Modelling how a macroscopic detector interacts with a quantum mechanical particle?

Background

I recently came up with a math formula and wondered if could be useful for any physical kind of situation. I then, realised it might be useful for the situation of particle detector.

P.S: Rather, than argue about the (controversial) maths I use to solve the PDE I would like to ask if the model can physically represent such a situation?

Experiment

When we fire a stream of particles with a momentum distribution, a particle detector "clicks" upon detecting a particle.Because of the quantum mechanical nature of the particle there is a chance the detector will not click.

Hence, using the conservation of momentum in the case the particle clicks:

$$dp_{\text{detector}} = dp_{\text{particle}}$$

In the case it does not hit the detector:

$$dp_{\text{detector}} = 0 = dp_{\text{particle}}$$

Concerning ourselves only to the detector we write:

$$dp_{\text{detector}} = \alpha_c dp_{\text{particle} }$$

where $\alpha_c$ can be $0$ or $1$ in different $\Delta p$ depending on if the detector clicks or not. Further if the momentum distribution of the particle:

$\int_{0}^\infty dp_{\text{particle}} = \text{constant}$ (In some sense it means the recoil momentum distribution doesn't blow up). Then I believe I can solve the above equation.

Note, I hesitate writing this as :

$$\frac{dp_{\text{detector}} }{dp_{\text{particle} }} = \alpha_c$$ as $dp_{\text{particle}} = 0$ when $dp_{\text{detector}} = 0$

Question

Does this model make sense? In a nutshell I think I'm modelling how a macroscopic detector is affected by a microscopic particle. I am not aware of particle detectors being modelled? If someone could point me to the relevant literature I would be grateful.

Edit: I agree that this is an oversimplified detector but I'm curious as to if the final theory would have an $\alpha_c$ parameter?

• Looks like you are trying to (re)invent stochastic calculus. In that context, your $\alpha_c$ is called a Poisson increment. I would recommend looking at a textbook such as "Stochastic Methods" by C. Gardiner. – Mark Mitchison Jul 17 '18 at 11:32
• Consider trying to detect photons with a reverse biased silicon diode. Photon enters device, creates an electron-hole pair, those are separated in the junction and give a little current pulse. Well, except that the generated carriers don't always make it out of the junction region (they can just recombine). So, the photon interacts with the detector, but you get no signal... – Jon Custer Jul 17 '18 at 14:01
• @MarkMitchison I did invent a (controversial) method of solving this ... Though it's not stochastic calculus (at least not yet) but perhaps in the future it might be interesting to show that both methods are some kinds of limits of each other? Essentially what I as I have memory of $\alpha_c$ so I can sum over arbitrary $\sum \alpha_c dp$. The method I'm referring to is: mathoverflow.net/questions/306068/… – More Anonymous Jul 17 '18 at 14:46
• @JonCuster I think my detector can possibly model something like that it just means we have more information about $\alpha$ going to $0$? – More Anonymous Jul 17 '18 at 14:51

• Any idea on what literature I should go through? Also, I agree this is an oversimplified model but my main point of contention is if anything in the final model will have the $\alpha_c$ parameter? – More Anonymous Jul 17 '18 at 1:54