Quantum observable measuring whether a particle is in a given spatial region

What would an operator look like that measures the following thing? We have a particle in one dimension. A lamp goes on if the particle is found in a certain interval on the real axis.

Your operator should have eigenvalue 1 (lamp on) if the particle is found within some interval $x\in [0,L]$, and eigenvalue 0 (lamp off) otherwise. Such an operator would look like $$P_{[0,L]} = \int_0^L\mathrm{d}x \; \lvert x\rangle \langle x \rvert,$$ i.e. it is a projector onto the subspace spanned by position eigenstates $\lvert x\rangle$ lying in the given interval. These position states are defined to be the eigenstates of the position operator $\hat{x}$ with eigenvalue $x$, i.e. $\hat{x}\lvert x\rangle = x\lvert x \rangle$, and they describe a particle that is perfectly localised at position $x$.

The eigenstates of the projection operator $P_{[0,L]}$ take the form $$\lvert \psi\rangle = \int_R\mathrm{d}x\; \psi(x) \lvert x\rangle,$$ where the integral extends over a region $R$ which is either completely contained within $[0,L]$, or completely outside $[0,L]$. The former corresponds to eigenvalue 1, the latter to eigenvalue 0. In other words, the eigenstates of $P_{[0,L]}$ have wave functions that vanish everwhere outside $[0,L]$ (eigenvalue 1) or everywhere inside $[0,L]$ (eigenvalue 0).

• Thank you! And how would the $\lvert x\rangle$ look like? Isn't the only eigenstate the wave function itself then? – Darius Oct 25 '15 at 13:33
• @Darius I have updated the answer. The eigenfunctions of the observable you are interested in must have support either completely inside or outside the interval, but are otherwise arbitrary. This makes sense: if the particle is definitely inside or outside the interval, it is in an eigenstate of the observable. In any other scenario it is not. – Mark Mitchison Oct 25 '15 at 15:50
• Thank you Mark! But is there than any special way the operator will look like (e.g. some matrix or something like that)? I fully understand what you wrote, but I somehow cannot really imagine how to build the operator out of the information I have now :/ – Darius Oct 25 '15 at 21:14
• @Darius It is difficult to give you advice without more detail on what problem you want to solve. In principle $P_{[0,L]}$ can only be represented by an infinite-dimensional matrix (i.e. it cannot be exactly represented by a matrix). However in practice I very much doubt you need to represent it as a matrix. If you merely want to calculate the probability of the lamp switching on given some state $\phi(x)$ of the particle, you only need the probability of finding the particle in the interval, i.e. $$\langle \phi\rvert P_{[0,L]}\lvert\phi\rangle=\int_0^L\mathrm{d}x\; \lvert\phi(x)\rvert^2.$$ – Mark Mitchison Oct 25 '15 at 22:59
• And can't I build an operator out of the eigenstates by using something like a projection operator? – Darius Oct 26 '15 at 16:00

In general, the expectation value for any observable quantity is found by putting the quantum mechanical operator for that observable in the integral of the wavefunction over space: $$\langle Q\rangle= \int_{-\infty}^\infty \psi^*\hat{Q}\;\psi\; dV.$$

• Thank you, I already knew that, but can this help me? – Darius Oct 25 '15 at 12:53
• This is a mathjax-enabled site. It supports $\LaTeX$; do take advantage of it:) – user36790 Oct 25 '15 at 15:58