Undergrad Projection Operators: Pi, Pj, δij in Quantum Mechanics

[RohanJ]

TL;DR

Product of two projection operators

In Principles of Quantum mechanics by shankar it is written that
Pi is a projection operator and Pi=|i> <i|.
Then PiPj= |i> <i|j> <j|= (δij)Pj.
I don't understand how we got from the second result toh the third one mathematically.I know that the inner product of i and j can be written as δijbut how do we get to Pj in the final result from the second result?

 

[DrClaude]

Once you get the $\delta_{ij}$, you can change $|i\rangle$ to $|j\rangle$:
$$
\delta_{ij} |i\rangle \langle j| = \delta_{ij} |j\rangle \langle j|
$$

 

[RohanJ]

Once you get the $\delta_{ij}$, you can change $|i\rangle$ to $|j\rangle$:
$$
\delta_{ij} |i\rangle \langle j| = \delta_{ij} |j\rangle \langle j|
$$

I was thinking that only. That means I can write Pi in place of Pj in the final result too and it won't make a difference.
Am I right?

 

[DrClaude]

I was thinking that only. That means I can write Pi in place of Pj in the final result too and it won't make a difference.
Am I right?

Right. This is not the only equality that is valid here. I don't have Shankar's book with me at the moment, but it can be that he uses that particular form later to make a point.