Bra–ket notation

Summary

Bra–ket notation, or Dirac notation, is the standard notation for vectors, linear functionals and operators on a complex Hilbert space in quantum mechanics. A vector is a ket $\lvert \psi \rangle$, and the functional of inner product with $\varphi$ is a bra $\langle \varphi \rvert$. Their pairing is the bracket $\langle \varphi \vert \psi \rangle$. By the Riesz representation theorem, bras are exactly the continuous linear functionals. The notation makes compositions such as $\lvert \psi \rangle\langle \varphi \rvert$ and $\langle \varphi \vert A \vert \psi \rangle$ easy to read and to compute.