Bra-ket notation

In quantum mechanics, each [quantum state]? corresponds to a vector in a Hilbert space. Paul Dirac introduced the following notation as a concise and convenient way to describe quantum states:

Each vector in the Hilbert space is known as a ket, and denoted

|ψ>

where ψ is a label for the ket.

Each ket |ψ> has a dual vector, called a bra, and denoted

<ψ|

A bra <ψ′| and a ket |ψ> may form an inner product, called a bra-ket, or bracket and denoted

(|ψ′> , |ψ>) = <ψ′|ψ>

[Outer products]? are written as |ψψ′><ψ|. One use of the outer product is to denote the [projection operator]? on the [linear subspace]? spanned by, say, a ket |ψ>. This is simply |ψ><ψ|.

Two Hilber spaces V and W may form a tensor product V × W. If |ψ> is a ket in V and |φ> is a ket in W, the tensor product of the two ket is a ket in V × W, and denoted

|ψ>|φ>