Bra–ket notation, also called Dirac notation, is a notation for linear algebra and linear operators on complex vector spaces together with their dual space both in the finite-dimensional and infinite-dimensional case. It is specifically designed to ease the types of calculations that frequently come up in quantum mechanics. Its use in quantum mechanics is quite widespread.
Bra-ket notation was created by Paul Dirac in his 1939 publication A New Notation for Quantum Mechanics. The notation was introduced as an easier way to write quantum mechanical expressions. The name comes from the English word "Bracket".
Quantum mechanics
In quantum mechanics, bra–ket notation is used ubiquitously to denote quantum states. The notation uses angle brackets, and , and a vertical bar , to construct "bras" and "kets".
A ket is of the form . Mathematically it denotes a vector, , in an abstract (complex) vector space , and physically it represents a state of some quantum system.
A bra is of the form . Mathematically it denotes a linear form , i.e. a linear map that maps each vector in to a number in the complex plane . Letting the linear functional act on a vector is written as .
Assume that on there exists an inner product with antilinear first argument, which makes an inner product space. Then with this inner product each vector can be identified with a corresponding linear form, by placing the vector in the anti-linear first slot of the inner product: . The correspondence between these notations is then . The linear form is a covector to , and the set of all covectors form a subspace of the dual vector space , to the initial vector space . The purpose of this linear form can now be understood in terms of making projections on the state , to find how linearly dependent two states are, etc.
For the vector space , 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 has the standard Hermitian inner product , under this identification, the identification of kets and bras and vice versa provided by the inner product is taking the Hermitian conjugate (denoted ).
It is common to suppress the vector or linear form from the bra–ket notation and only use a label inside the typography for the bra or ket. For example, the spin operator on a two dimensional space of spinors, has eigenvalues with eigenspinors . In bra–ket notation, this is typically denoted as , and . 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 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 (), boldface () or indices ().
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 of an abstract complex vector space as a ket , to refer to it as a "ket" rather than as a vector, and to pronounce it "ket-" or "ket-A" for .