# Why must energy be increased in a cyclotron to refine measured distance?

I have been listening to Nima Arkani-Hamed's messenger lectures at Cornell, and was confused by one point. He says that in order to measure at increasingly small distances energy must be increased (he was describing the LHC), and that this follows from quantum mechanics. I imagine this has to do with uncertainty, but I have trouble connecting the two. What is it about quantum mechanics that dictates this?

Measuring something with a particle beam involves scattering the beam off a target (which could be another beam, of course) and seeing what the distribution of scattered particles is. Think of Rutherford's alpha scattering experiment: send a lot of alphas end an sample the scattering angle to determine the structure of the atom.

Heisenberg's Uncertainty Principle imposes a relationship $$\sigma_x \sigma_p \ge \frac{\hbar}{2}$$ between the momentum of a probe and the size of the area probed.1 In Rutherford's case his alpha particles could easily probe regions smaller than an atom, but could not probe regions as small as a nucleus, so his distribution had two key features:

• Most alphas went through barely disturbed
• A small number scattered at large angles (even essentially straight back)

leading to the two conclusions that the atom is mostly "empty space" and that there is a small, heavy, charged core in their.

The same consideration apply to the LHC. To examine features of smaller physical extent (say the nucleus or the nucleons) you need a higher momentum probe. For any given particle mass the only way to get high momentum is to dial up the energy. The LHC provides the highest energy beams yet constructed on Earth and can therefore probe the smallest regions.

1 Those in the know will recognize that this should read "between the momentum transfer of the probe and the size of the region probed", but that detail is for another day.

• Thank you for this response, @dmckee. What Im still curious about is why momentum of the probe is related to the area probed. – Jonathan Basile Jan 30 '15 at 16:00
• @JonathanBasile The short answer is "Heisenberg's Uncertainty Principle", just like it says in the answer. The long answer involves explaining about quantum wave equations and doing some basic Fourier analysis---it's kind of mathy and results in a statement of HUP. The in-between version relies on my telling you that "particles are waves and they can't probe things smaller than their wavelength", which is imprecise. Any way 'round, it works out to be roughly equivalent to the Rayleigh criterion for telescopic optics. – dmckee --- ex-moderator kitten Jan 30 '15 at 17:12
• It occurs to me that that last comment may have been addressed to something other than your actual question. I'll try again in a while to answer the question "How does the momentum enter into the problem when HUP address the uncertainty of the momentum?", which is another matter entirely. – dmckee --- ex-moderator kitten Jan 30 '15 at 17:24
• thanks @dmckee - that sounds like it would get to the heart of my question – Jonathan Basile Jan 30 '15 at 19:48
• BTW, Jonathon, I haven't forgotten about you, I've just run into a lot of work in the "real" world at the same time as I realized that I don't have an answer on tap. DOn't know when I'll get back to it. – dmckee --- ex-moderator kitten Feb 1 '15 at 2:43

When detecting anything you are actually analyzing results of some collision. In quantum mechanics, a particle is something that has both, particle and wave characteristics. If you want to detect or see an object, wavelength of a particle must be smaller then this object. There is a connection between energy of a particle and its wavelength, higher the energy, lower the wavelength and higher the precision.

• ''If you want to detect or see an object, wavelength of a particle must be smaller then this object'' what does this even mean? – Paul Jan 30 '15 at 7:42
• If you want a detailed information about something small you need a smaller wavelength... – Žarko Tomičić Jan 30 '15 at 21:18
• So I was talking about resolution...small details need smaller wavelengths..even the uncertainty principle can be linked to the relation between wavelength and position, or wave number... – Žarko Tomičić Jan 30 '15 at 21:37
• fourier trasform position basis and momentum basis... – Žarko Tomičić Jan 30 '15 at 21:43
• Large spread in momentum means small spread in position..etc.. – Žarko Tomičić Jan 30 '15 at 21:44