This seems strange. what would be the hermitian operator for such an observable. It seems that the eigenfunctions isn't even well defined inside the detector??

@lucky-guess The operator is diag(1,1,...,0,0,...) in the position basis where the ones are the points inside the detector. It's more realistic than the usual position operator, but still not very realistic.

Probability density of wave function is zero at the edges of the detector then?

@lucky-guess The probability density is zero inside, nonzero outside, and at the boundary there's a discontinuity. As I said, it's unrealistic, but it's more realistic than the delta function you get after a standard position measurement, or the particle that's always detected at one slit or the other in the double slit experiment. It's just a thought experiment. All that really matters is that the measurement operator has two eigenvalues corresponding in some way to "inside" and "outside" (or click and no click).

So it is zero inside, then what gives us the measurement distribution

Have you read the answer above your own? Both conceptually the same, with different ‘theoretical’ solutions that don’t make sense

@lucky-guess It's nonzero inside before a measurement, zero after. The wave function spreads out and overlaps the detector again between measurements. The time between measurements is small but nonzero because of quantization of measurement time. Please don't ask me to construct a more realistic detector; this is the same level of abstraction as every other QM thought experiment.

Oh ok. Still, there’s an issue, the time of detection distribution will vary due to the time interval of measurement. If we assume the wave function instantly pops back, the number of trials taken within a time period will significantly alter the probability a particle is detected earlier than time t

Are you suggesting that the sensitivity of my screen determines when the particle arrives?

Let’s take a stationary free particle instead. Are you telling me if you wait long enough the particle will be detected anywhere in space?

Right, well this stuff isn’t taught very well

i mean, your suggestion kind of flies in the face of all the widely accepted forms of qm, which state that collapse must occur into an eigenstate

wavefunction collapse is not an event and therefore does not have a specific time. Just look at the quantum erasure experiment, the position and time of wavefuction collapse can vary due to actions years after the experiment has been done and dusted

@lucky-guess All of these collapses are to eigenstates. What you may be missing is that there's a big difference between an eigenstate and a basis state when the eigenvalues are degenerate. They're extremely degenerate here. Every state that is zero inside the detector is an eigenstate with eigenvalue 0. / In the delayed choice quantum eraser, there is a collapse when the particle is detected at the screen before the delayed choice, and another later.

eigenstates are basis states when we choose a basis of eigenstates

@Dirac'stwin No – a one-to-one correspondence between eigenvalues and (normalized) eigenfunctions only exists if the eigenvalues are all distinct. I covered this in my last comment. The space of eigenfunctions of this operator with eigenvalue 0 is very large: it consists of every state that vanishes inside the detector.

@lucky-guess When you choose an eigenbasis, the eigenstates are linear combinations of basis states that have the same eigenvalue. When the eigenvalues are all distinct, the eigenstates are the same as the basis states (up to scalar multiplication). When the eigenvalues are not all distinct, there are far more eigenstates than basis states. The eigenvalues are never all distinct in real life. "Complete" measurements, where the post-collapse state is completely determined by the measured value, never happen in reality.

@benrg i can see that you have posted about the quantum zeno effect. Would this be observable in this experiment? e.g. using a screen with a backlight would speed up the arrival times of the particle, as opposed to using a screen which is dark (does not measure arrival until after the experiment)

@benrg, I guess a purposeful comment I can make is, have you tried to use this sort of model to derive an arrival time distribution for any realistic scenario as a sanity check? I have learned that using an example of a gaussian wavepacket with momentum moving towards the detector quickly shows that such proposals do not give physical distributions. Some distributions need a potential to get weird results but I suspect that the "finite-$\Delta t$ approach" you write about would have issues even for $V(x)=0$.

Imagine a wave packet moving into the detector region. If $\Delta t$ is very small, so that the amount of probability moving into the detector in time $\Delta t$ is << 1, then I believe you will get a Zeno Effect. On the other hand if $\Delta t$ is any larger than this, then the time resolution of your prediction will be of a non-negligible size compared to the width of the arrival time distribution, so the distribution will be messed up. This is in contradiction with modern-day experiment which can measure the distribution to a high precision relative to its width/standard deviation.