Braket notation
In quantum mechanics, bra–ket notation is a standard notation for describing quantum states. It can also be used to denote abstract vectors and linear functionals in mathematics. The notation begins with using angle brackets, ⟨ and ⟩, and a vertical bar, |, to denote the scalar product of vectors or the action of a linear functional on a vector in a complex vector space. The scalar product or action is written as
The right part is called the ket /kɛt/; it is a vector, typically represented as a column vector and written
The left part is called the bra, /brɑː/; it is the Hermitian conjugate of the ket with the same label, typically represented as a row vector and is written
A combination of bras, kets, and operators is interpreted using matrix multiplication. A bra and a ket with the same label are Hermitian conjugates of each other.
Bra-ket notation was introduced in 1939 by Paul Dirac and is also known as the Dirac notation.
The bra-ket notation has a precursor in Hermann Grassmann's use of the notation for his inner products nearly 100 years earlier.
![[\phi \mid \psi ]](https://wikimedia.org/api/rest_v1/media/math/render/svg/f3cdbb361cd858109b3c4f847d21b989a73bacba)
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