Question

What is the uncertainty in locating the position of an electron with a speed to 25m/s , having a uncertainty of 0.1% ​

Answer 1

We need to find the Uncertainty in locating the position of the electron , having a speed of 25 m/s . And the uncertainty is 0.1 % . The Heisenberg's Uncertainty Principle is as follows

• It states that , the position and momentum of microscopic moving particles cannot be determined simultaneously with accuracy. If we try to measure any of the quantities with Greater accuracy then the other becomes less accurate.

$\qquad\underline{\boxed{\red{\sf \Delta x \times \Delta p_x \geqslant \dfrac{h}{4\pi} }}}$

• Where the symbols have usual meanings.
• Here the uncertainty in speed = 0.1%

$\sf\dashrightarrow \Delta v = 0.1\% \times 25m/s \\\\\sf\dashrightarrow \Delta v = \dfrac{1}{100}\times 25 m/s \\\\\sf\dashrightarrow \boxed{\red{\sf \Delta v = 2.5\times 10^{-2}\ m/s }}$

$\rule{200}2$

• According to Heisenberg's Principal

$\sf\dashrightarrow \Delta x \times \Delta p_x \geqslant \dfrac{h}{4\pi} \\\\\sf\dashrightarrow \Delta x \times m\Delta v \geqslant \dfrac{h}{4\pi} \\\\\sf\dashrightarrow \Delta x \geqslant \dfrac{h}{4\pi m\Delta v } \\\\\sf\dashrightarrow \Delta x \geqslant \dfrac{ 6.6 \times 10^{-34} kg\ m^2 s^{-1} }{4\times 3.14 \times (9.1\times 10^{-31} kg )\times (2.5\times 10^{-2} m/s ) } \\\\\sf\dashrightarrow\underset{\blue{\sf Required\ Answer}}{\underbrace{\boxed{\pink{\sf \Delta x = 2.317 \times 10^{-3} m }}}}$