# Quantum mechanics operators

## Homework Statement

Suppose a linear operator L satisfies <A|L|A> = 0 for every state A. Show that then all matrix elements <B|L|A> = 0, and hence L = 0.

## Homework Equations

##<A|L|A>=L_{AA} and <B|L|A>=L_{BA}##

## The Attempt at a Solution

It seems very straight forward and I don't know how to prove it but here is what I have tried:
##<B|L|A> \to##Using resolution of Identity ##\to \sum_{A} <B|A><A|L|A> \to <B|L|A>=0##
Is it right or do I need to write more.

Last edited:

Orodruin
Staff Emeritus
Homework Helper
Gold Member
It is not correct, you are summing over A and at the same time assuming that ##|A\rangle## is a fixed state (i.e., the one you started with in ##\langle B|L|A\rangle##).

since the problems says for every state A so should I write as following ##<A_i|L|A_i>=0 \to ## then as before
## <B_j|L|A_i>=\sum_{i}<B_j|A_i><A_i|L|A_i>=0##
is it right now?

Orodruin
Staff Emeritus