Question

Heisenberg's uncertainty principle can be expressed mathematically as Δx*Δp = h/4π, where Δx and Δp denote the uncertainty in position and momentum respectively and h is Planck's constant. What would be the uncertaintry in the position of a pitched baseball (mass = 0.2574 kg) that was traveling at a velocity of 84.50 m/s if the velocity has an uncertainty of 2.275%?

please explain step by step

Answer 1

According to Heisenberg uncertainty principle,

Δx*Δp = h/4π

where, Δp - the uncertainty in momentum;

Δx - the uncertainty in position

and Planck's Constant, h=6.626 X 10^-34 m²kgs^-1

Now, The uncertainty in momentum can be written as

Δp=m⋅Δv , where

Δv - the uncertainty in velocity;

=2.275% of 84.50 m/s i.e. 1.922 m/s
m - the mass of the particle.

In the given case, we are dealing with 0.2574 kg base ball having an uncertainty in velocity is 1.922 ms^-1

so, the uncertainty in momentum will be

Δp=0.2574 kg X 1.922 ms^-1

=0.4947 kgms^-1

Now, From the given equation we can calculate the uncertainty in position as

Δx = h/4π.(1/Δp)

=(6.626 X 10^-34 m²kg s^-1/4 x 3.14 ).(1/0.4947 kg^-1m^-1s¹)

=(6.626 X 10^-34 m²kg s^-1/12.56). (2.02 kg^-1m^-1s¹)

=(0.527 X 10^-34m²kg s^-1).(2.02 kg^-1m^-1s¹)

=1.064.X 10^-34 m