Bra-ket notation

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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>