Question

(a)

An electron and a 0.0260 kg bullet each have a velocity of magnitude 490 m/s, accurate to within 0.0100%. Within what lower limit could we determine the position of each object along the direction of the velocity? (Give the lower limit for the electron in mm and that for the bullet in m.)

for the electron mmf

or the bullet m

(b)

What If? Within what lower limit could we determine the position of each object along the direction of the velocity if the electron and the bullet were both relativistic, traveling at 0.300c measured with the same accuracy? (Give the lower limit for the electron in nm and that for the bullet in m.)

for the electron nm

for the bullet m

Answer 1

a] Use Uncertainty relation for position and momentum,

Here,

the uncertainty in momentum will be:

=> [for electron]

and [for bullet]

therefore, uncertainty in position will be atleast,

and .

b] The uncertainty in velocity will be:

The mass here will be the relativistic mass given by:

this will proportionately increase the uncertainty in momentum by a factor of 1.0483. Therefore, the uncertainty in position will decrease by a factor of 1.0483 in both the bullet and electron case. This gives,

and .