Now we move on a little with our understanding of operators.
A ket vector followed by a bra vector is an example of an operator.
For example the **operator which projects a vector onto the eigenstate** is

First the bra vector dots into the state, giving the coefficient of in the state, then its multiplied by the unit vector , turning it back into a vector, with the right length to be a projection. An operator maps one vector into another vector, so this is an operator.

The sum of the projection operators is 1, if we **sum over a complete set of states**,
like the eigenstates of a Hermitian operator.

The same is true for definite momentum states.

We can form a projection operator into a **subspace**.

We could use this to project out the odd parity states, for example.

Jim Branson 2013-04-22