Question

A proton is traveling with a speed of (4.010±0.018)× 10 5 m/s .

Part A

With what maximum precision can its position be ascertained?

Express your answer using two significant figures. ?x? ____ m

Answer 1

The principle used here is Hiesenberg's Uncertainty principle according to which

where is the uncertainty in position and is the uncerrtainty in momentum. Since the speed of the proton is given to be in the order of 10^5 m/s, this is less that 0.001c, so we can treat the momentum classically as p=mv. Since the mass of a proton is known as a constant, the uncertainty in momentum is due to uncertainty in velocity only.

So the uncertainty in momentum will just be the mass times the uncertainty in velocity. The mass of the proton is 1.67x10^-27 kg. The uncertainty in velocity is already given.

For the maximum precision, we use the equality instead of the inequality in Hiesenberg's principle. Using tthe values of h=6.626x10^-34 kg m^2/s, we get the right hand side

This means

This gives us  . (two significant figures)