# Bra-ket notation

 related topics {math, number, function} {math, energy, light}

Bra-ket notation is a standard notation for describing quantum states in the theory of quantum mechanics 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) of two states is denoted by a bracket, <Φ|Ψ>, consisting of a left part, <Φ|, called the bra (pronounced /ˈbrɑː/), and a right part, |Ψ>, called the ket (pronounced /ˈkɛt/). The notation was introduced in 1939 by Paul Dirac,[1] and is also known as Dirac notation.

Bra-ket notation is widespread in quantum mechanics: almost every phenomenon that is explained using quantum mechanics—including a large proportion of modern physics—is usually explained with the help of bra-ket notation. It is less common in mathematics.

## Contents

### Most common use: Quantum mechanics

In quantum mechanics, the state of a physical system is identified with a ray in a complex separable Hilbert space, $\mathcal{H}$, or, equivalently, by a point in the projective Hilbert space of the system. Each vector in the ray is called a "ket" and written as $|\psi\rangle$, which would be read as "ket psi". (The $\psi\!$ can be replaced by any symbols, letters, numbers, or even words—whatever serves as a convenient label for the ket.) The ket can be viewed as a column vector and (given a basis for the Hilbert space) written out in coordinates,