# [QM] Help understanding this bra-ket solution

1. Jun 27, 2013

### JBrandonS

Hello,

I am working my way though Sakurai's book on Quantum MEchanics and am having some problems understanding the bra-ket notation. I keep believing I understand everything there is to it but then he will do something in a single line that I cannot understand. This is one of them. If someone could help me out it would be great.

1. The problem statement, all variables and given/known data

Show why the following in correct: $<a''|A|a'>=<a'|A|a'>\delta_{a'a''}= a'\delta_{a'a''}$

A is an hermitian operator. a' and a'' are the eigenkets and eigenvalues of A.

2. Relevant equations

3. The attempt at a solution

The only method I can think of to coming up with the final solution is the following, which may not even be correct.

Use $A|a'> = a'|a'>$ since A is hermitian and rewrite as $<a''|a'|a'>$
Since a' is real $a'=a'^*$ so we can rewrite as $a'<a''|a'>$
From here we can use the fact that a'' and a' are orthonormal eigenkets from the same operator so $<a''|a'> = \delta_{a'a''}$ and we finally have $a'\delta_{a'a''}$

However this method does no provide the middle expression which has me really thrown off. I am not sure if I am doing everything correct and I do not know how Sakurai came to that. I am also not 100% on what all this means either.

2. Jun 27, 2013

### JBrandonS

Ugh I don't know how I just saw what they did. Multiply by the identity operator (a') between <a''| and A. It all falls together then. Still not 100% sure what all it means but I'll work on it.

3. Jun 27, 2013

### rude man

I think this post belongs under Advanced Physics.

4. Jun 27, 2013

### JBrandonS

I checked the rules for the advanced physics and it said that just because it's QM doesn't mean it belongs there. So I figured this would be a good place to put it. Either way this question can be closed now as I figured it out. Just cant find out how to mark it for closure.