How to Prove This Hermitian Operator Statement?

[abcs22]

1. Homework Statement prove the following statement:
Hello, can someone help me prove this statement

A is hermitian and {|Ψi>} is a full set of functions

Homework Equations

Σ<r|A|s> <s|B|c>[/B]

The Attempt at a Solution

Since the right term of the equation reminds of the standard deviation, I tried using its definition but it didn't yield any results. Also, I tried to use the hermicity of the operator A to get the complete set but after that I got stuck.

 

[Lucas SV]

This is prety simple if you assume $\psi_i$ form a basis for the Hilbert space of states. Just use the complex relation $|z|^2=z z^*$ for the left hand side, and use the completeness relation for $\psi_i$. Is this what you meant by a full set of functions?

 

[abcs22]

Yes, I did, I apologize, English is not my mother language. I tried but I can't get those two terms on the right side

 

[Lucas SV]

Yes, I did, I apologize, English is not my mother language. I tried but I can't get those two terms on the right side

No problem. The two terms come as a property of the summation. The completeness relation assumes you sum over all $i$. The first term comes from the completeness relation. The second term comes from the fact that you are missing the $i=j$ in the summation on the left hand side. The key equation you need to use is
$$
\langle \psi_j | A^2 | \psi_j \rangle=\sum_i \langle \psi_j | A | \psi_i \rangle \langle \psi_i | A | \psi_j \rangle
$$