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Chemistry LibreTexts

Bra-Ket Notation

Bra-ket notation is a standard notation for describing quantum states in the theory of quantum mechanics composed of angle brackets and vertical bars. It can also be used to denote abstract vectors and linear functionals in mathematics. It is so called because the inner product (or dot product) of two states is denoted by a bracket, \(\langle Φ|Ψ \rangle\), consisting of a left part, \(\langle Φ | \), called the bra, and a right part, \(|Ψ \rangle\), called the ket. The notation was introduced in 1939 by Paul Dirac, and is also known as Dirac notation. Bra-ket notation is widespread in quantum mechanics: almost every phenomenon that is explained using quantum mechanics—including a large proportion of modern physics—is usually explained with the help of bra-ket notation.