# The Physical Meaning of Projectors in Quantum Mechanics

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.