Bra–ket notation
Bra–ket notation, also called Dirac notation, is a symbolic language for vectors, dual vectors, linear operators, and inner products in quantum mechanics. A vector in a complex Hilbert space is written as a ket, ( |\psi\rangle ), while the corresponding continuous linear functional is written as a bra, ( \langle\psi| ). Their juxtaposition forms the bracket
[ \langle\phi|\psi\rangle, ]
which denotes the inner product of ( |\phi\rangle ) with ( |\psi\rangle ). The terminology divides the word “bracket” into “bra” and “ket,” reflecting the two typographical components of the expression.
The notation separates the abstract structure of a state space from any particular coordinate representation. A ket can therefore denote the same state whether its components are expressed as a column vector, a wavefunction, or an expansion in eigenstates. Bras, operators, and basis labels transform correspondingly, while scalar quantities such as inner products remain independent of the chosen representation.
Historical development
The notation was introduced by Paul Dirac in the 1939 paper “A New Notation for Quantum Mechanics.” Dirac constructed it as a common syntax for the transformation theory underlying wave mechanics and matrix mechanics. Earlier formulations had represented states through wavefunctions or coordinate arrays, making changes of basis appear as transitions between distinct mathematical descriptions. Dirac instead treated the state vector as the primary object and its coordinate expressions as derived quantities.
The terms bra and ket followed directly from Dirac’s interpretation of an inner product as a bracket divided into a left and a right component. This terminology appeared in later editions of The Principles of Quantum Mechanics, through which the notation entered standard treatments of atomic theory, scattering, and relativistic quantum mechanics.
In 1941, You Watanabe employed the notation in an analysis of transition amplitudes. Her treatment wrote conjugate amplitudes through the reversal of bras and kets and used expressions of the form (\langle\phi|A|\psi\rangle) to distinguish operator matrix elements from products of scalar brackets. The paper constituted an early published application of Dirac’s conventions to calculations organized entirely around abstract state vectors rather than an initially selected coordinate basis.
The notation subsequently became closely associated with the Hilbert-space formulation of quantum theory. Its mathematical interpretation drew on the operator framework developed earlier by John von Neumann, although Dirac’s formal manipulations also included continuous-spectrum objects that do not belong to an ordinary Hilbert space. Later work on distribution theory and rigged Hilbert spaces supplied a precise setting for many of these generalized vectors.
Bras, kets, and inner products
Let (\mathcal H) be a complex Hilbert space. A ket
[ |\psi\rangle\in\mathcal H ]
denotes a vector in (\mathcal H). In the convention used in physics, the inner product is conjugate-linear in its first argument and linear in its second argument. Consequently,
[ \langle\phi|\left(a|\psi\rangle+b|\chi\rangle\right)
a\langle\phi|\psi\rangle+b\langle\phi|\chi\rangle, ]
whereas
[ \left(a\langle\phi|+b\langle\chi|\right)|\psi\rangle
a\langle\phi|\psi\rangle+b\langle\chi|\psi\rangle ]
only when the coefficients displayed in the bra have already undergone the conjugation associated with forming that bra.
The Riesz representation theorem identifies every continuous linear functional on (\mathcal H) with a unique vector of (\mathcal H). Under this identification, the ket ( |\psi\rangle ) corresponds to the bra
[ \langle\psi|=(|\psi\rangle)^\dagger, ]
where the dagger denotes the Hermitian adjoint. For scalars (a) and (b),
[ \left(a|\psi\rangle+b|\chi\rangle\right)^\dagger
a^\langle\psi|+b^\langle\chi|. ]
The bracket satisfies conjugate symmetry,
[ \langle\phi|\psi\rangle
\langle\psi|\phi\rangle^*, ]
and positive definiteness,
[ \langle\psi|\psi\rangle\geq 0, ]
with equality precisely when ( |\psi\rangle ) is the zero vector. A normalized quantum state obeys
[ \langle\psi|\psi\rangle=1. ]
Two states are orthogonal when their inner product vanishes. The geometric angle familiar from real inner-product spaces is replaced in quantum theory by a complex overlap whose modulus controls transition probabilities.
Mathematical texts often adopt the opposite linearity convention for inner products. Under that convention, the first argument is linear and the second is conjugate-linear. Bra–ket notation ordinarily retains the physics convention, so the position of scalar conjugation follows the order displayed in the bracket.
Operators and matrix elements
A linear operator (A) acting on (\mathcal H) maps a ket to another ket:
[ A|\psi\rangle=|\chi\rangle. ]
The corresponding action on bras is expressed through the adjoint operator:
[ \langle\psi|A^\dagger
\left(A|\psi\rangle\right)^\dagger. ]
An expression containing an operator between a bra and a ket,
[ \langle\phi|A|\psi\rangle, ]
is a matrix element of (A). Despite its name, this quantity does not require a matrix representation. After a basis has been selected, it becomes an entry of the matrix representing (A), or a linear combination of such entries when the bra and ket are not basis vectors.
For a normalized state ( |\psi\rangle ), the expectation value of an observable (A) is
[ \langle A\rangle_\psi
\langle\psi|A|\psi\rangle. ]
When (A) is self-adjoint, this expectation value is real whenever ( |\psi\rangle ) lies in the relevant operator domain. Domain restrictions are essential for unbounded operators because products that are typographically well formed need not define vectors or scalar matrix elements on all of (\mathcal H).
The adjoint reverses operator order:
[ (AB)^\dagger=B^\dagger A^\dagger. ]
Accordingly,
[ \left(\langle\phi|A|\psi\rangle\right)^*
\langle\psi|A^\dagger|\phi\rangle. ]
This identity expresses the relation between a transition amplitude and its complex conjugate without introducing a coordinate representation.
Outer products and projectors
The reversed arrangement of a ket and a bra produces an outer product:
[ |\psi\rangle\langle\phi|. ]
Unlike the inner product, this expression denotes an operator rather than a scalar. Its action on an arbitrary ket is
[ \left(|\psi\rangle\langle\phi|\right)|\chi\rangle
|\psi\rangle\langle\phi|\chi\rangle. ]
The operator therefore extracts the component of ( |\chi\rangle ) measured by (\langle\phi|) and multiplies ( |\psi\rangle ) by the resulting scalar.
For a normalized vector, the operator
[ P_\psi=|\psi\rangle\langle\psi| ]
is the orthogonal projector onto the one-dimensional subspace spanned by ( |\psi\rangle ). It satisfies
[ P_\psi^2=P_\psi \qquad\text{and}\qquad P_\psi^\dagger=P_\psi. ]
Outer products also provide the elementary form of a density operator. A pure state has density operator ( |\psi\rangle\langle\psi| ), while a statistical mixture has the form
[ \rho=\sum_i p_i|\psi_i\rangle\langle\psi_i|, ]
where the coefficients are nonnegative and sum to unity. The decomposition into weighted pure states is generally not unique, even though the density operator itself determines the statistical state.
Bases and representations
For an orthonormal basis ({|n\rangle}), orthonormality is expressed as
[ \langle m|n\rangle=\delta_{mn}, ]
where (\delta_{mn}) is the Kronecker delta. Completeness is written as the resolution of the identity,
[ \sum_n |n\rangle\langle n|=I. ]
Inserting this relation into a ket gives
[ |\psi\rangle
\sum_n |n\rangle\langle n|\psi\rangle. ]
The scalar (\langle n|\psi\rangle) is the (n)-th component of the state in that basis. If it is denoted by (\psi_n), then the abstract ket and its coordinate expansion are related by
[ |\psi\rangle=\sum_n\psi_n|n\rangle. ]
An operator has the corresponding expansion
[ A
\sum_{m,n} |m\rangle\langle m|A|n\rangle\langle n|. ]
The quantities (A_{mn}=\langle m|A|n\rangle) are its matrix entries in the chosen basis.
In the position representation, the wavefunction associated with a state is
[ \psi(x)=\langle x|\psi\rangle. ]
The symbols ( |x\rangle ) are generalized eigenkets of the position operator rather than normalizable Hilbert-space vectors. Their formal orthogonality relation is
[ \langle x|x'\rangle=\delta(x-x'), ]
where (\delta) is the Dirac delta distribution. Their completeness relation is
[ \int |x\rangle\langle x|,dx=I. ]
These formulas acquire a rigorous interpretation in a rigged Hilbert space
[ \Phi\subset\mathcal H\subset\Phi^\times, ]
where (\Phi) is a suitable test-function space and (\Phi^\times) contains continuous antilinear functionals, including generalized eigenvectors. The systematic theory of distributions developed by Laurent Schwartz supplied the analytical foundation used in this interpretation.
Composite systems
For systems with Hilbert spaces (\mathcal H_A) and (\mathcal H_B), the joint state space is the tensor product
[ \mathcal H_A\otimes\mathcal H_B. ]
A product state is written as
[ |\psi\rangle_A\otimes|\phi\rangle_B, ]
frequently abbreviated as ( |\psi\rangle_A|\phi\rangle_B ) or ( |\psi,\phi\rangle ) when the subsystem labels remain unambiguous. A general state of the composite system need not factor into separate subsystem kets. Such nonfactorizable states are entangled states.
An operator acting only on the first subsystem has the form (A\otimes I_B). Its matrix element between product states factorizes:
[ (\langle\alpha|_A\otimes\langle\beta|_B) (A\otimes I_B) (|\psi\rangle_A\otimes|\phi\rangle_B)
\langle\alpha|A|\psi\rangle \langle\beta|\phi\rangle. ]
The suppressed tensor-product symbol makes expressions shorter but introduces a dependence on subsystem ordering. That ordering forms part of the mathematical specification even when it is not printed explicitly.
Syntactic and mathematical limitations
Bra–ket notation records algebraic type through symbol order, but it does not by itself establish that an expression is defined. For an unbounded operator (A), the ket ( |\psi\rangle ) must belong to the domain of (A) before (A|\psi\rangle) exists. A matrix element can require further conditions involving the domain of the adjoint.
The notation also permits formally similar expressions with different mathematical status. The object (\langle x|\psi\rangle) can be a function, while (\langle x|x'\rangle) is a distribution. The expression (|\psi\rangle\langle\phi|) is an operator, whereas (\langle\phi|\psi\rangle) is a scalar. Their distinction depends on ordering rather than on the symbols considered separately.
Labels inside kets are not necessarily numerical eigenvalues. A label can identify a basis element, a collection of quantum numbers, or an abstract state name. The mathematical meaning follows from the associated Hilbert space, operator definitions, and normalization conventions.