I Projection operators

RohanJ

Summary
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?

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DrClaude

Mentor
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

Mentor
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.

"Projection operators"

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