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Physics Question on Wave characteristics

According to Heisenberg's uncertainty principle, the product of uncertainties in position and velocities for an electron of mass 9.1×10−31 kg9.1\times {{10}^{-31}}\,kg is:

A

2.8×10−3 m2 s−12.8\times {{10}^{-3}}\,{{m}^{2}}\,{{s}^{-1}}

B

3.8×10−5 m2 s−13.8\times {{10}^{-5}}\,{{m}^{2}}\,{{s}^{-1}}

C

5.8×10−5 m2 s−15.8\times {{10}^{-5}}\,{{m}^{2}}\,{{s}^{-1}}

D

6.8×10−6 m2 s−16.8\times {{10}^{-6}}\,{{m}^{2}}\,{{s}^{-1}}

Answer

5.8×10−5 m2 s−15.8\times {{10}^{-5}}\,{{m}^{2}}\,{{s}^{-1}}

Explanation

Solution

We will use formula Δx×Δp=h4π\Delta x \times \Delta p=\frac{h}{4 \pi} to solve problem.
Δx×Δp=h4π\Delta x \times \Delta p=\frac{h}{4 \pi}
Δx×mΔv=h4π\Delta x \times m \Delta v=\frac{h}{4 \pi}
Δx×Δv=h4πm\Delta x \times \Delta v=\frac{h}{4 \pi m}
Δx=\Delta x= uncertainty in position
Δv=\Delta v= uncertainty in velocity
h=h= Planck's constant
=6.63×10−34 kg m2 s−1=6.63 \times 10^{-34} \,kg\, m ^{2} \,s ^{-1}
m=m= mass of electron
=9.1×10−31 kg=9.1 \times 10^{-31} \,kg
∴Δx×Δv=6.63×10−344×3.14×9.1×10−31\therefore {\Delta} x \times \Delta v=\frac{6.63 \times 10^{-34}}{4 \times 3.14 \times 9.1 \times 10^{-31}}
=5.8×10−5 m2 s−1=5.8 \times 10^{-5} \,m ^{2} \,s ^{-1}