Bra–ket notation

In quantum mechanics, bra–ket notation is a standard notation for describing quantum states, composed of angle brackets and vertical bars. It can also be used to denote abstract vectors and linear functionals in mathematics. It is so called because the inner product (or dot product on a complex vector space) of two states is denoted by

$⟨\varphi \mid \psi ⟩$,

consisting of a left part, $⟨\varphi |$ called the bra, and a right part, $|\psi ⟩$, called the ket. The notation was introduced in 1939 by Paul Dirac.

Quotes

• Admire the power of the Dirac notation!
  • J. J. Sakurai and Jim Napolitano, Modern Quantum Physics (2nd ed, 2011), Ch. 3 : Theory of Angular Momentum

• I do not use Dirac’s bra-ket notation, because for some purposes it is awkward […]
  • Steven Weinberg, Lectures on Quantum Mechanics (2012), Preface