**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

*H*(i.e. each continuous linear function from

*H*to the complex numbers

**C**) is known as a bra, and written as

*H*defined as follows:

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

*H*is given by the outer product: if ⟨φ| is a bra and |ψ⟩ is a ket, the outer product |φ⟩ ⟨ψ| denotes the operator which maps the ket |ρ⟩ to the ket |φ⟩ ⟨ψ|ρ⟩ (here the scalar ⟨ψ|ρ⟩ is written to the right of the vector |φ⟩). One use of the outer product is to construct projection operators. Given a ket |ψ⟩ of norm 1, the orthogonal projection onto the subspace spanned by |ψ⟩ is