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Concept
Bra–ket notation
Summary
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.
The name comes from the English word "Bracket".
Paul Dirac created Bra-ket notation in 1939 to make it easier to write quantum mechanical equations. In his 1939 publication, A New Notation for Quantum Mechanics, Dirac wrote:
A Hilbert-space vector, which was denoted in the old notation by the letter ψ, will now be denoted by a special new symbol . If we are concerned with a particular vector, specified by a label, a say, which would be used as a suffix to the ψ in the old notation, we write it .
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.
Official source
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