Question

Given : The mass of electron is 9.11x10-31 kg.
Planck's constant is 6.626x10-34 Js, the uncertainty
involved in the measurement of velocity within a
distance of 0.1 A is [AIPMT (Prelims)-2006)
(1) 5.79 * 10 ms (2) 5.79 x 10*7 ms-1
(3) 5.79 × 10*8 ms-1 (4) 5.79 ~ 10*5 ms -1​

Answer 1

Answer : The uncertainty involved in the measurement of velocity is, $5.79\times 10^{6}m/s$

Explanation :

According to the Heisenberg's uncertainty principle,

$\Delta x\times \Delta p=\frac{h}{4\pi}$          ...........(1)

where,

$\Delta x$ = uncertainty in position

$\Delta p$ = uncertainty in momentum

h = Planck's constant

And as we know that the momentum is the product of mass and velocity of an object.

$p=m\times v$

or,

$\Delta p=m\times \Delta v$      .......(2)

Equating 1 and 2, we get:

$\Delta x\times m\times \Delta v=\frac{h}{4\pi}$

As we are given that:

m = mass of electron = $9.11\times 10^{-31}kg$

h = Planck's constant = $6.626\times 10^{-34}Js$

$\Delta x$ = $0.1\AA=0.1\times 10^{-10}m$

conversion used : $(1\AA=10^{-10}m)$

Now put all the given values in the above formula, we get:

$(0.1\times 10^{-10}m)\times (9.1\times 10^{-31}kg)\times \Delta v=\frac{6.626\times 10^{-34}Js}{4\times 3.14}$

$\Delta v=5.79\times 10^{6}m/s$

Therefore, the uncertainty involved in the measurement of velocity is, $5.79\times 10^{6}m/s$