Question

The velocity of a proton in an accelerator is known to an accuracy of 0.250% of the speed of light. (This could be small compared with its velocity.) What is the smallest possible uncertainty in its position?

$42.1 \textrm{ fm}$

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This is College Physics Answers with Shaun Dychko. We know the uncertainty in the velocity of a proton that is 0.0025 times the speed of light and the mass of a proton, we know, is 1.6726 times 10 to the minus 27 kilograms and we'll assume that this is known with no uncertainty. The Heisenberg uncertainty principle; equation [29.43], says that the uncertainty in position mutliplied by uncertainty in momentum has to be more than or equal to Planck's constant divided by 4

*π*. And we'll assume that the uncertainty in momentum can be attributed entirely to the uncertainty in velocity assuming that we know mass perfectly, in which case,*ΔP*will equal*m Δv*and we'll substitute that in place of*Δp*here. And let's solve this for*Δx*by dividing both sides by*m Δv*. And we get that the uncertainty in position then is more than or equal to Planck's constant over 4*π*times mass times uncertainty in velocity. So that's Planck's constant divided by 4*π*times the mass of a proton times 0.0025 times the speed of light, which at a minimum then the uncertainty in position is 42.1 femtometers.