Question

If uncertainty in the measurement of position and momentum are equal, than uncertainty in the measurement of velocity is equal to:
A. \[\dfrac{1}{{2m}}\sqrt {\dfrac{{2h}}{\pi }} \]
B, \[\dfrac{1}{{2m}}\sqrt {\dfrac{h}{\pi }} \]
C. \[\dfrac{1}{{4m}}\sqrt {\dfrac{h}{\pi }} \]
D. \[\dfrac{1}{{2m}}\sqrt {\dfrac{h}{{2\pi }}} \]

Answer 1

Hint:
As we know about the Heisenberg uncertainty principle, it says that there is an inherent uncertainty to a particle and that we can’t measure both its position and momentum with infinite precision. Because of it, the wave nature of reality that is counterintuitive arises.

Complete step by step solution
The Heisenberg uncertainty principle is one of the major principles. It says that we can never measure the exact speed of a moving object simultaneously; this uncertainty can be explained using a simple theory. When we try to measure the position and exact speed of a moving object, it causes some change in its speed. But the real reason behind this is much deeper and interesting. It exists because everything in the universe behaves both as a particle and a wave at the same time.
Heisenberg uncertainty principle, its equation is something like that as shown below.
\[\Delta x \times \Delta P > = \dfrac{h}{{4\pi }}\]
Where, \[\Delta x\] is the change in the position of the particle and \[\Delta P\] is the change in the momentum of the particle and h is the “plank constant” which is equal to the energy of the photon released in one electromagnetic radiation.
The question says that position and momentum are equal and we have to find the value of velocity.
If uncertainty in the measurement of position and momentum are equal, then uncertainty in the measurement of velocity is equal to \[\dfrac{1}{{2m}}\sqrt {\dfrac{h}{\pi }} \].
By the Heisenberg uncertainty principle;
\[
\Delta x\;\Delta P = \dfrac{h}{{4\pi }}\\
{\rm{Here}},\;\Delta x = \Delta P\\
{\Rightarrow \left( {\Delta P} \right)^2} = \sqrt {\dfrac{h}{{4\pi }}}
\]
And we know the relation between the momentum and velocity that \[\Delta P = m\Delta u\].
Hence, \[\Delta u = \dfrac{1}{{2m}}\sqrt {\dfrac{h}{\pi }} \]

Therefore, the correct answer is B.

Note:
Heisenberg uncertainty principle is used to identify broadening of spectral lines, predict quantum fluctuations and fundamental limit to observations.