Projection Operators: Pi, Pj, δij in Quantum Mechanics

In summary, Shankar explains in Principles of Quantum Mechanics that Pi is a projection operator, represented as Pi=|i> <i|. Using this, we can show that PiPj=|i> <i|j> <j|= (δij)Pj. To reach the final result of (δij)Pj, we can change |i> to |j> in the second result using δij. It is also valid to switch Pi and Pj in the final result.
  • #1
RohanJ
18
2
TL;DR 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?
 
Physics news on Phys.org
  • #2
Once you get the ##\delta_{ij}##, you can change ##|i\rangle## to ##|j\rangle##:
$$
\delta_{ij} |i\rangle \langle j| = \delta_{ij} |j\rangle \langle j|
$$
 
  • #3
DrClaude said:
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?
 
  • #4
RohanJ said:
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.
 

1. What is the purpose of projection operators in quantum mechanics?

Projection operators, such as Pi, Pj, and δij, are used in quantum mechanics to project a state vector onto a subspace of interest. This allows us to focus on specific aspects of a quantum system and make predictions about its behavior.

2. How do projection operators work?

Projection operators work by taking a state vector and projecting it onto a subspace defined by the operator. This is done by multiplying the state vector by the projection operator, which results in a new state vector that only contains components within the subspace.

3. What is the difference between Pi and Pj in projection operators?

Pi and Pj are both projection operators, but they project onto different subspaces. Pi projects onto the subspace defined by the eigenvalues of the observable i, while Pj projects onto the subspace defined by the eigenvalues of the observable j. In other words, Pi and Pj are specific to the observables they represent.

4. What is the significance of the Kronecker delta (δij) in projection operators?

The Kronecker delta (δij) is a special case of a projection operator, where i and j are equal. This means that the projection operator will only project onto the subspace defined by the eigenvalue of the observable i, and not any other subspace. It is often used in quantum mechanics to simplify calculations and make predictions about a system.

5. Can projection operators be used for any quantum system?

Yes, projection operators can be used for any quantum system. They are a fundamental concept in quantum mechanics and are used to make predictions and calculations for a wide range of systems, from simple particles to complex molecules.

Similar threads

Replies
27
Views
2K
  • Quantum Physics
Replies
3
Views
780
  • Quantum Physics
Replies
9
Views
895
Replies
11
Views
1K
Replies
13
Views
990
  • Quantum Physics
Replies
3
Views
1K
  • Quantum Physics
Replies
14
Views
1K
Replies
14
Views
1K
  • Quantum Physics
Replies
16
Views
2K
Back
Top