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Everyone knows the standard probability interpretation of the quantum mechanics. For example, the wave function of some particle at some time $t$ is $\psi (x,t)$. Therefore, if the particle is detected at the time $t$, the probability density that the detector located at $x$ finds the particle will be $|\psi (x,t)|^2$. By the unitarity, the normalization $\int dx|\psi (x,t)|^2=1$ always holds, whatever $t$ is. Also this probability is irrelevant to the details how the detectors work.

However, if I want to know the probability when the detector will detect the particle, what should I do? To be more specific, I want to evaluate the probability density, denoted by $P(x,t)$, that the particle is detected by the detector located at $x$ and at the time $t$. It should satisfy the normalization $\int dxdt P(x,t)=1$.

According to the conditional probability principles, it should have the form $P(x,t)=f(t)|\psi(x,t)|^2$, for some function $f(t)$. $f(t)$ may be referred to as the probability when a detector anywhere detects the particle. How to evaluate $f(t)$? Does it depend on the details of the detectors? Or it’s just a constant but why?

I used to think this question doesn’t make sense, because the measurement happens once I put a detector, and then the wave function changes dramatically. But later I recognized that the detector is always somewhere in the universe, so it’s hard to talk about when I put this detector.

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    $\begingroup$ When the detector registers a hit the probability of the particle being in the detectors sensitive volume is essentially one. (Exactly one is you checked the detector out from the same stock room that provides the massless frictionless pulleys and the like.) What experimenters do is prepare a statistical number of initial states and look at the distribution of hits. $\endgroup$ – dmckee Mar 22 at 16:30
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A particle quantum mechanically is represented in various ways. If it is in a complex state, an atom or molecule, a wavefunction represents the system. If it is an interaction with another particle, as for example $e+e-$ at LEP quantum field theory is used and the interactions are calculable using Feynman diagrams , for crossections. A single decaying particle will also have Feynman diagrams that will give the decay probability into various channels'

A single free quantum mechanical particle traveling in space time is represented again in quantum field theory as a wavepacket made up of plane wave solutions that represent its energy momentum vector within the Heisenberg Uncertainty. This is necessary because a plane wave is not localized.

wavepacket

This is your Ψ(x,t) for a particle. Now you ask

How to evaluate the probability when a particle is detected?

Detection means an interaction. After an interaction a new wave function describes the system, the original wavefunction no longer applies. The probability of interaction will be calculable using Feynman diagrams and the details of the specific detector, density of targets, distances, etc. have to enter so as to predict a crossection for the scatter. You state :

the probability when a detector anywhere detects the particle

I do not think this can be evaluated without boundary conditions. If you mean for people setting up detectors , it is a specific experiment at a specific location for which the probability has to be evaluated. It is observer dependent, and the detector introduces interactions accordingly. "detector anywhere" cannot be defined, imo.

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