Bra–ket notation or Dirac notation is a mathematical notation for linear algebra and linear operators on complex vector spaces together with their dual spaces both in the finite- and infinite-dimensional cases. It is specifically designed to ease the types of calculations that frequently arise in quantum mechanics. It is now of ubiquitous usage in that subject. Bra–ket notation was created by Paul Dirac in his paper, "A New Notation for Quantum Mechanics" from 1939. 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 states. The notation uses angle brackets, ⟨ {\displaystyle \langle } and ⟩ {\displaystyle \rangle } , and a vertical bar | {\displaystyle |} , to construct "bras" and "kets". A ket is of the form | v ⟩ {\displaystyle |v\rangle } . Mathematically it denotes a vector, v {\displaystyle {\boldsymbol {v}}} , in an abstract (complex) vector space V {\displaystyle V} , and physically it represents a state of some quantum system. A bra is of the form ⟨ f | {\displaystyle \langle f|} . Mathematically it denotes a linear form f : V → C {\displaystyle f:V\to \mathbb {C} } , i.e. a linear map that maps each vector in V {\displaystyle V} to a number in the complex plane C {\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 on V {\displaystyle V} there exists an inner product ( ⋅ , ⋅ ) {\displaystyle (\cdot ,\cdot )} with antilinear first argument, which makes V {\displaystyle V} an inner product space. Then with this inner product each vector ϕ ≡ | 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 } . The linear form ⟨ ϕ | {\displaystyle \langle \phi |} is a covector to | ϕ ⟩ {\displaystyle |\phi \rangle } , and the set of all covectors forms a subspace of the dual vector space V ∨ {\displaystyle V^{\vee }} , to the initial vector space V {\displaystyle V} . The purpose of this linear form ⟨ ϕ | {\displaystyle \langle \phi |} can now be understood in terms of making projections onto the state ϕ , {\displaystyle {\boldsymbol {\phi }},} to find how linearly dependent two states are, etc. For the vector space C n {\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 using matrix multiplication. If C n {\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 the Hermitian conjugate (denoted † {\displaystyle \dagger } ). It is common to suppress the vector or linear form from bra–ket notation and only use a label inside the typography for the bra or ket. For example, the spin operator σ ^ z {\displaystyle {\hat {\sigma }}_{z}} on a two-dimensional space Δ {\displaystyle \Delta } of spinors has eigenvalues ± 1 2 {\textstyle \pm {\frac {1}{2}}} with eigenspinors ψ + , ψ − ∈ Δ {\displaystyle {\boldsymbol {\psi }}_{+},{\boldsymbol {\psi }}_{-}\in \Delta } . In bra–ket notation, this is typically denoted as ψ + = | + ⟩ {\displaystyle {\boldsymbol {\psi }}_{+}=|+\rangle } , and ψ − = | − ⟩ {\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 with Hermitian conjugate column and row vectors. Bra–ket notation was effectively established in 1939 by Paul Dirac; it is thus also known as Dirac notation, despite the notation having a precursor in Hermann Grassmann's use of [ ϕ ∣ ψ ] {\displaystyle [\phi {\mid }\psi ]} for inner products nearly 100 years earlier.
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 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 ( r → {\displaystyle {\vec {r}}} ), boldface ( p {\displaystyle \mathbf {p} } ) or indices ( v μ {\displaystyle v^{\mu }} ). 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 wavefunctions (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 ϕ {\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 labeling energy eigenkets in quantum mechanics through a listing of their quantum numbers. At its simplest, the label inside the ket is the eigenvalue of a physical operator, such as x ^ {\displaystyle {\hat {x}}} , p ^ {\displaystyle {\hat {p}}} , L ^ z {\displaystyle {\hat {L}}_{z}} , 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:
| A ⟩ = | B ⟩ + | C ⟩ | C ⟩ = ( − 1 + 2 i ) | D ⟩ | D ⟩ = ∫ − ∞ ∞ e − x 2 | x ⟩ d x . {\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 number x. Since the ket is an element of a vector space, a bra ⟨ A | {\displaystyle \langle A|} 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 ⟨ 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 with outer 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
Bra–ket notation is particularly useful in Hilbert spaces which have an inner product 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 ( , ) {\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 bra–ket notation: for a vector ket ψ = | ψ ⟩ {\displaystyle \psi =|\psi \rangle } define a functional (i.e. bra) f ϕ = ⟨ ϕ | {\displaystyle f_{\phi }=\langle \phi |} by
( ϕ , ψ ) = ( | ϕ ⟩ , | ψ ⟩ ) =: f ϕ ( 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 In the simple case where we consider the vector space C n {\displaystyle \mathbb {C} ^{n}} , 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 C n {\displaystyle \mathbb {C} ^{n}} , the bra corresponding to a ket, in particular a bra ⟨m| and a 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. 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 fixed orthonormal basis, the inner product can be written as a matrix multiplication of a row vector with a column vector:
⟨ A | B ⟩ ≐ A 1 ∗ B 1 + A 2 ∗ B 2 + ⋯ + A N ∗ B N = ( A 1 ∗ A 2 ∗ ⋯ A N ∗ ) ( B 1 B 2 ⋮ B N ) {\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 | ≐ ( A 1 ∗ A 2 ∗ ⋯ A N ∗ ) | B ⟩ ≐ ( B 1 B 2 ⋮ B N ) {\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. The conjugate transpose (also called Hermitian 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
( A 1 ∗ A 2 ∗ ⋯ A N ∗ ) , {\displaystyle {\begin{pmatrix}A_{1}^{*}&A_{2}^{*}&\cdots &A_{N}^{*}\end{pmatrix}}\,,}
then performs a complex conjugation, and then a matrix transpose, one ends up with the ket
( A 1 A 2 ⋮ A N ) {\displaystyle {\begin{pmatrix}A_{1}\\A_{2}\\\vdots \\A_{N}\end{pmatrix}}}
Writing elements of a finite dimensional (or mutatis mutandis, countably infinite) vector space as a column vector of numbers requires picking a 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 "|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 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, i.e. non-normalizable wavefunctions. Examples include states whose wavefunctions are Dirac delta functions or infinite plane waves. 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 spaces). 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 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 functions and other quantum states can be represented as vectors in a separable 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 rays of vectors in the Hilbert space, as c|ψ⟩ corresponds to the same state for any nonzero complex number c.) Quantum superpositions can be described as vector sums of the constituent states. For example, an electron in the state 1/√2|1⟩ + i/√2|2⟩ is in a quantum superposition of the states |1⟩ and |2⟩. Measurements are associated with linear operators (called observables) 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 U with the property that if an electron is in state |ψ⟩ right now, at a later time it will be in the state U|ψ⟩, the same U for every possible |ψ⟩. Wave function normalization is scaling a wave function so that its 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
The Hilbert space of a spin-0 point particle can be represented in terms of a "position basis" { |r⟩ }, where the label r extends over the set of all points in position space. These states satisfy the eigenvalue equation for the position 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 the domain of definition of its wavefunctions. That is, starting from any ket |Ψ⟩ in this Hilbert space, one may define a complex scalar function of r, known as a wavefunction,
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 ⟩ = ∫ d 3 r Ψ ( 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, by
A ^ ( 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, the momentum operator p ^ {\displaystyle {\hat {\mathbf {p} }}} has the following coordinate representation,
p ^ ( r ) Ψ ( r ) = def ⟨ r | p ^ | Ψ ⟩ = − i ℏ ∇ 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 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, ∇ ⟨ r | Ψ ⟩ , {\displaystyle \nabla \langle \mathbf {r} |\Psi \rangle \,,}
even though, in the momentum basis, this operator amounts to a mere multiplication operator (by iħp). That is, to say,
⟨ r | p ^ = − i ℏ ∇ ⟨ r | , {\displaystyle \langle \mathbf {r} |{\hat {\mathbf {p} }}=-i\hbar \nabla \langle \mathbf {r} |~,}
or
p ^ = ∫ d 3 r | r ⟩ ( − i ℏ ∇ ) ⟨ r | . {\displaystyle {\hat {\mathbf {p} }}=\int d^{3}\mathbf {r} ~|\mathbf {r} \rangle (-i\hbar \nabla )\langle \mathbf {r} |~.}
Overlap of states In quantum mechanics the expression ⟨φ|ψ⟩ is typically interpreted as the probability amplitude for the state ψ to collapse into the state φ. Mathematically, this means the coefficient for the projection of ψ onto φ. It is also described as the projection of state ψ onto state φ.
Changing basis for a spin-1/2 particle A stationary spin-1⁄2 particle has a two-dimensional Hilbert space. One orthonormal basis is:
| ↑ z ⟩ , | ↓ z ⟩ {\displaystyle |{\uparrow }_{z}\rangle \,,\;|{\downarrow }_{z}\rangle }
where |↑z⟩ is the state with a definite value of the spin operator Sz equal to +1⁄2 and |↓z⟩ is the state with a definite value of the spin operator Sz 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:
| ψ ⟩ = a ψ | ↑ z ⟩ + b ψ | ↓ z ⟩ {\displaystyle |\psi \rangle =a_{\psi }|{\uparrow }_{z}\rangle +b_{\psi }|{\downarrow }_{z}\rangle }
where aψ and bψ are complex numbers. A different basis for the same Hilbert space is:
| ↑ x ⟩ , | ↓ x ⟩ {\displaystyle |{\uparrow }_{x}\rangle \,,\;|{\downarrow }_{x}\rangle }
defined in terms of Sx rather than Sz. Again, any state of the particle can be expressed as a linear combination of these two:
| ψ ⟩ = c ψ | ↑ x ⟩ + d ψ | ↓ x ⟩ {\displaystyle |\psi \rangle =c_{\psi }|{\uparrow }_{x}\rangle +d_{\psi }|{\downarrow }_{x}\rangle }
In vector form, you might write
| ψ ⟩ ≐ ( a ψ b ψ ) or | ψ ⟩ ≐ ( c ψ d ψ ) {\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 us