# Is sole projection onto a subspace of a system associated with a quantum mechanical measurement?

I'm a little confused about a recent discussion, where a one dimensional wave-function $$\Psi(x)$$ was considered, associated with a state

$$|\Psi\rangle = \int dx \, \Psi(x) |x\rangle .$$

Now consider a projection operator, somehow narrowing x to be between a and b:

$$P = \int_a^b dx \, |x\rangle \langle x | .$$

The resulting state would be

$$P |\Psi\rangle = \int dx \int_a^b \, dx' |x'\rangle \langle x'|x\rangle \Psi(x)= \int_a^b d x' \Psi(x') |x'\rangle .$$

I would say the result of applying this operator is to project onto a reduced subspace of $$|x\rangle$$, defined by the interval $$[a,b]$$.

Now the confusion: is that operator at the same time a measurement? If yes, on what? More precise: what is the observable? I think it cannot define a "measurement" because in that case a reduction of the wave function must occur. Here there is just a reduction of the original Hilbert space to a space containing only vectors $$|x\rangle$$ with $$a \le x \le b$$, but not a collapse onto a particular wave function. So in my opinion the projection operator P itself can not be regarded as a measurement.

However, from textbook knowledge, after a measurement the state of the system must be an Eigenstate of a Hermitian operator.

Is it correct to say that P is Hermitean and has Eigenvalues 0 and 1, for vectors $$\int dx \Psi(x) |x\rangle$$ with $$\Psi(x)$$ vanishing for any x outside or inside $$[a,b]$$ respectively? Then the measurement we talk about would be the binary decision whether we find $$x$$ inside or outside $$[a,b]$$ and the measured value is the binary information "inside" or "outside" ? In my opinion such measurement does not need to end up in a single $$|x\rangle$$ as the reduced wave function as it would be in the case of position measurement; I can only say that after measurement the resulting state is with a $$\Psi(x)$$ either completely vanishing inside or outside $$[a,b]$$.

Can I say that such process of "incomplete" reduction is a measurement at all? I have some doubts, because this kind of measurement would end up in sub state, that is still a superposition of base vectors and does, in general, not reduce to a particular one.

Please correct me where I'm not correct.

Is it correct to say that $$P$$ is Hermitean and has Eigenvalues $$0$$ and $$1$$, for vectors $$\int dx\ \psi(x)|x\rangle$$ with $$\psi(x)$$ vanishing for any $$x$$ outside or inside $$[a,b]$$ respectively? Then the measurement we talk about would be the binary decision... "inside" or "outside"? ...does not need to end up in a single $$|x\rangle$$ ... I can only say that after measurement the resulting state is with a $$\psi(x)$$ either completely vanishing inside or outside $$[a,b]$$.
Real position-measurements don't have infinite precision, so this is actually a better model of a real position measurement than the typical textbook collapses-to-$$|x\rangle$$ model.
The projection operator $$P$$, together with its complement $$1-P$$, constitute a dichotomic observable — an observable that, when measured, has only two possible outcomes. For convenience, we can use a single-operator representation of this observable, like $$A=\alpha P+\beta(1-P)$$ where $$\alpha$$ and $$\beta$$ are distinct real numbers used to label the different outcomes, but that's just for convenience. Nature doesn't care how we label things.