Bra–ket notation

In quantum mechanics, bra–ket notation is a standard notation for describing quantum states, composed of angle brackets and vertical bars. It can also be used to denote abstract vectors and linear functionals in mathematics. In such terms, the scalar product, or action of a linear functional on a vector in a complex vector space, is denoted by

consisting of a left part, called the bra /brɑː/, and a right part, , called the ket /kɛt/. The notation was introduced in 1939 by Paul Dirac and is also known as Dirac notation, though the notation has precursors in Grassmann's use of the notation for his inner products nearly 100 years earlier.

Bra–ket notation is widespread in quantum mechanics: almost every phenomenon that is explained using quantum mechanics – including a large portion of modern physics – is usually explained with the help of bra-ket notation. Part of the appeal of the notation is the abstract representation-independence it encodes, together with its versatility in producing a specific representation (e.g. x, or p, or eigenfunction base) without much ado, or excessive reliance on the nature of the linear spaces involved. The overlap expression is typically interpreted as the probability amplitude for the state ψ to collapse into the state φ.