Ket vector

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 uses angle brackets ⟨ and ⟩, and vertical bars | to denote the scalar product, or action of a linear functional on a vector in a complex vector space.

The left part, called the bra /brɑː/, is an element of the dual space (e.g. a row vector):

The right part, called the ket /kɛt/, is an element of a vector space (e.g. a column vector):

Any combination of bras, kets, and operators are interpreted using matrix multiplication. A bra and a ket with the same label are Hermitian conjugates of each other.

The notation was introduced in 1939 by Paul Dirac and is also known as the Dirac notation, though the notation has precursors in Hermann Grassmann's use of the notation

for his inner products nearly 100 years earlier.

Bra–ket notation is a notation for linear algebra, particularly focused on vectors, inner products, linear operators, Hermitian conjugation, and the dual space, for both finite-dimensional and infinite-dimensional complex vector spaces. It is specifically designed to ease the types of calculations that frequently come up in quantum mechanics.

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