Operators and eigenstates/values

[nmsurobert]

Homework Statement

Let the Hermitian operator A^ corresponding to the observable A have two eigenstates |a1> and |a2> with eigenvalues a1 and a2, respectively, where a1 ≠ a2. Show that A^ can be written in the form A^ = a1|a1><a1| + a2|a2><a2|.

Homework Equations

The Attempt at a Solution

I reached out the instructor for some guidance but I am still confused.
To my understanding i should start with A^|ψ>. Where |ψ> is some arbitrary spin state.
and i don't know where to go from there.

 

[zhaos]

The identity operator can be written as

$$
1 = |1\rangle \langle 1| + |2\rangle \langle 2| \\
$$

For example suppose $|\psi\rangle = c_1 |1\rangle + c_2|2\rangle$
$$ |\psi \rangle = 1|\psi \rangle = |1\rangle \langle 1|\psi\rangle + |2\rangle \langle 2| \psi \rangle \\
= |1\rangle c1 + |2\rangle c2 \\
= |\psi \rangle
$$

Suppose you tried putting "1" on both sides of your operator?

 

[blue_leaf77]

Have you heard about completeness theorem?

 

[nmsurobert]

Thank you zhaos, that's actually a lot of help.

Blue leaf, I have not heard of completeness theorem. But I will give it a Google!

 

[blue_leaf77]

Completeness theorem is exactly what zhaos wrote in the first equation in his post.