**Bra-ket notation** is the standard notation used for describing quantum mechanical states. It was invented by Paul Dirac. It is so called because the inner product of two states is denoted by a **bracket**, ⟨φ|ψ⟩, consisting of a left part, ⟨φ|, called the **bra**, and a right part, |ψ⟩, called the **ket**.

In quantum mechanics, the state of a physical system is identified with a vector in a Hilbert space, *H*. Each vector is called a ket, and written as

The bra-ket operation has the following properties:

- Given any bra ⟨φ|, kets |ψ
_{1}⟩ and |ψ_{2}⟩, and complex numbers*c*_{1}and*c*_{2}, then, since bras are*linear*functionals,

- Given any ket |ψ⟩, bras ⟨φ
_{1}| and ⟨φ_{2}|, and complex numbers*c*_{1}and*c*_{2}, then, by the definition of addition and scalar multiplication of linear functionals,

- Given any kets |ψ
_{1}⟩ and |ψ_{2}⟩, and complex numbers*c*_{1}and*c*_{2}, from the properties of the inner product (with "*" denoting the complex conjugate),

- Given any bra ⟨φ| and ket |ψ⟩, the inner product axiom gives

If *A* : *H* `->` *H* is a linear operator, we can apply *A* to the ket |ψ⟩ to obtain the ket (*A*|ψ⟩). The operator also acts on bras: applying the operator *A* to the bra ⟨φ| results in the bra (⟨φ|*A*), defined as a linear functional on *H* by the rule