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
Quantum mechanics is the fundamental physical Scientific theory, theory that describes the behavior of matter and of light; its unusual characteristics typically occur at and below the scale of atoms. Reprinted, Addison-Wesley, 1989, It is ...
. 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
Quantum mechanics is the fundamental physical Scientific theory, theory that describes the behavior of matter and of light; its unusual characteristics typically occur at and below the scale of atoms. Reprinted, Addison-Wesley, 1989, It is ...
and
quantum computing, bra–ket notation is used ubiquitously to denote
quantum state
In quantum physics, a quantum state is a mathematical entity that embodies the knowledge of a quantum system. Quantum mechanics specifies the construction, evolution, and measurement of a quantum state. The result is a prediction for the system ...
s. The notation uses
angle bracket
A bracket is either of two tall fore- or back-facing punctuation marks commonly used to isolate a segment of text or data from its surroundings. They come in four main pairs of shapes, as given in the box to the right, which also gives their n ...
s, 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
In mathematics and physics, a vector space (also called a linear space) is a set (mathematics), set whose elements, often called vector (mathematics and physics), ''vectors'', can be added together and multiplied ("scaled") by numbers called sc ...
, 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
In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that p ...
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 forms 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 onto 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