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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?

Thanks for reading.

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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.

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This is only answering your question 1.

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.

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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 $\vert \psi\rangle$.

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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, \begin{equation} \left< \hat{P}_L\right>=\left < \psi \right | \hat{P}_L \left | \psi \right > \end{equation}

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.

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