Bra-ket notation

In quantum mechanics, each [quantum state]? is identified with a vector in a Hilbert space. Paul Dirac introduced the bra-ket notation as a concise and convenient way to describe quantum states. The terminology comes from the fact that the central operation looks like a "bracket" <φ|ψ> consisting of a left part, the "bra" <φ|, and a right part, the "ket" |ψ>.

We start with a Hilbert space H. Each vector in H is known as a ket, and written as

|ψ>

where ψ is an arbitrary label for the ket. Each element of the dual space of H (i.e. each continuous linear function from H to the complex numbers C) is known as a bra, and written as

<φ|

where φ is an arbitrary label for the bra. Applying the bra <φ| to the ket |ψ> results in a complex number, called a bra-ket, which we write as

<φ|ψ>

Every ket |ψ> has a dual bra, written as <ψ|, a continuous linear function on H defined as follows:

<ψ|x> = ( |ψ> , |x> )

for all bras |x>, where the right hand side ( , ) denotes the inner product given on the Hilbert space. The notation is justified by the [Riesz representation theorem]?, which states that every bra in the dual space arises from one and only one ket in this fashion.

[Outer products]? are written as |φ><ψ|. One use of the outer product is to construct [projection operators]?. Given a ket |ψ> of norm 1, the projection operator onto the subspace spanned by |ψ> is

|ψ><ψ|

Two Hilbert spaces V and W may form a third space V × W by a tensor product. If |ψ> is a ket in V and |φ> is a ket in W, the tensor product of the two kets is a ket in V × W. This is written variously as

|ψ>|φ>   or   |ψ> × |φ>   or   |ψ φ>