Projection operators

Armed with the bra-ket formalism we can construct any operator in Hilbert space. The projection operator P_a is defined as

 

P_a = | a><a|, (1.37)

Note that $P^{2}_{a} \equiv P_{a} P_{a} = \vert a><a\vert \vert a><a\vert=\vert a><a\vert=P_{a}$and in general any projection operator P has the property P²=P. Consider operator X whose eigenstates are given by the set $\{ \vert a_{1} > \, \vert a_{2}> ... \vert a_{n}> \}$. If we define the projection operators P_an = | a_n><a_n|, show that operator X can be expressed as a sum of projection operators, i.e.

$$X=\sum_{n} P_{a_{n}} a_{n}$$ (1.38)