# 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?

Related Quantum Physics News on Phys.org

#### 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"

### Physics Forums Values

We Value Quality
• Topics based on mainstream science
• Proper English grammar and spelling
We Value Civility
• Positive and compassionate attitudes
• Patience while debating
We Value Productivity
• Disciplined to remain on-topic
• Recognition of own weaknesses
• Solo and co-op problem solving