Bra-ket notation, or the Dirac notation, is a notation for complex vector spaces together with their dual space.

Looks like this: The name comes from the English word “Bracket”.

A ket is of the form that represents a column vector. For example:

Warning

Don’t confuse with the vector. It is instead a unit vector.

A bra is of the form represent a row vector. It is the conjugate transpose of .

Example

If then

This notation is used ubiquitously in quantum mechanics.

We sometimes combine the bra and ket: the notation means the inner product .

Usage

We can write the length of the vector as following:

We can also use the bra-ket notation to write the th component of the matrix in terms of unit vectors and

References