# The Physical Meaning of Projectors in Quantum Mechanics

Let $$O$$ be a single-particle observable for a system, and $$|L\rangle$$ and $$|R\rangle$$ two orthonormal eigenstates of $$O$$. You may imagine that the system consists in two photons, and $$|L\rangle$$ and $$|R\rangle$$ represent some orthogonal polarization states of each photon.

1. Then what is the physical meaning of the operator $$|L\rangle\langle L|$$ (or $$|R\rangle\langle R|$$ for that matter)? Is it an observable on $$H$$? If so, what kind of observable is it? More precisely, what is its relation with respect to $$O$$, $$|L\rangle$$, or $$|R\rangle$$ from a physical point of view?

2. Relatedly, let $$\hat{L} \equiv |L\rangle\langle L|$$ and $$\hat{L}^{2} \equiv \hat{L} \otimes I + I \otimes \hat{L}$$. Then is it physically meaningful to think about the expected value of $$L^{2}$$ with respect to, say, $$|L\rangle|L\rangle$$? In other words, does $$\langle L|\langle L|\hat{L}^{2}|L\rangle|L\rangle$$ have any non-trivial physical meaning?

A projector is an observable - you can directly check that it is Hermitian $$|L\rangle\langle L|^\dagger = |L\rangle \langle L|$$. As to interpretation - a projector onto a single state will measure the value $$1$$ for definite if the system is in that state. If the system is in an orthogonal state it will measure $$0$$. Therefore you can think of projectors as operators whose measurement corresponds to asking a binary question. Any measurement you can think of can be approximated by a series of binary questions and so its not surprising that any observable can be decomposed into such projectors.

As for your second question: I don't see why not. The notation $$L^2$$ is confusing though - I'd stick to calling this $$L_1+L_2$$ or similar. Note that this operator is not a projector. It's still Hermitian, and it's a reasonable thing to consider if you have two subsystems on which $$L$$ is itself sensible to consider.

Like all Hermitian operators, the operator $$P_L=|L\rangle\langle L|$$ represents a physical observable. It is easy to verify that this operator has the two eigenvalues:
• $$1$$, with eigenstate $$|L\rangle$$
• $$0$$, with eigenstate $$|R\rangle$$
So the corresponding physical observable is a Boolean observable, for the property "the system is in state $$|L\rangle$$". The measurement result will be either true (1) or false (0). And after this measurement the system will be in state $$|L\rangle$$ or $$|R\rangle$$, respectively.
Projectors are used to model detection in the sense that the average value of $$\vert L\rangle\langle L\vert$$ for a system described by $$\vert\psi\rangle$$ is $$\langle \psi\vert L\rangle\langle L\vert \psi\rangle=\vert\langle L\vert \psi\rangle\vert^2$$ and thus is the probability of detecting the system in the state $$\vert L\rangle$$ having been prepared in $$\vert \psi\rangle$$.
As jacob1729 said any projector is an observable, since it is an hermitian operator. The physical meaning of the projector $$\hat{P}_L=\left |L\right> \left < L\right|$$ is simple. $$\hat{P}_L$$ gives you the projection of an arbitrary state, e.g. $$\left | \psi \right >$$ onto the subspace spanned by $$\left | L\right >$$. Let's put it in other words. The probability of measuring the eigenvalue $$O_L=\hat{O}\left|L\right>$$ (i.e. of finding the system in the state $$\left| L\right >$$) is equal to the average of the projector $$\hat{P}_L$$ on $$\left| L\right >$$, if the system is initially in the state $$\left | \psi \right >$$, that is, $$$$\left< \hat{P}_L\right>=\left < \psi \right | \hat{P}_L \left | \psi \right >$$$$
Concerning your point 2, I do not understand why you defined $$\hat{L}^2$$ that way. Since you have already defined $$\hat{L}=\left |L\right> \left < L\right|$$, then $$\hat{L}^2=\left |L\right> \left < L\right|\left |L\right> \left < L\right|=\hat{L}$$. Therefore, asking for the expectation value of $$\hat{L}^2$$ is the same as asking for $$\left< \hat{P}_L\right>$$, which is the probability of measuring the eigenvalue $$O_L=\hat{O}\left|L\right>$$. If you like to learn more about this, I suggest you read Quantum Mechanics by Messiah, in particular, the parts concerning projectors, that will help you a lot to better understand this topic.