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In quantum mechanics there is a law that states we cannot know a position and the momentum of a particle and so in a particle accelerator, we need to know the position of the particles we are going to crash into each other because or else would they not basically miss each other because we do not know the location of these. Next, if we do not know the particle's position how can we operate the magnetic fields without crashing these particles into the walls of the particle accelerator. As even a millimeter of inaccuracy might extrapolate and lead to a massive diversion from it desired path?

Furthermore, if we do not know the velocity of these particles assuming we know a very high accuracy of the particles position how can we even crash them into each other if they technically could be left behind or already past a location?

Or is it just done on basis of statistics and likeliness? If so is it impossible to collide ONLY 2 particles together at high energy levels without have certain amounts of pass-bys rather than a crash?

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    $\begingroup$ I'm not a particle nor high-energy physicist, but do know that you can create a directed beam using alternating electrical fields and magnetic confinement. So if you point two of these beams towards one another, it's likely that you will have some, if not many colliding particles - even though you can't precisely predict exactly which one's will collide. $\endgroup$ – docscience Jul 24 '15 at 23:46
  • $\begingroup$ I can send a stream of accelerated particles into a solid target, but I can't tell which particle will hit which atom. But I do know (or can measure) the cross section for some reaction of interest. $\endgroup$ – Jon Custer Jul 25 '15 at 0:02
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To answer your first point, as far as I know, a test beam is used to calibrate the position and accuracy of the system, before the main beam is sent, to avoid, as you point out, any damage to the equipment.

An indication of how few actual collisions there are is given by the fact that in 1978, a scientist did get a high energy beam right in the face, not at the LHC , but a lower energy facility in Russia. Although injured, he survived, showing how relativity few collisions actually occur.

So most of the time, the particles miss each other, although we know the velocity of them accurately enough, 99.999996 percent of the speed of light, around 670, 616, 603 mph, or 1,079,252,806 kph.

Take the LHC, for example. There are about 14 hundred bunches of protons in each beam, each bunch carries more than 100 billion particles, each time the bunches cross, we might get 20 or so interactions, it adds up if you do it enough times.

So, as you say, it's statistics, 20 collisions produces a spray of maybe a hundred hadrons, so that's enough events for the system to handle.

If they all collided, we could not cope with the data produced.

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With the uncertainty priciple, you cannot know the position and momentum of a particle (or anything for that matter) beyond a maximum precision. You can know where and how fast a particle is moving, just not with infinite precision in both variables. It's not just a measurement issue, but a fundamental fact. So a particle accelerator can still focus a beam to a fairly fine point with a fairly well know velocity. It is just impossible to know both with absolute precision. In practice, measurement is the real issue as the uncertainty principle limit is very small. They do require a bit of luck to get collisions.

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