Question

The speed of a certain proton is 350 km/s. If the uncertainty in its momentum is 0.100%, what is the necessary uncertainty in its location?

Answer 1

Answer:

necessary uncertainty is 0.90 nm

Explanation:

given data

speed v = 350 km/s = 350000 m/s

momentum = 0.1 %

to find out

necessary uncertainty

solution

we know here mass of proton m  is 1.6726 × $10^{-24}$ g

so momentum will be  = m × v  

momentum = 1.6726 × $10^{-24}$   × 350000

momentum = 5.85 × $10^{-19}$ gm/s

momentum = 5.85 × $10^{-17}$ gcm/s

and

uncertainty momentum is = 0.1 % ×  momentum

uncertainty momentum is = 0.1 % ×  5.85 × $10^{-17}$

uncertainty momentum is = 5.85 × $10^{-20}$ gcm/s

uncertainty momentum is = 5.85 × $10^{-26}$ kgm/s

and we consider here dx is uncertainty position

and dp is uncertainty momentum

so dp × dx = 1/2× h     ..................1

h is plank constant that is = 1.055 × $10^{-34}$ kgm²/s

so put here value in equation 1 and find dx

dx × 5.85 × $10^{-26}$   = 1/2× 1.055 × $10^{-34}$

dx = 9.0 × $10^{-10}$  m

dx = 0.90 nm

so necessary uncertainty is 0.90 nm