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Bra–ket notation

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Notation for quantum states
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Quantum mechanics
iℏddt|Ψ⟩=H^|Ψ⟩{\displaystyle i\hbar {\frac {d}{dt}}|\Psi \rangle ={\hat {H}}|\Psi \rangle }

Bra–ket notation, also calledDirac notation, is a notation forlinear algebra andlinear operators oncomplex vector spaces together with theirdual space both in the finite-dimensional and infinite-dimensional case. It is specifically designed to ease the types of calculations that frequently come up inquantum mechanics. Its use in quantum mechanics is quite widespread.

Bra–ket notation was created byPaul Dirac in his 1939 publicationA New Notation for Quantum Mechanics. The notation was introduced as an easier way to write quantum mechanical expressions.[1] The name comes from the English word "bracket".

Quantum mechanics

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Inquantum mechanics andquantum computing, bra–ket notation is used ubiquitously to denotequantum states. The notation usesangle brackets,⟨{\displaystyle \langle } and⟩{\displaystyle \rangle }, and avertical bar|{\displaystyle |}, to construct "bras" and "kets".

Aket is of the form|v⟩{\displaystyle |v\rangle }. Mathematically it denotes avector,v{\displaystyle {\boldsymbol {v}}}, in an abstract (complex)vector spaceV{\displaystyle V}, and physically it represents a state of some quantum system.

Abra is of the form⟨f|{\displaystyle \langle f|}. Mathematically it denotes alinear formf:V→C{\displaystyle f:V\to \mathbb {C} }, i.e. alinear map that maps each vector inV{\displaystyle V} to a number in the complex planeC{\displaystyle \mathbb {C} }. Letting the linear functional⟨f|{\displaystyle \langle f|} act on a vector|v⟩{\displaystyle |v\rangle } is written as⟨f|v⟩∈C{\displaystyle \langle f|v\rangle \in \mathbb {C} }.

Assume that onV{\displaystyle V} there exists an inner product(⋅,⋅){\displaystyle (\cdot ,\cdot )} withantilinear first argument, which makesV{\displaystyle V} aninner product space. Then with this inner product each vectorϕ≡|ϕ⟩{\displaystyle {\boldsymbol {\phi }}\equiv |\phi \rangle } can be identified with a corresponding linear form, by placing the vector in the anti-linear first slot of the inner product:(ϕ,⋅)≡⟨ϕ|{\displaystyle ({\boldsymbol {\phi }},\cdot )\equiv \langle \phi |}. The correspondence between these notations is then(ϕ,ψ)≡⟨ϕ|ψ⟩{\displaystyle ({\boldsymbol {\phi }},{\boldsymbol {\psi }})\equiv \langle \phi |\psi \rangle }. Thelinear form⟨ϕ|{\displaystyle \langle \phi |} is acovector to|ϕ⟩{\displaystyle |\phi \rangle }, and the set of all covectors forms a subspace of thedual vector spaceV∨{\displaystyle V^{\vee }}, to the initial vector spaceV{\displaystyle V}. The purpose of this linear form⟨ϕ|{\displaystyle \langle \phi |} can now be understood in terms of making projections onto the stateϕ,{\displaystyle {\boldsymbol {\phi }},} to find how linearly dependent two states are, etc.

For the vector spaceCn{\displaystyle \mathbb {C} ^{n}}, kets can be identified with column vectors, and bras with row vectors. Combinations of bras, kets, and linear operators are interpreted usingmatrix multiplication. IfCn{\displaystyle \mathbb {C} ^{n}} has the standard Hermitian inner product(v,w)=v†w{\displaystyle ({\boldsymbol {v}},{\boldsymbol {w}})=v^{\dagger }w}, under this identification, the identification of kets and bras and vice versa provided by the inner product is taking theHermitian conjugate (denoted†{\displaystyle \dagger }).

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 operatorσ^z{\displaystyle {\hat {\sigma }}_{z}} on a two-dimensional spaceΔ{\displaystyle \Delta } ofspinors haseigenvalues±12{\textstyle \pm {\frac {1}{2}}} with eigenspinorsψ+,ψ−∈Δ{\displaystyle {\boldsymbol {\psi }}_{+},{\boldsymbol {\psi }}_{-}\in \Delta }. In bra–ket notation, this is typically denoted asψ+=|+⟩{\displaystyle {\boldsymbol {\psi }}_{+}=|+\rangle }, andψ−=|−⟩{\displaystyle {\boldsymbol {\psi }}_{-}=|-\rangle }. 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 withHermitian conjugate column and row vectors.

Bra–ket notation was effectively established in 1939 byPaul Dirac;[1][2] it is thus also known as Dirac notation, despite the notation having a precursor inHermann Grassmann's use of[ϕ∣ψ]{\displaystyle [\phi {\mid }\psi ]} for inner products nearly 100 years earlier.[3][4]

Vector spaces

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Vectors vs kets

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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 likedisplacement orvelocity, which have components that relate directly to the three dimensions ofspace, or relativistically, to the four ofspacetime. Such vectors are typically denoted with over arrows (r→{\displaystyle {\vec {r}}}), boldface (p{\displaystyle \mathbf {p} }) or indices (vμ{\displaystyle v^{\mu }}).

In quantum mechanics, a quantum state is typically represented as an element of a complexHilbert space, for example, the infinite-dimensional vector space of all possiblewavefunctions (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 elementϕ{\displaystyle \phi } of an abstract complex vector space as a ket|ϕ⟩{\displaystyle |\phi \rangle }, to refer to it as a "ket" rather than as a vector, and to pronounce it "ket-ϕ{\displaystyle \phi }" or "ket-A" for|A⟩.

Symbols, letters, numbers, or even words—whatever serves as a convenient label—can be used as the label inside a ket, with the| ⟩{\displaystyle |\ \rangle } making clear that the label indicates a vector in vector space. In other words, the symbol "|A⟩" has a recognizable mathematical meaning as to the kind of variable being represented, while just the "A" by itself does not. For example,|1⟩ +|2⟩ is not necessarily equal to|3⟩. Nevertheless, for convenience, there is usually some logical scheme behind the labels inside kets, such as the common practice of labelingenergy eigenkets in quantum mechanics through a listing of theirquantum numbers. At its simplest, the label inside the ket is the eigenvalue of a physical operator, such asx^{\displaystyle {\hat {x}}},p^{\displaystyle {\hat {p}}},L^z{\displaystyle {\hat {L}}_{z}}, etc.

Notation

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

|A⟩=|B⟩+|C⟩|C⟩=(−1+2i)|D⟩|D⟩=∫−∞∞e−x2|x⟩dx.{\displaystyle {\begin{aligned}|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{aligned}}}

Note how the last line above involves infinitely many different kets, one for each real numberx.

Since the ket is an element of a vector space, abra⟨A|{\displaystyle \langle A|} is an element of itsdual 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⟨ϕ|{\displaystyle \langle \phi |} and a ket|ψ⟩{\displaystyle |\psi \rangle } (i.e. a functional and a vector), can be combined to an operator|ψ⟩⟨ϕ|{\displaystyle |\psi \rangle \langle \phi |} of rank one withouter product

|ψ⟩⟨ϕ|:|ξ⟩↦|ψ⟩⟨ϕ|ξ⟩ .{\displaystyle |\psi \rangle \langle \phi |\colon |\xi \rangle \mapsto |\psi \rangle \langle \phi |\xi \rangle ~.}

Inner product and bra–ket identification on Hilbert space

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Main article:Inner product

The bra–ket notation is particularly useful in Hilbert spaces which have an inner product[5] that allowsHermitian conjugation and identifying a vector with a continuous linear functional, i.e. a ket with a bra, and vice versa (seeRiesz representation theorem). The inner product on Hilbert space( , ){\displaystyle (\ ,\ )} (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 ketψ=|ψ⟩{\displaystyle \psi =|\psi \rangle } define a functional (i.e. bra)fϕ=⟨ϕ|{\displaystyle f_{\phi }=\langle \phi |} by

(ϕ,ψ)=(|ϕ⟩,|ψ⟩)=:fϕ(ψ)=⟨ϕ|(|ψ⟩)=:⟨ϕ∣ψ⟩{\displaystyle (\phi ,\psi )=(|\phi \rangle ,|\psi \rangle )=:f_{\phi }(\psi )=\langle \phi |\,{\bigl (}|\psi \rangle {\bigr )}=:\langle \phi {\mid }\psi \rangle }

Bras and kets as row and column vectors

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In the simple case where we consider the vector spaceCn{\displaystyle \mathbb {C} ^{n}}, a ket can be identified with acolumn vector, and a bra as arow vector. If, moreover, we use the standard Hermitian inner product onCn{\displaystyle \mathbb {C} ^{n}}, the bra corresponding to a ket, in particular a bra⟨m| and a ket|m⟩ with the same label areconjugate transpose. Moreover, conventions are set up in such a way that writing bras, kets, and linear operators next to each other simply implymatrix multiplication.[6] In particular the outer product|ψ⟩⟨ϕ|{\displaystyle |\psi \rangle \langle \phi |} 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 fixedorthonormal basis, the inner product can be written as a matrix multiplication of a row vector with a column vector:⟨A|B⟩≐A1∗B1+A2∗B2+⋯+AN∗BN=(A1∗A2∗⋯AN∗)(B1B2⋮BN){\displaystyle \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}}}Based on this, the bras and kets can be defined as:⟨A|≐(A1∗A2∗⋯AN∗)|B⟩≐(B1B2⋮BN){\displaystyle {\begin{aligned}\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{aligned}}}and then it is understood that a bra next to a ket implies matrix multiplication.

Theconjugate transpose (also calledHermitian conjugate) of a bra is the corresponding ket and vice versa:⟨A|†=|A⟩,|A⟩†=⟨A|{\displaystyle \langle A|^{\dagger }=|A\rangle ,\quad |A\rangle ^{\dagger }=\langle A|}because if one starts with the bra(A1∗A2∗⋯AN∗),{\displaystyle {\begin{pmatrix}A_{1}^{*}&A_{2}^{*}&\cdots &A_{N}^{*}\end{pmatrix}}\,,}then performs acomplex conjugation, and then amatrix transpose, one ends up with the ket(A1A2⋮AN){\displaystyle {\begin{pmatrix}A_{1}\\A_{2}\\\vdots \\A_{N}\end{pmatrix}}}

Writing elements of a finite dimensional (ormutatis mutandis, countably infinite) vector space as a column vector of numbers requires picking abasis. 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 "|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 "|−⟩" and "|+⟩".

Non-normalizable states and non-Hilbert spaces

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Bra–ket notation can be used even if the vector space is not aHilbert space.

In quantum mechanics, it is common practice to write down kets which have infinitenorm, i.e. non-normalizable wavefunctions. Examples include states whose wavefunctions areDirac delta functions or infiniteplane waves. These do not, technically, belong to theHilbert space itself. However, the definition of "Hilbert space" can be broadened to accommodate these states (see theGelfand–Naimark–Segal construction orrigged Hilbert spaces). 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 spaceB, the vectors may be notated by kets and the continuouslinear functionals by bras. Over any vector space without a giventopology, 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 theRiesz representation theorem does not apply.

Usage in quantum mechanics

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The mathematical structure of quantum mechanics is based in large part onlinear algebra:

  • Wave functions and other quantum states can be represented as vectors in aseparablecomplexHilbert 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 arerays of vectors in the Hilbert space, asc|ψ⟩ corresponds to the same state for any nonzero complex numberc.)
  • Quantum superpositions can be described as vector sums of the constituent states. For example, an electron in the state⁠1/√2⁠|1⟩ +⁠i/√2⁠|2⟩ is in a quantum superposition of the states|1⟩ and|2⟩.
  • Measurements are associated with linear operators (calledobservables) on the Hilbert space of quantum states.
  • Dynamics are also described by linear operators on the Hilbert space. For example, in theSchrödinger picture, there is a lineartime evolution operatorU with the property that if an electron is in state|ψ⟩ right now, at a later time it will be in the stateU|ψ⟩, the sameU for every possible|ψ⟩.
  • Wave function normalization is scaling a wave function so that itsnorm 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

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Discrete componentsAk of a complex vector|A⟩ = ΣkAk|ek⟩.
Continuous componentsψ(x) of a complex vector|ψ⟩ = ∫ dxψ(x)|x⟩.
Components of complex vectors plotted against index number; discretek and continuousx. Two particular components out of infinitely many are highlighted.

The Hilbert space of aspin-0 point particle can be represented in terms of a "positionbasis"{|r⟩ }, where the labelr extends over the set of all points inposition space. These states satisfy the eigenvalue equation for theposition operator:r^|r⟩=r|r⟩.{\displaystyle {\hat {\mathbf {r} }}|\mathbf {r} \rangle =\mathbf {r} |\mathbf {r} \rangle .}The position states are "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 thedomain of definition of its wavefunctions. That is, starting from any ket|Ψ⟩ in this Hilbert space, one maydefine a complex scalar function ofr, known as awavefunction,Ψ(r) =def ⟨r|Ψ⟩.{\displaystyle \Psi (\mathbf {r} )\ {\stackrel {\text{def}}{=}}\ \langle \mathbf {r} |\Psi \rangle \,.}

On the left-hand side,Ψ(r) is a function mapping any point in space to a complex number; on the right-hand side,|Ψ⟩=∫d3rΨ(r)|r⟩{\displaystyle \left|\Psi \right\rangle =\int d^{3}\mathbf {r} \,\Psi (\mathbf {r} )\left|\mathbf {r} \right\rangle } 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, byA^(r) Ψ(r) =def ⟨r|A^|Ψ⟩.{\displaystyle {\hat {A}}(\mathbf {r} )~\Psi (\mathbf {r} )\ {\stackrel {\text{def}}{=}}\ \langle \mathbf {r} |{\hat {A}}|\Psi \rangle \,.}

For instance, themomentum operatorp^{\displaystyle {\hat {\mathbf {p} }}} has the following coordinate representation,p^(r) Ψ(r) =def ⟨r|p^|Ψ⟩=−iℏ∇Ψ(r).{\displaystyle {\hat {\mathbf {p} }}(\mathbf {r} )~\Psi (\mathbf {r} )\ {\stackrel {\text{def}}{=}}\ \langle \mathbf {r} |{\hat {\mathbf {p} }}|\Psi \rangle =-i\hbar \nabla \Psi (\mathbf {r} )\,.}

One occasionally even encounters an expression such as∇|Ψ⟩{\displaystyle \nabla |\Psi \rangle }, though this is something of anabuse 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,∇⟨r|Ψ⟩,{\displaystyle \nabla \langle \mathbf {r} |\Psi \rangle \,,}even though, in the momentum basis, this operator amounts to a mere multiplication operator (byiħp). That is, to say,⟨r|p^=−iℏ∇⟨r| ,{\displaystyle \langle \mathbf {r} |{\hat {\mathbf {p} }}=-i\hbar \nabla \langle \mathbf {r} |~,}orp^=∫d3r |r⟩(−iℏ∇)⟨r| .{\displaystyle {\hat {\mathbf {p} }}=\int d^{3}\mathbf {r} ~|\mathbf {r} \rangle (-i\hbar \nabla )\langle \mathbf {r} |~.}

Overlap of states

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In quantum mechanics the expression⟨φ|ψ⟩ is typically interpreted as theprobability amplitude for the stateψ tocollapse into the stateφ. Mathematically, this means the coefficient for the projection ofψ ontoφ. It is also described as the projection of stateψ onto stateφ.

Changing basis for a spin-1/2 particle

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A stationaryspin-1⁄2 particle has a two-dimensional Hilbert space. Oneorthonormal basis is:|↑z⟩,|↓z⟩{\displaystyle |{\uparrow }_{z}\rangle \,,\;|{\downarrow }_{z}\rangle }where|↑z⟩ is the state with a definite value of thespin operatorSz equal to +1⁄2 and|↓z⟩ is the state with a definite value of thespin operatorSz equal to −1⁄2.

Since these are a basis,any quantum state of the particle can be expressed as alinear combination (i.e.,quantum superposition) of these two states:|ψ⟩=aψ|↑z⟩+bψ|↓z⟩{\displaystyle |\psi \rangle =a_{\psi }|{\uparrow }_{z}\rangle +b_{\psi }|{\downarrow }_{z}\rangle }whereaψ andbψ are complex numbers.

Adifferent basis for the same Hilbert space is:|↑x⟩,|↓x⟩{\displaystyle |{\uparrow }_{x}\rangle \,,\;|{\downarrow }_{x}\rangle }defined in terms ofSx rather thanSz.

Again,any state of the particle can be expressed as a linear combination of these two:|ψ⟩=cψ|↑x⟩+dψ|↓x⟩{\displaystyle |\psi \rangle =c_{\psi }|{\uparrow }_{x}\rangle +d_{\psi }|{\downarrow }_{x}\rangle }

In vector form, you might write|ψ⟩≐(aψbψ)or|ψ⟩≐(cψdψ){\displaystyle |\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}}}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 betweenaψ{\displaystyle a_{\psi }},bψ{\displaystyle b_{\psi }},cψ{\displaystyle c_{\psi }} anddψ{\displaystyle d_{\psi }}; seechange of basis.

Pitfalls and ambiguous uses

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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

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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 asψ{\displaystyle {\boldsymbol {\psi }}}, and(⋅,⋅){\displaystyle (\cdot ,\cdot )} for the inner product. Consider the following dual space bra-vector in the basis{|en⟩}{\displaystyle \{|e_{n}\rangle \}}, where{ψn}{\displaystyle \{\psi _{n}\}} are the complex number coefficients of⟨ψ|{\displaystyle \langle \psi |}:⟨ψ|=∑n⟨en|ψn{\displaystyle \langle \psi |=\sum _{n}\langle e_{n}|\psi _{n}}

It has to be determined by convention if the complex numbers{ψn}{\displaystyle \{\psi _{n}\}} are inside or outside of the inner product, and each convention gives different results.

⟨ψ|≡(ψ,⋅)=∑n(en,⋅)ψn{\displaystyle \langle \psi |\equiv ({\boldsymbol {\psi }},\cdot )=\sum _{n}({\boldsymbol {e}}_{n},\cdot )\,\psi _{n}}⟨ψ|≡(ψ,⋅)=∑n(enψn,⋅)=∑n(en,⋅)ψn∗{\displaystyle \langle \psi |\equiv ({\boldsymbol {\psi }},\cdot )=\sum _{n}({\boldsymbol {e}}_{n}\psi _{n},\cdot )=\sum _{n}({\boldsymbol {e}}_{n},\cdot )\,\psi _{n}^{*}}

Reuse of symbols

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It is common to use the same symbol forlabels andconstants. For example,α^|α⟩=α|α⟩{\displaystyle {\hat {\alpha }}|\alpha \rangle =\alpha |\alpha \rangle }, where the symbolα{\displaystyle \alpha } is used simultaneously as thename of the operatorα^{\displaystyle {\hat {\alpha }}}, itseigenvector|α⟩{\displaystyle |\alpha \rangle } and the associatedeigenvalueα{\displaystyle \alpha }. Sometimes thehat is also dropped for operators, and one can see notation such asA|a⟩=a|a⟩{\displaystyle A|a\rangle =a|a\rangle }.[7]

Hermitian conjugate of kets

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It is common to see the usage|ψ⟩†=⟨ψ|{\displaystyle |\psi \rangle ^{\dagger }=\langle \psi |}, where the dagger (†{\displaystyle \dagger }) corresponds to the Hermitian conjugate. This is however not correct in a technical sense, since the ket,|ψ⟩{\displaystyle |\psi \rangle }, represents avector in a complex Hilbert-spaceH{\displaystyle {\mathcal {H}}}, and the bra,⟨ψ|{\displaystyle \langle \psi |}, is alinear functional on vectors inH{\displaystyle {\mathcal {H}}}. In other words,|ψ⟩{\displaystyle |\psi \rangle } is just a vector, while⟨ψ|{\displaystyle \langle \psi |} is the combination of a vector and an inner product.

Operations inside bras and kets

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This is done for a fast notation of scaling vectors. For instance, if the vector|α⟩{\displaystyle |\alpha \rangle } is scaled by1/2{\displaystyle 1/{\sqrt {2}}}, it may be denoted|α/2⟩{\displaystyle |\alpha /{\sqrt {2}}\rangle }. This can be ambiguous sinceα{\displaystyle \alpha } 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 movedoutside the designed slot, e.g.|α⟩=|α/2⟩1⊗|α/2⟩2{\displaystyle |\alpha \rangle =|\alpha /{\sqrt {2}}\rangle _{1}\otimes |\alpha /{\sqrt {2}}\rangle _{2}}.

Linear operators

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See also:Linear operator

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 havecertain properties.) In other words, ifA^{\displaystyle {\hat {A}}} is a linear operator and|ψ⟩{\displaystyle |\psi \rangle } is a ket-vector, thenA^|ψ⟩{\displaystyle {\hat {A}}|\psi \rangle } is another ket-vector.

In anN{\displaystyle N}-dimensional Hilbert space, we can impose a basis on the space and represent|ψ⟩{\displaystyle |\psi \rangle } in terms of its coordinates as aN×1{\displaystyle N\times 1}column vector. Using the same basis forA^{\displaystyle {\hat {A}}}, it is represented by anN×N{\displaystyle N\times N} complex matrix. The ket-vectorA^|ψ⟩{\displaystyle {\hat {A}}|\psi \rangle } can now be computed by matrix multiplication.

Linear operators are ubiquitous in the theory of quantum mechanics. For example, observable physical quantities are represented byself-adjoint operators, such asenergy ormomentum, whereas transformative processes are represented byunitary linear operators such as rotation or the progression of time.

Linear operators acting on bras

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Operators can also be viewed as acting on brasfrom the right hand side. Specifically, ifA is a linear operator and⟨φ| is a bra, then⟨φ|A is another bra defined by the rule(⟨ϕ|A)|ψ⟩=⟨ϕ|(A|ψ⟩),{\displaystyle {\bigl (}\langle \phi |{\boldsymbol {A}}{\bigr )}|\psi \rangle =\langle \phi |{\bigl (}{\boldsymbol {A}}|\psi \rangle {\bigr )}\,,}(in other words, afunction composition). This expression is commonly written as (cf.energy inner product)⟨ϕ|A|ψ⟩.{\displaystyle \langle \phi |{\boldsymbol {A}}|\psi \rangle \,.}

In anN-dimensional Hilbert space,⟨φ| can be written as a1 ×Nrow vector, andA (as in the previous section) is anN ×N matrix. Then the bra⟨φ|A can be computed by normal matrix multiplication.

If the same state vector appears on both bra and ket side,⟨ψ|A|ψ⟩,{\displaystyle \langle \psi |{\boldsymbol {A}}|\psi \rangle \,,}then this expression gives theexpectation value, or mean or average value, of the observable represented by operatorA for the physical system in the state|ψ⟩.

Outer products

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A convenient way to define linear operators on a Hilbert spaceH is given by theouter product: if⟨ϕ| is a bra and|ψ⟩ is a ket, the outer product|ϕ⟩⟨ψ|{\displaystyle |\phi \rangle \,\langle \psi |}denotes therank-one operator with the rule(|ϕ⟩⟨ψ|)(x)=⟨ψ|x⟩|ϕ⟩.{\displaystyle {\bigl (}|\phi \rangle \langle \psi |{\bigr )}(x)=\langle \psi |x\rangle |\phi \rangle .}

For a finite-dimensional vector space, the outer product can be understood as simple matrix multiplication:|ϕ⟩⟨ψ|≐(ϕ1ϕ2⋮ϕN)(ψ1∗ψ2∗⋯ψN∗)=(ϕ1ψ1∗ϕ1ψ2∗⋯ϕ1ψN∗ϕ2ψ1∗ϕ2ψ2∗⋯ϕ2ψN∗⋮⋮⋱⋮ϕNψ1∗ϕNψ2∗⋯ϕNψN∗){\displaystyle |\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}}}The outer product is anN ×N matrix, as expected for a linear operator.

One of the uses of the outer product is to constructprojection operators. Given a ket|ψ⟩ of norm 1, the orthogonal projection onto thesubspace spanned by|ψ⟩ is|ψ⟩⟨ψ|.{\displaystyle |\psi \rangle \,\langle \psi |\,.}This is anidempotent in the algebra of observables that acts on the Hilbert space.

Hermitian conjugate operator

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Main article:Hermitian conjugate

Just as kets and bras can be transformed into each other (making|ψ⟩ into⟨ψ|), the element from the dual space corresponding toA|ψ⟩ is⟨ψ|A†, whereA† denotes the Hermitian conjugate (or adjoint) of the operatorA. In other words,|ϕ⟩=A|ψ⟩if and only if⟨ϕ|=⟨ψ|A†.{\displaystyle |\phi \rangle =A|\psi \rangle \quad {\text{if and only if}}\quad \langle \phi |=\langle \psi |A^{\dagger }\,.}

IfA is expressed as anN ×N matrix, thenA† is its conjugate transpose.

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,c1 andc2 denote arbitrarycomplex numbers,c* denotes thecomplex conjugate ofc,A andB denote arbitrary linear operators, and these properties are to hold for any choice of bras and kets.

Linearity

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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., theassociative property holds). For example:

⟨ψ|(A|ϕ⟩)=(⟨ψ|A)|ϕ⟩=def⟨ψ|A|ϕ⟩(A|ψ⟩)⟨ϕ|=A(|ψ⟩⟨ϕ|)=defA|ψ⟩⟨ϕ|{\displaystyle {\begin{aligned}\langle \psi |{\bigl (}A|\phi \rangle {\bigr )}={\bigl (}\langle \psi |A{\bigr )}|\phi \rangle \,&{\stackrel {\text{def}}{=}}\,\langle \psi |A|\phi \rangle \\{\bigl (}A|\psi \rangle {\bigr )}\langle \phi |=A{\bigl (}|\psi \rangle \langle \phi |{\bigr )}\,&{\stackrel {\text{def}}{=}}\,A|\psi \rangle \langle \phi |\end{aligned}}}

and so forth. The expressions on the right (with no parentheses whatsoever) are allowed to be written unambiguouslybecause of the equalities on the left. Note that the associative property doesnot hold for expressions that include nonlinear operators, such as theantilineartime reversal operator in physics.

Hermitian conjugation

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Bra–ket notation makes it particularly easy to compute the Hermitian conjugate (also calleddagger, and denoted†) 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.,(x†)†=x.{\displaystyle \left(x^{\dagger }\right)^{\dagger }=x\,.}
  • 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:

Composite bras and kets

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Two Hilbert spacesV andW may form a third spaceV ⊗W by atensor product. In quantum mechanics, this is used for describing composite systems. If a system is composed of two subsystems described inV andW 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 actuallyidentical particles. In that case, the situation is a little more complicated.)[citation needed]

If|ψ⟩ is a ket inV and|φ⟩ is a ket inW, the tensor product of the two kets is a ket inV ⊗W. This is written in various notations:

|ψ⟩|ϕ⟩,|ψ⟩⊗|ϕ⟩,|ψϕ⟩,|ψ,ϕ⟩.{\displaystyle |\psi \rangle |\phi \rangle \,,\quad |\psi \rangle \otimes |\phi \rangle \,,\quad |\psi \phi \rangle \,,\quad |\psi ,\phi \rangle \,.}

Seequantum entanglement and theEPR paradox for applications of this product.

The unit operator

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Consider a completeorthonormal system (basis),{ei | i∈N},{\displaystyle \{e_{i}\ |\ i\in \mathbb {N} \}\,,}for a Hilbert spaceH, with respect to the norm from an inner product⟨·,·⟩.

From basicfunctional analysis, it is known that any ket|ψ⟩{\displaystyle |\psi \rangle } can also be written as|ψ⟩=∑i∈N⟨ei|ψ⟩|ei⟩,{\displaystyle |\psi \rangle =\sum _{i\in \mathbb {N} }\langle e_{i}|\psi \rangle |e_{i}\rangle ,}with⟨·|·⟩ the inner product on the Hilbert space.

From the commutativity of kets with (complex) scalars, it follows that∑i∈N|ei⟩⟨ei|=I{\displaystyle \sum _{i\in \mathbb {N} }|e_{i}\rangle \langle e_{i}|=\mathbb {I} }must be theidentity operator, which sends each vector to itself.

This, then, can be inserted in any expression without affecting its value; for example⟨v|w⟩=⟨v|(∑i∈N|ei⟩⟨ei|)|w⟩=⟨v|(∑i∈N|ei⟩⟨ei|)(∑j∈N|ej⟩⟨ej|)|w⟩=⟨v|ei⟩⟨ei|ej⟩⟨ej|w⟩,{\displaystyle {\begin{aligned}\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{aligned}}}where, in the last line, theEinstein summation convention has been used to avoid clutter.

In quantum mechanics, it often occurs that little or no information about the inner product⟨ψ|φ⟩ of two arbitrary (state) kets is present, while it is still possible to say something about the expansion coefficients⟨ψ|ei⟩ =⟨ei|ψ⟩* and⟨ei|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, seeResolution of the identity,[9]I=∫dx |x⟩⟨x|=∫dp |p⟩⟨p|,{\displaystyle {\mathbb {I} }=\int \!dx~|x\rangle \langle x|=\int \!dp~|p\rangle \langle p|,} where|p⟩=∫dxeixp/ℏ|x⟩2πℏ.{\displaystyle |p\rangle =\int dx{\frac {e^{ixp/\hbar }|x\rangle }{\sqrt {2\pi \hbar }}}.}

Since⟨x′|x⟩ =δ(x −x′), plane waves follow,⟨x|p⟩=eixp/ℏ2πℏ.{\displaystyle \langle x|p\rangle ={\frac {e^{ixp/\hbar }}{\sqrt {2\pi \hbar }}}.}

In his book (1958), Ch. III.20, Dirac defines thestandard ket which, up to a normalization, is the translationally invariant momentum eigenstate|ϖ⟩=limp→0|p⟩{\textstyle |\varpi \rangle =\lim _{p\to 0}|p\rangle } in the momentum representation, i.e.,p^|ϖ⟩=0{\displaystyle {\hat {p}}|\varpi \rangle =0}. Consequently, the corresponding wavefunction is a constant,⟨x|ϖ⟩2πℏ=1{\displaystyle \langle x|\varpi \rangle {\sqrt {2\pi \hbar }}=1}, and|x⟩=δ(x^−x)|ϖ⟩2πℏ,{\displaystyle |x\rangle =\delta ({\hat {x}}-x)|\varpi \rangle {\sqrt {2\pi \hbar }},} as well as|p⟩=exp⁡(ipx^/ℏ)|ϖ⟩.{\displaystyle |p\rangle =\exp(ip{\hat {x}}/\hbar )|\varpi \rangle .}

Typically, when all matrix elements of an operator such as⟨x|A|y⟩{\displaystyle \langle x|A|y\rangle } are available, this resolution serves to reconstitute the full operator,∫dxdy|x⟩⟨x|A|y⟩⟨y|=A.{\displaystyle \int dx\,dy\,|x\rangle \langle x|A|y\rangle \langle y|=A\,.}

Notation used by mathematicians

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The object physicists are considering when using bra–ket notation is a Hilbert space (acomplete inner product space).

Let(H,⟨⋅,⋅⟩){\displaystyle ({\mathcal {H}},\langle \cdot ,\cdot \rangle )} be a Hilbert space andh ∈H a vector inH. What physicists would denote by|h⟩ is the vector itself. That is,|h⟩∈H.{\displaystyle |h\rangle \in {\mathcal {H}}.}

LetH* be the dual space ofH. This is the space of linear functionals onH. TheembeddingΦ:H↪H∗{\displaystyle \Phi :{\mathcal {H}}\hookrightarrow {\mathcal {H}}^{*}} is defined byΦ(h)=φh{\displaystyle \Phi (h)=\varphi _{h}}, where for everyh ∈H the linear functionalφh:H→C{\displaystyle \varphi _{h}:{\mathcal {H}}\to \mathbb {C} } satisfies for everyg ∈H the functional equationφh(g)=⟨h,g⟩=⟨h∣g⟩{\displaystyle \varphi _{h}(g)=\langle h,g\rangle =\langle h\mid g\rangle }.Notational confusion arises when identifyingφh andg with⟨h| and|g⟩ respectively. This is because of literal symbolic substitutions. Letφh=H=⟨h∣{\displaystyle \varphi _{h}=H=\langle h\mid } and letg = G =|g⟩. This givesφh(g)=H(g)=H(G)=⟨h|(G)=⟨h|(|g⟩).{\displaystyle \varphi _{h}(g)=H(g)=H(G)=\langle h|(G)=\langle h|{\bigl (}|g\rangle {\bigr )}\,.}

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 anasterisk but an overline (which the physicists reserve for averages and theDirac spinor adjoint) to denotecomplex conjugate numbers; i.e., for scalar products mathematicians usually write⟨ϕ,ψ⟩=∫ϕ(x)ψ(x)¯dx,{\displaystyle \langle \phi ,\psi \rangle =\int \phi (x){\overline {\psi (x)}}\,dx\,,}whereas physicists would write for the same quantity⟨ψ|ϕ⟩=∫dxψ∗(x)ϕ(x) .{\displaystyle \langle \psi |\phi \rangle =\int dx\,\psi ^{*}(x)\phi (x)~.}

See also

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Notes

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  1. ^abDirac 1939
  2. ^Shankar 1994, Chapter 1
  3. ^Grassmann 1862
  4. ^Lecture 2 | Quantum Entanglements, Part 1 (Stanford), Leonard Susskind on complex numbers, complex conjugate, bra, ket. 2006-10-02.
  5. ^Lecture 2 | Quantum Entanglements, Part 1 (Stanford), Leonard Susskind on inner product, 2006-10-02.
  6. ^"Gidney, Craig (2017). Bra–Ket Notation Trivializes Matrix Multiplication".
  7. ^Sakurai & Napolitano 2021 Sec 1.2, 1.3
  8. ^Lecture notes by Robert LittlejohnArchived 2012-06-17 at theWayback Machine, eqns 12 and 13
  9. ^Sakurai & Napolitano 2021 Sec 1.2, 1.3

References

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External links

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