],<ref>{{harvnb|Sakurai|Napolitano|2021}} Sec 1.2, 1.3</ref>
<math display="block"> {\mathbb I} = \int\!  dx~ | x \rangle \langle x |= \int\! dp ~| p \rangle \langle p |,</math> where <math display="block">|p\rangle = \int dx \frac{e^{ixp / \hbar} |x\rangle}{\sqrt{2\pi\hbar}}.</math>

Since {{math|1={{bra-ket|''x''{{prime}}|''x''}} = ''δ''(''x'' − ''x''{{prime}})}}, plane waves follow, <math display="block"> \langle x | p \rangle = \frac{e^{ixp / \hbar}}{\sqrt{2\pi\hbar}}.</math>

In his book (1958), Ch. III.20, Dirac defines the ''standard ket'' which, up to a normalization, is the translationally invariant momentum eigenstate <math display="inline">|\varpi\rangle=\lim_{p\to 0} |p\rangle</math> in the momentum representation, i.e., <math>\hat{p}|\varpi\rangle=0</math>. Consequently, the corresponding wavefunction is a constant, <math> \langle x|\varpi\rangle \sqrt{2\pi \hbar}=1</math>, and 
<math display="block">|x\rangle= \delta(\hat{x}-x) |\varpi\rangle \sqrt{2\pi \hbar},</math> as well as <math display="block">|p\rangle= \exp (ip\hat{x}/\hbar ) |\varpi\rangle.</math>

Typically, when all matrix elements of an operator such as <math display="block"> \langle x| A |y\rangle </math> are available, this resolution serves to reconstitute the full operator,
<math display="block"> \int dx \, dy \, |x\rangle \langle x| A |y\rangle \langle y | = A \,. </math>

== Notation used by mathematicians ==

The object physicists are considering when using bra–ket notation is a Hilbert space (a [[Complete metric space|complete]] inner product space).

Let <math>(\mathcal H, \langle\cdot,\cdot\rangle)</math> be a Hilbert space and {{math|''h'' ∈ {{mathcal|H}}}} a vector in {{math|{{mathcal|H}}}}. What physicists would denote by {{math|{{ket|''h''}}}} is the vector itself. That is,
<math display="block"> |h\rangle\in \mathcal{H} .</math>

Let {{math|{{mathcal|H}}*}} be the dual space of {{math|{{mathcal|H}}}}. This is the space of linear functionals on {{math|{{mathcal|H}}}}. The [[embedding]] <math>\Phi:\mathcal H \hookrightarrow \mathcal H^*</math> is defined by <math>\Phi(h) = \varphi_h</math>, where for every {{math|''h'' ∈ {{mathcal|H}}}} the linear functional <math>\varphi_h:\mathcal H\to\mathbb C</math> satisfies for every {{math|''g'' ∈ {{mathcal|H}}}} the functional equation <math>\varphi_h(g) = \langle h, g\rangle = \langle h\mid g\rangle</math>.
Notational confusion arises when identifying {{math|''φ<sub>h</sub>''}} and {{math|''g''}} with {{math|{{bra|''h''}}}} and {{math|{{ket|''g''}}}} respectively. This is because of literal symbolic substitutions. Let <math>\varphi_h = H = \langle h\mid</math> and let {{math|1=''g'' = G = {{ket|''g''}}}}. This gives
<math display="block"> \varphi_h(g) = H(g) = H(G)=\langle h|(G) = \langle h|\bigl(|g\rangle\bigr) \,. </math>

One ignores the parentheses and removes the double bars.

Moreover, mathematicians usually write the dual entity not at the first place, as the physicists do, but at the second one, and they usually use not an [[asterisk]] but an overline (which the physicists reserve for averages and the [[Dirac equation#Conservation of probability current|Dirac spinor adjoint]]) to denote [[complex conjugate]] numbers; i.e., for scalar products mathematicians usually write
<math display="block">\langle\phi ,\psi\rangle=\int \phi (x)\overline{\psi(x)}\, dx \,,</math>
whereas physicists would write for the same quantity
<math display="block"> \langle\psi |\phi \rangle = \int dx \, \psi^*(x) \phi(x)~.</math>

==See also==
{{Portal|Mathematics|Physics}}
* [[Angular momentum diagrams (quantum mechanics)]]
* [[N-slit interferometric equation|{{mvar|n}}-slit interferometric equation]]
* [[Quantum state]]
* [[Inner product space]]

== Notes ==
{{reflist}}

== References ==
* {{cite journal |last=Dirac |first=P. A. M. |year=1939 |title=A new notation for quantum mechanics |journal=[[Mathematical Proceedings of the Cambridge Philosophical Society]] |volume=35 |issue=3 |pages=416–418 |doi=10.1017/S0305004100021162 |bibcode=1939PCPS...35..416D |s2cid=121466183}} Also see his standard text, ''The Principles of Quantum Mechanics'', IV edition, Clarendon Press (1958), {{isbn|978-0198520115}} 
* {{cite book |last=Grassmann |first=H. |year=1862 |title=Extension Theory |series=History of Mathematics Sources |publisher=American Mathematical Society, London Mathematical Society |others=2000 translation by Lloyd C. Kannenberg}}
* {{cite book |last=Cajori |first=Florian |author-link=Florian Cajori |title=A History Of Mathematical Notations Volume II |year=1929 |publisher=[[Open Court Publishing Company|Open Court Publishing]] |url=https://archive.org/details/historyofmathema00cajo_0/page/134 |isbn=978-0-486-67766-8 |page=[https://archive.org/details/historyofmathema00cajo_0/page/134 134]}}
* {{cite book |last=Shankar |first=R. |year=1994 |title=Principles of Quantum Mechanics |edition=2nd |isbn=0-306-44790-8}}
* {{cite book |last1=Feynman |first1=Richard P. |last2=Leighton |first2=Robert B. |last3=Sands |first3=Matthew |year=1965 |title=The Feynman Lectures on Physics |volume=III |location=Reading, MA |publisher=Addison-Wesley |isbn=0-201-02118-8 |url=https://feynmanlectures.caltech.edu/III_toc.html}}
*{{cite book |last1=Sakurai |first1=J J |last2=Napolitano |first2=J |title=[[Modern Quantum Mechanics]] |edition=3rd |date=2021 |publisher=Cambridge University Press |isbn=978-1-108-42241-3}}

==External links==
*Richard Fitzpatrick, [https://farside.ph.utexas.edu/teaching/qm/lectures/ "Quantum Mechanics: A graduate level course"], The University of Texas at Austin. Includes:
** 1. [http://farside.ph.utexas.edu/teaching/qm/lectures/node7.html Ket space]
** 2. [http://farside.ph.utexas.edu/teaching/qm/lectures/node8.html Bra space]
** 3. [http://farside.ph.utexas.edu/teaching/qm/lectures/node9.html Operators]
** 4. [http://farside.ph.utexas.edu/teaching/qm/lectures/node10.html The outer product]
** 5. [http://farside.ph.utexas.edu/teaching/qm/lectures/node11.html Eigenvalues and eigenvectors]
*Robert Littlejohn, [http://bohr.physics.berkeley.edu/classes/221/0708/notes/hilbert.pdf Lecture notes on "The Mathematical Formalism of Quantum mechanics", including bra–ket notation.] University of California, Berkeley.
*{{cite journal |first=F. |last=Gieres |title=Mathematical surprises and Dirac's formalism in quantum mechanics |journal=Rep. Prog. Phys. |volume=63 |issue=12 |year=2000 |pages=1893&ndash;1931 |arxiv=quant-ph/9907069 |bibcode=2000RPPh...63.1893G |doi=10.1088/0034-4885/63/12/201 |s2cid=10854218}}

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