Question

A proton is accelerated to one-tenth of the velocity of light. If its velocity can be measured with a precision $\pm 1\%$, then its uncertainty in position is:
A. $1.05 \times {10^{ - 13}}$
B. $1.13 \times {10^{ - 13}}$
C. $1.15 \times {10^{ - 13}}$
D. $1.25 \times {10^{ - 13}}$

Answer 1

The uncertainty in momentum or position can be determined from results derived from the Heisenberg’s Uncertainty principle. Thus, if one measurement is known, the uncertainty in the other can be easily calculated using the uncertainty formula.
Formulas used: $\Delta x.\Delta p = \dfrac{h}{{4\pi }}$
Where $\Delta x$ is the uncertainty in position, $\Delta p$ is the uncertainty in momentum and $h$ is the Planck’s constant.
$\Delta p = m\Delta v$
Where $m$ is the mass of the particle and $\Delta v$ is the uncertainty in velocity.

Complete step by step answer:
The Heisenberg’s Uncertainty Principle states that it is impossible to simultaneously predict both the momentum and position of a particle accurately. From his principle, an empirical relation was devised which relates the uncertainty in position to the uncertainty in momentum:
$\Delta x.\Delta p = \dfrac{h}{{4\pi }}$
Where $\Delta x$ is the uncertainty in position, $\Delta p$ is the uncertainty in momentum and $h$ is the Planck’s constant.
Velocity is given as one-tenth of the speed of light ($3 \times {10^8}m/s$). Therefore,
$v = \dfrac{{3 \times {{10}^8}}}{{10}} = 3 \times {10^7}m/s$
The uncertainty (precision) is given as $\pm 1\%$ (one-hundredth) of this value. Hence, we have:
$\Delta v = 3 \times {10^7} \times \dfrac{1}{{100}} = 3 \times {10^5}m/s$
As we know, momentum of the product of mass and velocity. Thus, the uncertainty caused due to change in momentum:
$\Delta p = m\Delta v$
Where $m$ is the mass of the particle and $\Delta v$ is the uncertainty in velocity.
Substituting this in our first equation, we get:
$\Delta x.m\Delta v = \dfrac{h}{{4\pi }}$
Rearranging this, we get:
$\Delta x = \dfrac{h}{{4\pi \times m\Delta v}}$
Uncertainty in velocity as we found out is $3 \times {10^5}m/s$ . Mass of a proton is $1.673 \times {10^{ - 27}}kg$ and the Planck’s constant, $h = 6.626 \times {10^{ - 34}}Js$
Substituting these values, we get:
$\Delta x = \dfrac{{6.626 \times {{10}^{ - 34}}}}{{4 \times 3.14 \times 1.673 \times {{10}^{ - 27}} \times 3 \times {{10}^5}}}$
On solving this, we get:
$\Delta x = 1.05 \times {10^{ - 13}}m$
Hence, the correct option to be marked is A.

Note:
The Uncertainty Principle is a direct cause of the wave-particle dual nature of matter, as described by quantum mechanics. This uncertainty is not very visible for macroscopic objects, but for subatomic particles it is very evident. The reason for uncertainty is that when matter exhibits its wave nature, the position of the particle corresponding to the wave becomes difficult to predict, as waves are continuous packets of energy. On the other hand, when it exhibits particle nature, momentum and wave velocity becomes impossible to predict.