Question

The uncertainty in the position of an electron (mass = 9.41 x ${{10}^{-29}}$g) moving with a velocity of 3.0 X ${{10}^{4}}c{{m}^{-1}}$ accurate up to 0.011 percent will be
(A).1.92 cm
(B).7.66 cm
(C).0.175 cm
(D).3.84 cm

Answer 1

Hint: Heisenberg uncertainty principle states that it is impossible to calculate the position and momentum of an object. This principle is based on the wave particle duality of matter. Another implication is that it is impossible to accurately measure the energy of a system with some finite amount of time.

Complete answer:
In quantum mechanics the Heisenberg uncertainty principle is a fundamental theory which explains why it is important to measure more than one variable simultaneously.
According to the formula, is $\Delta x$ is the error in the position measurement and $\Delta p$ is the error in measurements of the momentum then$\Delta x.\Delta p\ge \dfrac{h}{4\pi }$.
Given in the question:
Mass of electron = mass = 9.41 x ${{10}^{-29}}$g
Velocity= 3.0 X ${{10}^{4}}c{{m}^{-1}}$
\[\Delta V\]=0.3
So the value of $\Delta x$will be
\[\Delta x=\dfrac{h}{4\pi m\Delta v}=\dfrac{(6.6)({{10}^{-27}})}{4(3.14)(9.1)({{10}^{-28}})(0.3)}=(1.92)({{10}^{-2}})m\]
= 1.92 cm

Hence the correct answer is option (A) i.e. the uncertainty in the position of an electron (mass = 9.41 x ${{10}^{-29}}$g) moving with a velocity of 3.0 X ${{10}^{4}}c{{m}^{-1}}$accurate up to 0.011 percent will be \[(1.92)({{10}^{-2}})m\].

Note:
The electromagnetic radiation and the microscopic matter waves exhibit a dual nature of mass and momentum and the wave character. Position and velocity or momentum of a macroscopic matter wave can be determined accurately simultaneously. For example the location and the speed of a moving car can be determined at the same time with minimum error. But in the case of microscopic particles, it will not be possible to measure the velocity and position simultaneously.