The de Broglie-Bohm (dBB) theory has a wave (a function from configuration space) and a particle (a point in configuration space) and both evolve in time.
The evolution of the particle does not influence the evolution of the wave, but the wave does influence the particle. Since the particle doesn't do anything except be bossed around, it's basically a marker, nothing more. If you have a wave that has multiple packets that don't overlap, the particle marks one of them as occupied and the others as empty. But that marking affects nothing whatsoever.
Configuration space tells you where absolutely everything is. For instance if I had two particles in a 1d universe then I could specify a point $(x_1,x_2)$ and that tells you there is a particle at $x_1$ and another at $x_2$, if I had 8 particles in 1d I could specific a point in an 8d space $(x_1, x_2, x_3, x_4, x_5, x_6, x_7, x_8)$ and it tells me where al the particles are at. So the point in configuration space in principle tells you perfectly where every single particle is.
But the point in configuration space doesn't affect a single thing. So it can't ever be known, and its value has zero testable consequences, because it is just along for the ride and can't affect anything. It's like you said you had the location of a ghost, and made an equation that tell you how the ghost moves around. If thinking about it helps you to focus on the wave, that's super. The wave is important, it helps you to make predictions. If thinking about the location of the ghost (or the particle) helps you to recognize a liar when someone says there can't be a position, that's super. You don't want to pay attention to people that are more interested in wrongly saying what you can or can't do than focusing on how to make a prediction using an existing theory or come up with a new theory.
The trajectory evolves, but in a passive wave determined by the wave evolution and the wave evolution actually determines what happens.
Is the particle position a is detected at related to the particle's actual position?
People like to say detection or measurement because it sounds cool. But there isn't a magic box where a real number with an infinite number of decimal digits appears. What you can do is separate a wave into disjoint parts such that they act independently. And once that happens in a way where they will forever more act independently. These packets are the actual results. For instance a stern-gerlach device is called a spin-measurement device. What it really does is split a wave into two or fewer parts, one going left, the other right. But it also polarizes the spin of the wave on the separate parts so that if you took the part that went left and sent it through a similarly calibrated machine again it will just go left (no split to a part going right), and similarly if you took the part that went right and sent it through a similarly calibrated machine again it will just go right (no split to a part going left). It split a way into potentially more than one disjoint wave, and it does so in a reproducible way.
That's what is called measurement. It doesn't measure some preexisting thing, it splits it into disjoint wavepackets. And yes sometimes it doesn't split, so after the measurement it is definitely polarized to not split under certain circumstances. But it wasn't necessarily polarized like that before you sent it through so it is misleading to say you measured it (the first time).
So how would you measure position? You have to separate wavepackets. And wavepackets push particles around, so you move the particles. Moving it isn't measuring where it was. You can't measure where it was, you are only going to break the wave into a finite number of packets, so you'll never get some infinite decimal expansion of where it was.
All the dBB theory does is tell you that it can have a position. In the stern-gerlach example we can also conclude that the ones that were farthest left ended up going left and the ones farthest right ended up going right. But you didn't know if the particle was on one edge of the wavepacket or the other edge, so you didn't know which way it was going to go. So all you had was a probabilty of different wavepackets being occupied. But dBB theory tells you that this probability is the regular of not knowing where the particle is located. Which is very different than the claims some people make about quantum theories. But those claims are usually so wild because they think the so-called measurements are about some prexisting property of a particle rather than just a splitting of a wave.
In the example of a beam-splitter experiment, does the particle and its wave function evolves over time culminating in an electron being ejected from the surface of one of two CCD detectors?
No. The wave has amplitudes for both paths, and if you separate those paths to make the different wavepackets disjoint, then you might be able to get a "measurement". But you can only get that if they separate in a way where they will never ever ever overlap ever again ever. Then they are forever more independent. Then, both mathematically and practically they can each live in their own little world where the other doesn't exist because the other one doesn't affect it. (Just like the wave can ignore the particle because the particle doesn't affect it, so these wavepackets may now ignore each other, the other parts of the wave, since they don't affect each other).
The wave is always taking all the options available, whenever the wave can split it splits, and the math tells you the probability that the particle will occupy one wavepacket versus the other, but the wave spilts, so they both happened. The fact that one wavepacket is occupied by the particle and one wavepacket is not really doesn't change anything ever.
Worse, this idea that they are independent is really an engineering issue not a philosophical one. Because what makes them be independent is like calling a random phone number and trying to call it again, technically you might. practically the chance is too small. Those separate wavepackets if reflected backwards and aimed really really carefully could overlap again. But it would be easier to shoot a laser beam from the Earth, bounce off it off of a mirror on the moon and back into itself (so hit it at just the perfect angle).
So the segregation is really just approximate and/or temporary. Meas
Does the copenhagen interpretation say the location of that electron is where the wave-function "collapses"
Copenhagen is forced to say the same thing as dBB about the waves, otherwise they will disagree with experiments. They simply don't have a particle in Copenhagen (or in Ithaca). The modern Cophenhagen gives you serendipity at best for a story, and they only own up to the existence of the disjoint waves.
There are some (non Copenhagen) that go for a true collapse postulate (stochastic theories), but their predictions actually disagree with the predictions of regular quantum mechanics.
are only two options
Does the dBB interpretation say the location of that electron determined by teh particle position?
Not quite. It says that there is one. But that is not something that is ever revealed or known. Not now. Not ever. What you measure and observe and predict is the wavepackets. You can have a statistical uncertainty of where the particle is. And if that uncertainty follows the square of the wavefunction intitially, it will later. And you can use that to compute the probability that different wavepackets will be the occupied on. And those probabilities (of which wavepackets are occupied) are the same probabilities everyone wants touse quantum theory to compute.
The dBB can simply interpret those probabilities as the probabilities of different wavepackets having the particles in them. And the dBB theory can say that the particles have positions that are consistent with the probabilities. Consistent in the sense that the probability of the wavepacket being the occupied one agrees with the quantum probabilities (for any measurement, not just position). Thus teh dBB theory tells us that the experimental verifications of quantum mechanics are
1) Consistent with particles having positions
2) Can be the normal kind of probabilties based on ignorance (ignorance of which of many disjoint wavepackets are occupied by a particle whose position is not known)
3) And together this reveals which results and saying in quantum mechanics are based about what properties can or can't exist. For instance whether a particle has a spin up or a spin down before measurement. Usually it does not.
Let's talk about #3 some more. You could make a spin gerlach with different calibrations. Each calibration send a left going one to the left again, and one that went right goes right again. But by basically flipping it upside down (or flipping yourself upside down, either way) you can can realize how arbitrary each is, and make one or the other, the upside-up and the upside-down version are equally reliable, and equally practical, and they measure the same thing, and they measure it by separating/splitting wavepackets. But the one that goes left or right depends on the wave (so we know which percentage of particles would go left or right) and on where the particle is (the ones farthest to the left go left, the ones farthest to the right go right)
So whether a particle goes left or right depends not only on the wave but on the unknown location of the particle and on arbitrary matter like whether you choose to make one spin go left and the other right or vice versa
Yes, whether you get left or right depends on whether you used the upside-down machine or the upside-up machine. So it doesn't measure a preexisting property of the particle
My understanding is that the shape of the wave-function is almost identical at both detectors regardless of which detector actually detects the particle.
Absolutely nothing is different for the two detectors. Whether the wavepacket has the hypothetical-and-you-never-ever-see-it-or-see-a-testable-consequence-particle or not, affects nothing whatsoever. There are two wavepackets, and since they don't affect each other, they can ignore each other. If you want to have a favorite, you can root for the one with the particle. But you don't know which one that is. But for any one of them you can compute the probability that it is your secret favorite one. And you'll get the correct probability.
So why does the electron get ejected from the detector at the location of the bohmian particle, rather than at the other detector?
There is only one particle, so if the wavepackets don't overlap it is stuck in one (the particle never travels through regions where the wave is zero). So it has to be in one. It's not a big deal, because you don't know which one it is in. The point is that the wavepackets now act independently if you've done your separation well enough. And that is not because the particle is stuck in one. It's the other way around. The fact hat they will never overlap again is why the particle is stuck in one.
And that's a super advantage of the dBB theory conceptionally. If you say you want to find the probability that the particle gets stuck in a particular wavepacket, then you know to not ask until the wavepackets are fully separated and will stay that way. You can intuitively tell when ciomputing a probability makes sense and when it would be silly. The Copenhagen theory doesn't give you that because it gives you nothing intuitive to think about. But it does compute the same probabilities in exactly the same situations. And avoids computing them in the same situations. But makes it unclear when or why you would do either. The dBB theory makes it clear when the question makes sense to ask.
Shouldn't their either be some interaction between the particle itself and the electron (perhaps via their quantum potentials) OR that a mutual cause correlates the position of the bohmian particle with the position of the ejected electron.
No, nope, and nada. All wave. All the time. The particle is not a causal agent, it doesn't cause things to happen, it's like a tracer particle in the atmosphere or a tracking device on stolen car, it's not moving the atmosphere or driving the car. Except its one that we never ever see. So it's more like a hypothetical tracer.
Why is the bohmian particle necesary?
Depends on your goal. If your goal is to catch people in lies, it helps to have a particle whose motion and the probabilities you can compute (probabilities about whether a wavepacket has the particle or not) and deal with intuitively, where you can fix a sample space (locations of particles) and translate other questions into probabilties (questions of which wavepackets have the tracer). It can help you recognize when someone made a mistake.
Or alternatively you could be using the theory to inspire you to make other theories, and the particle might help with that.
It could be numerically useful to do an approximation, this sometimes happens in computational chemistry.
The first one is worth a tremendous amount by itself, even to stop you from accidentally misleading yourself. If you focus on the clear probabilistic question of identifying which of distinct disjoint wavepackets have the particle, then you can compute probablities correctly. And if youknow that a a measurement gives different results based on stuff you don't know (like where exactly the particle is) then you won't delude yourself into thinking there is an element of reality that isn't there. For instance you seem to want to think there "is" some momentum and that it "is" being transferred "here" or "there" maybe even at a particular "when" and that is all just plain wrong. A Copenhagenist might go overboard and not imagine that anything happens ever. But if you study the dBB theory you can have a picture of the particle located somewhere and the wave smoothly changing over time, and be patient to wait until it separates and then compute the only thing you can, the probabilty that different packets are occupied, and you can know (from effort) how to not to read too much into it because the dynamics of the particle can tell you how unrelated the current position of the particle is to the later computed probability that a wavepacket is occupied.
So I have to assume that the particle itself interacts in some way.
Only do that if you want to risk disagreeing with the predictions of quantum mechanics
