Bra-ket notation

In quantum mechanics, each [quantum state]? is identified with 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 written as

|ψ>

where ψ is an arbitrary label for the ket.

Each ket |ψ> has a dual vector, called a bra, written as

<ψ|

A bra <ψ′| and a ket |ψ> may form an inner product, called a bra-ket, or simply bracket. This is written as

(|ψ′> , |ψ>) ≡ <ψ′|ψ>

[Outer products]? are written as |ψ′><ψ|. One use of the outer product is to construct a [projection operator]?. Given a ket |ψ>, 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>