I believe your questions stems from the misunderstanding of quantum superposition and quantum measurement. First of all, you should understand that “photon goes through both slits” is a simplification – it uses words from classical physics to describe a non-classical phenomenon. I'll try to explain the subtlety here, but before that – to make sure we're both on the same page – let us explore the classical case.
Classical analogy
Imagine you have a rubber ball cannon pointing on a wall with two holes. The rubber ball cannon isn't very precise and shoots the rubber balls with various speeds and under various angles and there's a 50:50 chance the rubber ball will pass through each hole. Now imagine you've set up a camera to take a picture of your rubber ball one second after firing. Can you predict, where that ball would be?
Since you don't know the exact position and velocity of the rubber ball, you'd have to track every single point where the ball can be, assign it a probability maybe, and let those points evolve in time. If you understand how phase space works, you could assign a probability amplitude to each point in the rubber ball's phase space and evolve them in time. Then, if you wanted to compute the probability that the ball is in such and such volume, you'd just integrate over it.
Having constructed this analogy, let's forget the ball's actual trajectory (since we don't know it anyway). Instead, let's focus on the probabilistic description and try to answer several questions about it.
- When we fire a ball, through which slit does it go?
We don't have the information to answer this. Based on our probabilistic description, we can say the ball went through both slits with the same probability. Does that mean that we could see it passing through both slits at the same time? No, of course it doesn't. But it means we could see it passing through either slit, both options are possible.
- When the ball hits the wall on the other side, can it hit on several places at once?
Our probabilistic description tells us the ball can be in either of many places, and when it hits the wall in the back, it can hit it on one point out of many. But can it hit the wall on two places at once? Of course not, it's just a single ball! If two boys were standing behind the slits, both boys could get hit with the same probability. But they would never get hit at the same time. If we include the boys into our probabilistic model, we can say that either one of them got hit, or that both got possibly hit, but we know they never got hit at the same time. A detail that will become important later: if one boy tells us he got hit, we know with certainty that the other didn't get hit.
The quantum case
All the phenomena I described in the classical analogy carry surprisingly well to the quantum case. Wavefunction is essentially just a probability distribution on the phase space (with several built-in limitations, like the uncertainty principle). There are just two crucial differences. The first difference is that in the classical analogy, there existed one actual trajectory, we just didn't know it. Meanwhile in the quantum case, a single trajectory simply doesn't exist and the probabilistic description is as close as we can get. I'll get to the second difference in a minute.
Now, equipped with a better intuition, let's try answering a few questions about the double-slit experiment. Assume we have a photon gun pointing at a plate with two slits and we measure where the photon lands using a second plate in the back.
- When we detect that a photon landed, through which slit did it go?
The most natural answer is that it could have gone through either. It has gone through both slits with the same probability – same as the rubber ball.
- Can the photon hit two places at once?
No, it can't. It's the same as with the ball, one photon can't hit two places at once. However, an interesting thing can happen if we place two perfectly isolated atoms in the space behind the slits, let the photon pass and then try to test which atom got hit.
- Which atom got hit by the photon?
Same as with the boys – either could get hit. Either one of the atoms is excited by the photon. Both atoms got hit with the same probability, but both never got hit at the same time. And what's really curious: if we measure one of the atoms and find that it was excited by the photon, we know with certainty that the other atom wasn't excited. This is the famous entanglement – the state of one particle depends on the state of a different particle. In fact, a similar procedure called the Rydberg blockade is used to create entangled atoms in a lab.
Hopefully you now understand the analogy with classical physics and it isn't surprising to you that one photon cannot be detected on two places at once.
Now the second difference that I promissed. In the classical model, the probabilities of different outcomes could only add up. If there's a 20 % chance that the balls flies this way and ends here, and a 10 % chance it flies a different way and ends here, the probability of the ball ending here is 30 %. In the quantum case, the probability amplitudes aren't positive real numbers, they are actually complex. That means that probabilities of different outcomes can not only add up, but also cancel each other out. This is the reason why you see an interference pattern in the quantum case, but not in the classical one.
The measurement problem
I know this answer is quite long already, there's just one elephant in this room that needs to be adressed. How do we know when to use the wavefunction description and when to resort to definite description? If somebody placed detectors in front of the two slits, even if they didn't tell us, the probabilistic description would fail and the interference pattern would be lost. It seems that in order to use the probabilistic description, the results must not only be unknown to you as an experimentator, but fundamentally unknowable.
This is called the measurement problem and there is a lot of interpretations and hypotheses about it. My personal favorite interpretation is the relational QM.
