What are some examples of the projection operator being used in quantum mechanics? In Quantum Mechanics, the projection operator is defined as $\hat{P}_\psi = |\psi\rangle\langle\psi|$, so it acts on quantum states $|\Psi\rangle$ as
$$\hat{P}_\psi|\Psi\rangle = |\psi\rangle\langle\psi|\Psi\rangle = \langle\psi|\Psi\rangle|\psi\rangle=c|\psi\rangle$$
I see that it behaves in a similar way of the projection operation between euclidean vectors: it returns a new ket in the $|\psi\rangle$ direction whose lenght $c$ is that of the projection of $|\Psi\rangle $ onto the $|\psi\rangle$ direction.
However, what is the importance of this operator in Quantum Mechanics? What are some examples of the cases in which it is necessary to make use of it?
 A: There are lots of places you can find this object to be acted on. The best way would be to open a Quantum Mechanics book on pc and search for the word Projection and see the section and use of the Operator.
These are few or two examples.

The projection operator look  like
$$\mathcal{P}=|n\rangle \langle n|$$
A very nice property of the operator is rather obvious from the interpretation of it. The operator project a part of the vector along a specific basis vector. So If I project a vector along with all the bases, I should get the vector back.
$$\sum_n\mathcal{P}_n|\psi\rangle=\sum_n |n\rangle \langle n|\psi\rangle =|\psi\rangle $$
$$\sum_n|n\rangle \langle n|=I$$
That's the crucial property and use the time to when changing basis. Whenever we need to write a vector on some basis, We insert a complete set. It's also used in a change of basis.
A nice example of optics can be found on Principle of Quantum Mechanics R. Shankar: Section 1.6 Matrix element of a linear operator.

In the context of the density matrix, A pure ensemble given by
$$\rho=|\alpha^{(n)}\rangle \langle \alpha^{(n)}|$$
You can represent complete polarization beam with $S_z+$ as
$$\rho=|+\rangle \langle +|$$

It's also used in the context of perturbation theory. Look for section 5.1 Modern Quantum Mechanics Sakurai : Formal development of perturbation expansion.
