Dirac notation for quantum states

Quantum states are usually described using bra–ket notation, also known as Dirac notation.

In Dirac notation, a ket represents a state of a quantum system and is a vector in a complex vector space . A bra denotes a linear map , and the application of to is written as .

The ket can be represented as a column vector:

and the bra as a row vector:

In this representation, a bra next to a ket simply denotes matrix multiplication of a row vector with a column vector: