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 |ψ1〉 and |ψ2〉, and complex numbers c1 and c2, then, since bras are linear functionals,
- Given any ket |ψ〉, bras 〈φ1| and 〈φ2|, and complex numbers c1 and c2, then, by the definition of addition and scalar multiplication of linear functionals,
- Given any kets |ψ1〉 and |ψ2〉, and complex numbers c1 and c2, 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