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| {{Quantum mechanics|cTopic=Background}}
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| {{cleanup|reason= The article uses a wrong notation. Vectors with specific components may represent a ket, but they are not the same mathematical object. Therefore no equal sign should be used between them. See Modern Quantum Mechanics Revised Revision, Sakurai, p. 20. for reading and discussion |date=February 2014}}
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| In [[quantum mechanics]], '''bra–ket notation''' is a standard notation for describing [[quantum state]]s, composed of [[bracket|angle brackets]] and [[vertical bar]]s. It can also be used to denote abstract [[vector space|vectors]] and [[linear functional]]s in [[mathematics]]. It is so called because the [[inner product]] (or [[dot product]] on a complex vector space) of two states is denoted by a ⟨'''bra'''|'''ket'''⟩,
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| :<math>\langle\phi|\psi\rangle</math> ,
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| consisting of a left part, ⟨''φ''|, called the '''bra''' {{IPAc-en|b|r|ɑː}}, and a right part, |''ψ''⟩, called the '''ket''' {{IPAc-en|k|ɛ|t}}. The notation was introduced in 1939 by [[Paul Dirac]]<ref name=Dirac>{{cite news
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| |author=PAM Dirac
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| |title=A new notation for quantum mechanics
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| |journal=Mathematical Proceedings of the Cambridge Philosophical Society
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| |year=1939
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| |volume=35|issue=3|pages=416–418|doi=10.1017/S0305004100021162
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| |url=http://journals.cambridge.org/action/displayAbstract?fromPage=online&aid=2031476}}
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| </ref> and is also known as '''Dirac notation''', though the notation has precursors in [[Hermann Grassmann|Grassmann]]'s use of the notation [''φ''|''ψ''] for his inner products nearly 100 years previously.<ref name=Grassmann>{{cite book
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| |author=H. Grassmann
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| |title=Extension Theory
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| |series=History of Mathematics Sources
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| |publisher=American Mathematical Society, London Mathematical Society, 2000 translation by Lloyd C. Kannenberg
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| |year=1862}}</ref><ref>{{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=http://www.archive.org/details/historyofmathema027671mbp | isbn=978-0-486-67766-8 | page= 134 | ref=harv | postscript=<!--None-->}}</ref>
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| Bra–ket notation is widespread in [[quantum mechanics]]: almost every phenomenon that is explained using quantum mechanics—including a large portion of [[modern physics]] — is usually explained with the help of bra–ket notation. Part of the appeal of the notation is the abstract representation-independence it encodes, together with its versatility in producing a specific representation (e.g. ''x'', or ''p'', or eigenfunction base) without much ado, or excessive reliance on the nature of the linear spaces involved. The overlap expression ⟨''φ''|''ψ''⟩ is typically interpreted as the [[probability amplitude]] for the [[Quantum state|state]] ''[[ψ]]'' to [[wavefunction collapse|collapse]] into the state ''[[Φ#Use as a symbol|ϕ]]''.
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| ==Vector spaces==
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| ===Background: Vector spaces===
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| {{main|Vector space}}
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| In physics, [[basis vector]]s allow any [[Euclidean vector]] to be represented geometrically using [[angle]]s and [[length]]s, in different directions, i.e. in terms of the [[Orientation (geometry)|spatial orientations]]. It is simpler to see the notational equivalences between ordinary notation and bra–ket notation; so, for now, consider a vector '''A''' starting at the origin and ending at an [[Set element|element]] of 3-d [[Euclidean space]]; the vector then is specified by this end-point, a triplet of elements in the [[field (mathematics)|field]] of [[real number]]s, symbolically dubbed as '''A''' ∈ ℝ<sup>3</sup>.
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| The vector '''A''' can be written using any set of [[basis vector]]s and corresponding [[coordinate system]]. Informally basis vectors are like "building blocks of a vector": they are added together to compose a vector, and the [[Coordinate vector|coordinate]]s are the numerical coefficients of basis vectors in each direction. Two useful representations of a vector are simply a [[linear combination]] of [[basis vectors]], and [[Column vector|column]] [[matrix (mathematics)|matrices]]. Using the familiar [[Cartesian coordinate|Cartesian]] basis, a vector '''A''' may be written as
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| [[File:Vector components bases projection.svg|400px|thumb|[[three-dimensional space|3d]] [[real number|real]] vector components and bases projection; similarities between [[vector calculus]] notation and '''Dirac notation'''. Projection is an important feature of the Dirac notation.]]
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| :<math> \begin{align} | |
| \mathbf{A} & = A_x \mathbf{e}_x + A_y \mathbf{e}_y + A_z \mathbf{e}_z \\
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| & = A_x \begin{pmatrix} 1 \\ 0 \\ 0 \end{pmatrix} +
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| A_y \begin{pmatrix} 0 \\ 1 \\ 0 \end{pmatrix} +
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| A_z \begin{pmatrix} 0 \\ 0 \\ 1 \end{pmatrix} \\
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| & = \begin{pmatrix} A_x \\ 0 \\ 0 \end{pmatrix} +
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| \begin{pmatrix} 0 \\ A_y \\ 0 \end{pmatrix} +
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| \begin{pmatrix} 0 \\ 0 \\ A_z \end{pmatrix} \\
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| & = \begin{pmatrix}
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| A_x \\
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| A_y \\
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| A_z \\
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| \end{pmatrix}
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| \end{align}</math>
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| respectively, where '''e'''<sub>''x''</sub>, '''e'''<sub>''y''</sub>, '''e'''<sub>''z''</sub> denotes the Cartesian [[basis vector]]s (all are [[Orthonormality|orthogonal]] [[unit vector]]s) and ''A<sub>x</sub>'', ''A<sub>y</sub>'', ''A<sub>z</sub>'' are the corresponding [[coordinate vector|coordinate]]s, in the ''x'', ''y'', ''z'' directions. In a more general notation, for any basis in 3-d space one writes
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| :<math>\mathbf{A} = A_1 \mathbf{e}_1 + A_2 \mathbf{e}_2 + A_3 \mathbf{e}_3 = \begin{pmatrix}
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| A_1 \\
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| A_2 \\
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| A_3 \\
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| \end{pmatrix} </math>
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| Generalizing further, consider a vector '''A''' in an ''N'' dimensional vector space over the field of [[complex number]]s ℂ, symbolically stated as '''A''' ∈ ℂ<sup>''N''</sup>. The vector '''A''' is still conventionally represented by a linear combination of basis vectors or a column matrix:
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| :<math>\mathbf{A} = \sum_{n=1}^N A_n \mathbf{e}_n = \begin{pmatrix}
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| A_1 \\
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| A_2 \\
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| \vdots \\
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| A_N \\
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| \end{pmatrix} </math>
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| though the coordinates are now all complex-valued.
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| Even more generally, '''A''' can be a vector in a [[complex number|complex]] [[Hilbert space]]. Some Hilbert spaces, like ℂ<sup>''N''</sup>, have finite dimension, while others have infinite dimension. In an infinite-dimensional space, the column-vector representation of '''A''' would be a list of infinitely many complex numbers.
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| ===Ket notation for vectors===
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| Rather than boldtype, over arrows, underscores etc. conventionally used elsewhere; <math>\mathbf{A},\,\vec{A},\,\underline{A}</math>, Dirac's notation for a vector uses vertical bars and angular brackets: |''A''⟩. When this notation is used, these vectors are called "ket", read as "ket-A".<ref>Quantum Mechanics Demystified, D. McMahon, Mc Graw Hill (USA), 2006, ISBN(10-) 0-07-145546 9</ref> This applies to all vectors, the resultant vector and the basis. The previous vectors are now written
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| :<math> |A \rangle = A_x|e_x \rangle + A_y|e_y \rangle + A_z|e_z \rangle =
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| \begin{pmatrix} A_x \\ A_y \\ A_z \end{pmatrix},</math>
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| or in a more easily generalized notation,
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| :<math> |A \rangle = A_1|e_1 \rangle + A_2|e_2 \rangle + A_3|e_3 \rangle = | |
| \begin{pmatrix} A_1 \\ A_2 \\ A_3 \end{pmatrix},</math>
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| The last one may be written in short as
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| :<math>|A \rangle = A_1|1 \rangle + A_2|2 \rangle + A_3|3 \rangle </math> | |
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| Notice how any symbols, letters, numbers, or even words — whatever serves as a convenient label — can be used as the label inside a ket. In other words, the symbol |''A''⟩ has a specific and universal mathematical meaning, but just the "''A''" by itself does not. 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 with a list of their [[quantum number]]s.
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| ===Inner products and bras===
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| {{main|Inner product}}
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| An [[inner product]] is a generalization of the [[dot product]]. The inner product of two vectors is a complex number. Bra–ket notation uses a specific notation for inner products:
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| :<math> \langle A | B \rangle = \text{the inner product of ket } | A \rangle \text{ with ket } | B \rangle</math> | |
| For example, in three-dimensional complex [[Euclidean space]],
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| :<math>\langle A | B \rangle = A_x^*B_x + A_y^*B_y + A_z^*B_z </math>
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| where <math>A_i^*</math> denotes the [[complex conjugate]] of {{math|''A''<sub>i</sub>}}. A special case is the inner product of a vector with itself, which is the square of its [[norm (mathematics)|norm]] (magnitude):
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| :<math>\langle A | A \rangle = |A_x|^2 + |A_y|^2 + |A_z|^2 </math>
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| Bra–ket notation splits this inner product (also called a "bracket") into two pieces, the "bra" and the "ket":
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| :<math> \langle A | B \rangle = \left( \, \langle A | \, \right) \,\, \left( \, | B \rangle \, \right)</math>
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| where ⟨''A''| is called a bra, read as "bra-A", and |''B''⟩ is a ket as above.
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| The purpose of "splitting" the inner product into a bra and a ket is that ''both'' the bra ⟨''A''| and the ket |''B''⟩ are meaningful ''on their own'', and can be used in other contexts besides within an inner product. There are two main ways to think about the meanings of separate bras and kets:
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| ====Bras and kets as row and column vectors====
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| 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:
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| :<math> \langle A | B \rangle = A_1^* B_1 + A_2^* B_2 + \cdots + A_N^* B_N =
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| \begin{pmatrix} A_1^* & A_2^* & \cdots & A_N^* \end{pmatrix}
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| \begin{pmatrix} B_1 \\ B_2 \\ \vdots \\ B_N \end{pmatrix}</math>
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| Based on this, the bras and kets can be defined as:
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| :<math> \langle A | = \begin{pmatrix} A_1^* & A_2^* & \cdots & A_N^* \end{pmatrix}</math>
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| :<math> | B \rangle = \begin{pmatrix} B_1 \\ B_2 \\ \vdots \\ B_N \end{pmatrix}</math>
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| and then it is understood that a bra next to a ket implies [[matrix multiplication]]. | |
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| The [[conjugate transpose]] (also called ''Hermitian conjugate'') of a bra is the corresponding ket and vice versa:
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| :<math>\langle A |^\dagger = |A \rangle, \quad |A \rangle^\dagger = \langle A |</math>
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| because if one starts with the bra
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| :<math>\begin{pmatrix} A_1^* & A_2^* & \cdots & A_N^* \end{pmatrix},</math>
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| then performs a [[complex conjugation]], and then a [[matrix transpose]], one ends up with the ket
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| :<math>\begin{pmatrix} A_1 \\ A_2 \\ \vdots \\ A_N \end{pmatrix}</math>
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| ====Bras as linear operators on kets====
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| {{main|Dual space|Riesz representation theorem}}
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| A more abstract definition, which is equivalent but more easily generalized to infinite-dimensional spaces, is to say that bras are [[linear functional]]s on kets, i.e. operators that input a ket and output a complex number. The bra operators are defined to be consistent with the inner product.
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| In mathematics terminology, the [[vector space]] of bras is the [[dual space]] to the vector space of kets, and corresponding bras and kets are related by the [[Riesz representation theorem]].
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| === Non-normalizable states and non-Hilbert spaces ===
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| Bra–ket notation can be used even if the vector space is not a [[Hilbert space]].
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| In quantum mechanics, it is common practice to write down kets which have infinite [[norm (mathematics)|norm]], i.e. non-[[normalisable wavefunction]]s. Examples include states whose [[wavefunction]]s 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 broader context.
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| For a rigorous treatment of the Dirac inner product of non-normalizable states, see the definition given by D. Carfì.<ref name=Carfi-Dirac>{{cite journal|last=Carfì|first=David|title=Dirac-orthogonality in the space of tempered distributions|journal=Journal of Computational and Applied Mathematics|date=April 2003|volume=153|issue=1–2|pages=99–107|doi=10.1016/S0377-0427(02)00634-9|url=http://portal.acm.org/citation.cfm?id=774918|bibcode = 2003JCoAM.153...99C }}</ref><ref name=Carfi-Germanà>{{cite journal|last=Carfì|first=David|title=Some properties of a new product in the space of tempered distributions|journal=Journal of Computational and Applied Mathematics|date=April 2003|volume=153|issue=1–2|pages=109–118|doi=10.1016/S0377-0427(02)00635-0|url=http://portal.acm.org/citation.cfm?id=774919|bibcode = 2003JCoAM.153..109C }}</ref> For a rigorous definition of basis with a continuous set of indices and consequently for a rigorous definition of position and momentum basis, see.<ref name=Carfi-Characterizations>{{cite journal|last=Carfì|first=David|title=TOPOLOGICAL CHARACTERIZATIONS OF S-LINEARITY|journal=AAPP-PHYSICAL, MATHEMATICAL AND NATURAL SCIENCES|year=2007|volume=85|issue=2|pages=1–16|doi=10.1478/C1A0702005|url=http://cab.unime.it/journals/index.php/AAPP/article/view/407}}</ref> For a rigorous statement of the expansion of an S-diagonalizable operator, or observable, in its eigenbasis or in another basis, see.<ref name=Carfi-Glasnik>{{cite journal|last=Carfì|first=David|title=S-DIAGONALIZABLE OPERATORS IN QUANTUM MECHANICS|journal=Glasnik Matematicki|year=2005|volume=40|issue=2|pages=261–301|doi=10.3336/gm.40.2.08|url=http://web.math.hr/glasnik/vol_40/no2_08.html}}</ref>
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| [[Banach space]]s are a different generalization of Hilbert spaces. In a Banach space ''B'', the vectors may be notated by kets and the continuous [[linear functional]]s by bras. Over any vector space without [[topology]], we may also 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.
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| ==Usage in quantum mechanics==
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| The mathematical structure of [[quantum mechanics]] is based in large part on [[linear algebra]]:
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| *[[Wave function]]s and other [[quantum state]]s can be represented as vectors in a 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" |''ψ''⟩. (Technically, the quantum states are ''[[Line (geometry)#Ray|rays]]'' of vectors in the Hilbert space, as ''c''|''ψ''⟩ corresponds to the same state for any nonzero complex number ''c''.)
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| *[[Quantum superposition]]s can be described as vector sums of the constituent states. For example, an electron in the state |1⟩+i |2⟩ is in a quantum superposition of the states |1⟩ and |2⟩.
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| *[[Measurement in quantum mechanics|Measurements]] are associated with [[linear operator]]s (called [[observable]]s) on the Hilbert space of quantum states.
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| *Dynamics are also described by linear operators on the Hilbert space. For example, in the [[Schrödinger picture]], there is a linear [[time evolution]] operator ''U'' with the property that if an electron is in state |''ψ''⟩ right now, then in one second it will be in the state ''U''|''ψ''⟩, the same ''U'' for every possible |''ψ''⟩.
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| *[[Normalisable wave function|Wave function normalization]] is scaling a wave function so that its [[norm (mathematics)|norm]] is 1.
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| 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:
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| ===Spinless position–space wave function===
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| {{multiple image
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| | left
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| | footer = Components of complex vectors plotted against index number; discrete ''k'' and continuous ''x''. Two particular components out of infinitely many are highlighted.
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| | width1 = 225
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| | image1 = Discrete complex vector components.svg
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| | caption1 = Discrete components ''A''<sub>''k''</sub> of a complex vector {{braket|ket|''A''}} = ∑<sub>''k''</sub> ''A''<sub>''k''</sub>{{braket|ket|''e<sub>k</sub>''}}, which belongs to a ''countably infinite''-dimensional Hilbert space; there are countably infinitely many ''k'' values and basis vectors {{braket|ket|''e<sub>k</sub>''}}.
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| | width2 = 230
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| | image2 = Continuous complex vector components.svg
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| | caption2 = Continuous components ''ψ''(''x'') of a complex vector {{braket|ket|''ψ''}} = ∫''dx'' ''ψ''(''x''){{braket|ket|''x''}}, which belongs to an ''uncountably infinite''-dimensional [[Hilbert space]]; there are infinitely many ''x'' values and basis vectors {{braket|ket|''x''}}.
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| }}
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| The Hilbert space of a [[spin (physics)|spin]]-0 point particle is spanned by a "position [[basis (linear algebra)|basis]]" <math>\{ \, |\mathbf{r}\rangle \,\} </math>, where the label '''r''' extends over the set of all points in [[position space]]. Since there are [[uncountably infinite]]ly many vectors in the basis, this is an uncountably infinite-dimensional Hilbert space. The dimensions of the Hilbert space (usually infinite) and position space (usually 1, 2 or 3) are not to be conflated.
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| Starting from any ket |''Ψ''⟩ in this Hilbert space, we can ''define'' a complex scalar function of '''r''', known as a [[wavefunction]]:
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| :<math>\Psi(\mathbf{r}) \ \stackrel{\text{def}}{=}\ \lang \mathbf{r}|\Psi\rang</math> .
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| On the left side, ''Ψ''('''r''') is a function mapping any point in space to a complex number; on the right side, |''Ψ''⟩ = ∫d<sup>3</sup>'''r''' ''Ψ''('''r''') |'''r'''⟩ is a ket.
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| It is then customary to define linear operators acting on wavefunctions in terms of linear operators acting on kets, by
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| :<math>A \Psi(\mathbf{r}) \ \stackrel{\text{def}}{=}\ \lang \mathbf{r}|A|\Psi\rang </math> .
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| For instance, the [[momentum]] operator '''p''' has the following form,
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| :<math>\mathbf{p} \Psi(\mathbf{r}) \ \stackrel{\text{def}}{=}\ \lang \mathbf{r} |\mathbf{p}|\Psi\rang = - i \hbar \nabla \Psi(\mathbf{r})</math> .
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| One occasionally encounters a sloppy expression like
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| :<math>\nabla |\Psi\rang </math>,
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| though this is something of a (common) [[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 into the position basis,
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| :<math>\nabla \lang\mathbf{r}|\Psi\rang</math> ,
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| even though, in the momentum basis, the operator amounts to a mere multiplication operator (by iħ'''''p''''').
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| ===Overlap of states===
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| In quantum mechanics the expression ⟨''φ''|''ψ''⟩ is typically interpreted as the [[probability amplitude]] for the state ''ψ'' to [[wavefunction collapse|collapse]] into the state ''φ''. Mathematically, this means the coefficient for the projection of ''ψ'' onto ''φ''.
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| ===Changing basis for a spin-1/2 particle===
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| A stationary [[spin-½]] particle has a two-dimensional Hilbert space. One [[orthonormal basis]] is:
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| :<math>|\uparrow_z \rangle, \; |\downarrow_z \rangle</math>
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| where <math>|\uparrow_z \rangle</math> is the state with a definite value of the [[angular momentum operator|spin operator ''S<sub>z</sub>'']] equal to +1/2 and <math>|\downarrow_z \rangle</math> is the state with a definite value of the [[angular momentum operator|spin operator ''S<sub>z</sub>'']] equal to −1/2.
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| Since these are a [[basis (linear algebra)|basis]], ''any'' [[quantum state]] of the particle can be expressed as a [[linear combination]] (i.e., [[quantum superposition]]) of these two states:
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| :<math>|\psi \rangle = a_{\psi} |\uparrow_z \rangle + b_{\psi} |\downarrow_z \rangle</math>
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| where ''a''<sub>ψ</sub>, ''b''<sub>ψ</sub> are complex numbers.
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| A ''different'' basis for the same Hilbert space is:
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| :<math>|\uparrow_x \rangle, \; |\downarrow_x \rangle</math>
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| defined in terms of ''S<sub>x</sub>'' rather than ''S<sub>z</sub>''.
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| Again, ''any'' state of the particle can be expressed as a linear combination of these two:
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| :<math>|\psi \rangle = c_{\psi} |\uparrow_x \rangle + d_{\psi} |\downarrow_x \rangle</math>
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| In vector form, you might write
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| :<math>|\psi\rangle = \begin{pmatrix} a_\psi \\ b_\psi \end{pmatrix}, \;\; \text{OR} \;\; |\psi\rangle = \begin{pmatrix} c_\psi \\ d_\psi \end{pmatrix} </math>
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| depending on which basis you are using. In other words, the "coordinates" of a vector depend on the basis used.
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| There is a mathematical relationship between ''a''<sub>ψ</sub>, ''b''<sub>ψ</sub>,''c''<sub>ψ</sub>, ''d''<sub>ψ</sub>; see [[change of basis]].
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| ===Misleading uses===
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| There are a few conventions and abuses of notation that are generally accepted by the physics community, but which might confuse the non-initiated.
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| It is common among physicists to use the same symbol for ''labels'' and ''constants'' in the same equation. It supposedly becomes easier to identify that the constant is related to the labeled object, and is claimed that the divergent nature of each will
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| eliminate any ambiguity and no further differentiation is required. For example, α̂|''α''⟩= ''α''|''α''⟩, where the symbol ''α'' is used '''simultaneously''' as the ''name of the operator'' α̂, its ''eigenvector'' |''α''⟩ and the associated ''eigenvalue'' ''α''.
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| Something similar occurs in component notation of vectors. While ''Ψ'' (uppercase) is traditionally associated with wavefunctions, ''ψ'' (lowercase) may be used to denote a ''label'', a ''wave function'' or ''complex constant'' in the same context, usually differentiated only by a subscript.
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| The main abuses are including operations inside the vector labels. This is usually done for a fast notation of scaling vectors. E.g. if the vector |''α''⟩ is scaled by {{math|1/{{radical|2}}}}, it might be denoted by <math>| \alpha/\sqrt{2} \rangle</math>, which makes no sense since ''α'' is a label, not a function or a number, so you can't perform operations on it.
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| This is especially 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>. Here
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| part of the labeling that should state that all three vectors are different was moved outside the kets, as subscripts 1 and 2. And a further abuse occurs, since ''α'' is meant to refer to the norm of the first vector – which is a ''label'' is denoting a ''value''.
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| ==Linear operators==
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| {{see also|Linear operator}}
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| ===Linear operators acting on kets===
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| 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 '''''A''''' is a linear operator and |''ψ''⟩ is a ket, then '''''A'''''|''ψ''⟩ is another ket.
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| In an ''N''-dimensional Hilbert space, |''ψ''⟩ can be written as an ''N''×1 [[column vector]], and then '''''A''''' is an ''N''×''N'' matrix with complex entries. The ket '''''A'''''|''ψ''⟩ can be computed by normal [[matrix multiplication]].
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| 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.
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| ===Linear operators acting on bras===
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| Operators can also be viewed as acting on bras ''from the right hand side''. Specifically, if '''''A''''' is a linear operator and ⟨''φ''| is a bra, then ⟨''φ''|'''''A''''' is another bra defined by the rule
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| :<math>\bigg(\langle\phi|A\bigg) \; |\psi\rangle = \langle\phi| \; \bigg(A|\psi\rangle\bigg)</math> ,
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| (in other words, a [[function composition]]). This expression is commonly written as (cf. [[energy inner product]])
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| :<math>\langle\phi|A|\psi\rangle</math> .
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| In an ''N''-dimensional Hilbert space, ⟨''φ''| can be written as a 1×N [[row vector]], and '''''A''''' (as in the previous section) is an ''N''×''N'' matrix. Then the bra ⟨''φ''|'''''A''''' can be computed by normal [[matrix multiplication]].
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| If the same state vector appears on both bra and ket side,
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| :<math>\langle\psi|A|\psi\rangle </math> ,
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| then this expression gives the [[expectation value (quantum mechanics)|expectation value]], or mean or average value, of the observable represented by operator '''''A''''' for the physical system in the state |''ψ''⟩.
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| ===Outer products===
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| A convenient way to define linear operators on ''H'' is given by the [[outer product]]: if ⟨''φ''| is a bra and |''ψ''⟩ is a ket, the outer product
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| :<math> |\phi\rang \lang \psi| </math>
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| denotes the [[finite-rank operator|rank-one operator]] that maps the ket |''ρ''⟩ to the ket |''φ''⟩⟨''ψ''|''ρ''⟩ (where ⟨''ψ''|''ρ''⟩ is a scalar multiplying the vector |''φ''⟩).
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| For a finite-dimensional vector space, the outer product can be understood as simple matrix multiplication:
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| :<math> |\phi \rangle \, \langle \psi | =
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| \begin{pmatrix} \phi_1 \\ \phi_2 \\ \vdots \\ \phi_N \end{pmatrix}
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| \begin{pmatrix} \psi_1^* & \psi_2^* & \cdots & \psi_N^* \end{pmatrix}
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| = \begin{pmatrix}
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| \phi_1 \psi_1^* & \phi_1 \psi_2^* & \cdots & \phi_1 \psi_N^* \\
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| \phi_2 \psi_1^* & \phi_2 \psi_2^* & \cdots & \phi_2 \psi_N^* \\
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| \vdots & \vdots & \ddots & \vdots \\
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| \phi_N \psi_1^* & \phi_N \psi_2^* & \cdots & \phi_N \psi_N^* \end{pmatrix}
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| </math>
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| The outer product is an N×N matrix, as expected for a linear operator.
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| One of the uses of the outer product is to construct [[projection operator]]s. Given a ket |''ψ''⟩ of norm 1, the orthogonal projection onto the [[Linear subspace|subspace]] spanned by |''ψ''⟩ is
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| :<math>|\psi\rangle\langle\psi|.</math> | |
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| ===Hermitian conjugate operator===
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| {{main|Hermitian conjugate}}
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| Just as kets and bras can be transformed into each other (making <math>|\psi\rangle</math> into <math>\langle\psi|</math>) the element from the dual space corresponding with <math>A|\psi\rangle</math> is <math>\langle \psi | A^\dagger</math> where ''A<sup>†</sup>'' denotes the [[Hermitian conjugate]] (or adjoint) of the operator ''A''. In other words,
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| :<math> |\phi\rangle = A |\psi\rangle</math> if and only if <math> \langle\phi| = \langle \psi | A^\dagger</math>.
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| If ''A'' is expressed as an N×N matrix, then ''A''<sup>†</sup> is its [[conjugate transpose]].
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| [[Self-adjoint]] operators, where ''A''=''A''<sup>†</sup>, play an important role in quantum mechanics; for example, an [[observable]] is always described by a self-adjoint operator. If ''A'' is a self-adjoint operator, then <math> \langle \psi | A | \psi \rangle</math> is always a real number (not complex). This implies that [[Expectation value (quantum mechanics)|expectation values]] of observables are real.
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| ==Properties==
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| 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, ''c''<sub>1</sub> and ''c''<sub>2</sub> denote arbitrary [[complex number]]s, c* denotes the [[complex conjugate]] of c, ''A'' and ''B'' denote arbitrary linear operators, and these properties are to hold for any choice of bras and kets.
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| ===Linearity===
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| * Since bras are linear functionals,
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| ::<math>\langle\phi| \; \bigg( c_1|\psi_1\rangle + c_2|\psi_2\rangle \bigg) = c_1\langle\phi|\psi_1\rangle + c_2\langle\phi|\psi_2\rangle. </math>
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| * 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], eqns 12 and 13</ref>
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| ::<math>\bigg(c_1 \langle\phi_1| + c_2 \langle\phi_2|\bigg) \; |\psi\rangle = c_1 \langle\phi_1|\psi\rangle + c_2 \langle\phi_2|\psi\rangle. </math>
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| ===Associativity===
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| 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:
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| :<math> \lang \psi| (A |\phi\rang) = (\lang \psi|A)|\phi\rang \, \stackrel{\text{def}}{=} \, \lang \psi | A | \phi \rang </math>
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| :<math> (A|\psi\rang)\lang \phi| = A(|\psi\rang \lang \phi|) \, \stackrel{\text{def}}{=} \, A | \psi \rang \lang \phi |</math>
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| 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 non-linear operators, such as the [[antilinear]] [[T-symmetry|time reversal operator]] in physics.
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| ===Hermitian conjugation===
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| Bra–ket notation makes it particularly easy to compute the Hermitian conjugate (also called ''dagger'', and denoted {{mvar|†}}) of expressions. The formal rules are:
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| * The Hermitian conjugate of a bra is the corresponding ket, and vice-versa.
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| * The Hermitian conjugate of a complex number is its complex conjugate.
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| * The Hermitian conjugate of the Hermitian conjugate of anything (linear operators, bras, kets, numbers) is itself—i.e.,
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| ::{{math| (''x''<sup>†</sup>)<sup>†</sup> {{=}} ''x''}}.
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| * 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.
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| These rules are sufficient to formally write the Hermitian conjugate of any such expression; some examples are as follows:
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| * Kets:
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| ::<math>
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| \left(c_1|\psi_1\rangle + c_2|\psi_2\rangle\right)^\dagger = c_1^* \langle\psi_1| + c_2^* \langle\psi_2| ~.
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| </math>
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| * Inner products:
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| ::<math>\langle \phi | \psi \rangle^* = \langle \psi|\phi\rangle ~.</math>
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| * Matrix elements:
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| ::<math>\langle \phi| A | \psi \rangle^* = \langle \psi | A^\dagger |\phi \rangle</math>
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| ::<math>\langle \phi| A^\dagger B^\dagger | \psi \rangle^* = \langle \psi | BA |\phi \rangle ~.</math>
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| * Outer products:
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| ::<math>\left((c_1|\phi_1\rangle\langle \psi_1|) + (c_2|\phi_2\rangle\langle\psi_2|)\right)^\dagger = (c_1^* |\psi_1\rangle\langle \phi_1|) + (c_2^*|\psi_2\rangle\langle\phi_2|)~.</math>
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| ==Composite bras and kets==
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| Two Hilbert spaces ''V'' and ''W'' may form a third space ''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 ''V'' and ''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.)
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| If |''ψ''⟩ is a ket in ''V'' and |''φ''⟩ is a ket in ''W'', the direct product of the two kets is a ket in ''V''⊗''W''. This is written in various notations:
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| :<math>|\psi\rangle|\phi\rangle \,,\quad |\psi\rangle \otimes |\phi\rangle\,,\quad|\psi \phi\rangle\,,\quad|\psi ,\phi\rangle\,.</math>
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| See [[quantum entanglement#Quantum mechanical framework|quantum entanglement]] and the [[EPR paradox#Mathematical formulation|EPR paradox]] for applications of this product.
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| ==The unit operator==
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| Consider a complete [[orthonormal]] system (''[[Basis (linear algebra)|basis]]''), <math>\{ e_i \ | \ i \in \mathbb{N} \}</math>, for a Hilbert space ''H'', with respect to the norm from an inner product <math>\langle\cdot,\cdot\rangle</math>. From basic [[functional analysis]] we know that any ket |''ψ''⟩ can also be written as
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| :<math>|\psi\rangle = \sum_{i \in \mathbb{N}} \langle e_i | \psi \rangle | e_i \rangle,</math>
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| with <math>\langle\cdot|\cdot\rangle</math> the inner product on the Hilbert space.
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| From the commutativity of kets with (complex) scalars now follows that
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| :<math>\sum_{i \in \mathbb{N}} | e_i \rangle \langle e_i | = \hat{1}</math>
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| must be the identity operator, which sends each vector to itself. This can be inserted in any expression without affecting its value, for example
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| :<math> \langle v | w \rangle = \langle v | \sum_{i \in \mathbb{N}} | e_i \rangle \langle e_i | w \rangle = \langle v | \sum_{i \in \mathbb{N}} | e_i \rangle \langle e_i | \sum_{j \in \mathbb{N}} | e_j \rangle \langle e_j | w \rangle = \langle v | e_i \rangle \langle e_i | e_j \rangle \langle e_j | w \rangle </math> ,
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| where, in the last identity, the [[Einstein summation convention]] has been used.
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| In [[quantum mechanics]], it often occurs that little or no information about the inner product <math>\langle\psi|\phi\rangle</math> of two arbitrary (state) kets is present, while it is still possible to say something about the expansion coefficients <math>\langle\psi|e_i\rangle = \langle e_i|\psi\rangle^*</math> and <math>\langle e_i|\phi\rangle</math> 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.
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| For more information, see [[Resolution of the identity]], 1 = ∫''dx'' |''x''⟩⟨''x''| = ∫''dp'' |''p''⟩⟨''p''|, where |''p''⟩ = ∫''dx'' e<sup>i''xp''/ħ</sup>|''x''⟩/√{{overline|''2πħ''}}; since ⟨''x' '' |''x''⟩= δ(''x'' − ''x' ''), plane waves follow, ⟨''x''|''p''⟩= exp(i''xp''/ħ)/√{{overline|''2πħ''}}.
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| == Notation used by mathematicians ==
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| The object physicists are considering when using the "bra–ket" notation is a [[Hilbert space]] (a [[Complete metric space|complete]] [[inner product space]]).
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| Let <math> \mathcal{H} </math> be a Hilbert space and <math> h\in\mathcal{H} </math> is a vector in <math> \mathcal{H} </math>. What physicists would denote as |''h''⟩ is the vector itself. That is
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| :<math> (|h\rangle)\in \mathcal{H} </math>.
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| Let <math> \mathcal{H}^* </math> be the [[dual space]] of <math> \mathcal{H} </math>. This is the space of linear functionals on <math>\mathcal{H}</math>. The isomorphism <math> \Phi:\mathcal{H}\to\mathcal{H}^* </math> is defined by <math> \Phi(h) = \phi_h </math> where for all <math> g\in\mathcal{H} </math> we have
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| :<math> \phi_h(g) = \mbox{IP}(h,g) = (h,g) = \langle h,g \rangle = \langle h|g \rangle </math>,
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| where <math> \mbox{IP}(\cdot,\cdot), (\cdot,\cdot),\langle \cdot,\cdot \rangle</math> and <math>\langle \cdot | \cdot \rangle </math>
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| are just different notations for expressing an inner product between two elements in a Hilbert space (or for the first three, in ''any'' inner product space). Notational confusion arises when identifying <math> \phi_h </math> and <math> g </math> with <math> \langle h | </math> and <math>|g \rangle </math> respectively. This is because of literal symbolic substitutions. Let <math> \phi_h = H = \langle h| </math> and let <math> g=G=|g\rangle </math>. This gives
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| :<math> \phi_h(g) = H(g) = H(G)=\langle h|(G) = \langle h|(
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| |g\rangle). </math>
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| One ignores the parentheses and removes the double bars. Some properties of this notation are convenient since we are dealing with linear operators and composition acts like a [[Ring (mathematics)|ring]] multiplication.
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| Moreover, mathematicians usually write the dual entity not at the first place, as the physicists do, but at the second one, and they don't use the *-symbol, but an overline (which the physicists reserve for averages and Dirac conjugation) to denote conjugate-complex numbers, i.e. for scalar products mathematicians usually write
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| :<math>(\phi ,\psi )=\int \phi (x)\cdot \overline{\psi(x)}\, {\rm d}x \,,</math>
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| whereas physicists would write for the same quantity
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| :<math> \langle\psi |\phi \rangle=\int {\rm d}x\,\psi^*(x)\cdot\phi(x)\,.</math>
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| ==See also==
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| * [[Angular momentum diagrams (quantum mechanics)]]
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| == References and notes ==
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| <references/>
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| == Further reading ==
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| * {{cite book|author=Feynman, Leighton and Sands|title=The Feynman Lectures on Physics Vol. III|publisher= Addison-Wesley|year=1965|isbn=0-201-02115-3}}
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| ==External links==
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| *Richard Fitzpatrick, [http://farside.ph.utexas.edu/teaching/qm/lectures/lectures.html "Quantum Mechanics: A graduate level course"], The University of Texas at Austin.
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| ** 1. [http://farside.ph.utexas.edu/teaching/qm/lectures/node7.html Ket space]
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| ** 2. [http://farside.ph.utexas.edu/teaching/qm/lectures/node8.html Bra space]
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| ** 3. [http://farside.ph.utexas.edu/teaching/qm/lectures/node9.html Operators]
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| ** 4. [http://farside.ph.utexas.edu/teaching/qm/lectures/node10.html The outer product]
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| ** 5. [http://farside.ph.utexas.edu/teaching/qm/lectures/node11.html Eigenvalues and eigenvectors]
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| *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.]
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| {{DEFAULTSORT:Bra-Ket Notation}}
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| [[Category:Quantum mechanics]]
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| [[Category:Information theory]]
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| [[Category:Quantum information science]]
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| [[Category:Mathematical notation]]
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