Question

If the uncertainty in velocity and position is same then the uncertainty in momentum will be:
(A) $ \sqrt {\dfrac{{{\text{hm}}}}{{4\pi }}} $
(B) $ {\text{m}}\sqrt {\dfrac{{\text{h}}}{{4\pi }}} $
(C) $ \sqrt {\dfrac{{\text{h}}}{{4\pi {\text{m}}}}} $
(D) $ \dfrac{1}{{\text{m}}}\sqrt {\dfrac{{\text{h}}}{{4\pi }}} $

Answer 1

Hint: Momentum is termed as the product of mass and velocity. We will first calculate the change in momentum and substitute it in the Heisenberg uncertainty principle. We will calculate the change in momentum.

Formula used: $ \Delta {\text{u}}.\Delta {\text{x}} = \dfrac{{\text{h}}}{{4{\text{m}}\pi }} $
Here $ \Delta {\text{u}} $ is change in velocity, $ \Delta {\text{x}} $ is change in position, h is planck's constant and m is mass.

Complete step by step solution:
The actual expression for Heisenberg uncertainty principle is:
 $ \Delta {\text{p}}.\Delta {\text{x}} = \dfrac{{\text{h}}}{{4\pi }} $
Here p is momentum.
We know that $ {\text{p}} = {\text{m}} \times {\text{u}} $
Here p is momentum, m is mass and u is velocity. The change in momentum will be:
 $ \Delta {\text{p}} = {\text{m}} \times \Delta {\text{u}} $ Mass always remains constant.
We will substitute the value of momentum in the uncertainty formula and hence will get the formula:
 $ \Delta {\text{u}}.\Delta {\text{x}} = \dfrac{{\text{h}}}{{4{\text{m}}\pi }} $
Now it is given to us that the uncertainty in velocity and uncertainty in position are same that is $ \Delta {\text{u}} = \Delta {\text{x}} $
Hence the formula will become:
 $ \Delta {\text{u}}.\Delta {\text{u}} = {\left( {\Delta {\text{u}}} \right)^2} = \dfrac{{\text{h}}}{{4{\text{m}}\pi }} $
The value of $ \Delta {\text{u}} $ will be $ \Delta {\text{u}} = \sqrt {\dfrac{{\text{h}}}{{4{\text{m}}\pi }}} $
Now,
 $ \Delta {\text{p}} = {\text{m}} \times \Delta {\text{u}} $
Substituting the value of uncertainty in the above formula we will get uncertainty in momentum:
 $ \Delta {\text{p}} = {\text{m}} \times \sqrt {\dfrac{{\text{h}}}{{4{\text{m}}\pi }}} = \sqrt {\dfrac{{{\text{mh}}}}{{4{\text{m}}\pi }}} $
Hence, the correct option is A.

Note:
Heisenberg uncertainty principle was given by Werner Heisenberg in 1927. According to uncertainty principle it is impossible to accurately measure the position and momentum of a moving electron simultaneously. If we measure the position accurately then momentum will not be accurate and vice versa. We can see from the formula that if we make change in momentum zero then the change in momentum will be infinity which is very large and the reverse is true as well. According to Bohr the position and momentum can be measured because it considered the electron as material particle by Heisenberg also proves the wave nature of the particle.