Question

If the uncertainty in position and velocity are equal, then uncertainty in momentum will be:
A.$\dfrac{1}{2}\sqrt {\dfrac{{{\text{mh}}}}{\pi }} $
B. $\dfrac{1}{2}\sqrt {\dfrac{{\text{h}}}{{\pi {\text{m}}}}} $
C.$\dfrac{h}{{4\pi {\text{m}}}}$
D.\[\dfrac{{{\text{m}}h}}{{4\pi }}\]

Answer 1

Hint:This question is based on Heisenberg's Uncertainty principle, which states that the position and the velocity of a subatomic particle like an electron cannot be measured with absolute certainty.
Formula Used:
 \[\Delta {\text{x}} \times \Delta {\text{p}} \geqslant \dfrac{{\text{h}}}{{4\pi }}\]
where, $\Delta {\text{x}}$is the change in position while $\Delta {\text{p}}$is the change is momentum of the electron.Change in Momentum of the electron, $\Delta {\text{p}}$ = ${{m\Delta v}}$

Complete step by step answer:
From the Heisenberg’s Uncertainty principle, we have,
\[\Delta {\text{x}} \times \Delta {\text{p}} \geqslant \dfrac{{\text{h}}}{{4\pi }}\]
Since the mass of the electron is a fixed quantity, the change in the momentum is actually the change in the velocity of the electron.
Therefore, putting the value of $\Delta {\text{p}}$ in the above equation we get,
\[\Delta {\text{x}} \times {\text{m}}\Delta {\text{v}} \geqslant \dfrac{{\text{h}}}{{4\pi }}\]
\[ \Rightarrow \Delta {\text{x}} \times \Delta {\text{v}} \geqslant \dfrac{{\text{h}}}{{4\pi {\text{m}}}}\]
Where, $\Delta {\text{v}}$is the change in velocity of the electron
As per the question, the change in the position and the velocity are equal. Therefore, $\Delta {\text{x}}$= $\Delta {\text{v}}$.
So,
\[{\left( {\Delta {\text{v}}} \right)^2} \geqslant \dfrac{{\text{h}}}{{4\pi m}}\]
\[ \Rightarrow \Delta {\text{v}} \geqslant \sqrt {\dfrac{{\text{h}}}{{4\pi m}}} \]
Since the change in momentum, $\Delta {\text{p}}$ = ${{m\Delta v}}$
Therefore, $\Delta {\text{p}}$=${\text{m}}\sqrt {\dfrac{{\text{h}}}{{{{4\pi m}}}}} $=$\sqrt {\dfrac{{{\text{mh}}}}{{{{4\pi }}}}} $=$\dfrac{1}{2}\sqrt {\dfrac{{{\text{mh}}}}{{{\pi }}}} $


Hence, the correct answer is option A.

Note:
The pair of position and momentum in the Heisenberg’s Uncertainty principle are called complementary variables.The measurement of the position of an electron is done with the help of an electron microscope and the accuracy of such measurement is limited by the wavelength of the light illuminating the electron. Thus, it is possible in principle, to make such measurements accurate by using light of very short wavelength like the gamma rays. But, for gamma rays, the Compton Effect cannot be ignored, the interaction of the electron and the interaction of the radiation should be considered as a collision of at least one electron with the photons of the light. In such a collision the electron suffers recoil which disturbs the momentum of the electron.