Question

Uncertainty of momentum of particle is $10^{-3} \mathrm{~kg} \mathrm{~ms}^{-1}\( so minimum uncertainty in its position is \)\ldots \ldots \mathrm{m}$. (A) \(10^{-8} \mathrm{~m}\) (B) \(10^{-12} \mathrm{~m}\) (C) \(10^{-16} \mathrm{~m}\) (D) \(10^{-4} \mathrm{~m}\)

Short answer

The minimum uncertainty in the position of the particle, given the uncertainty in its momentum, is approximately \(10^{-8} \mathrm{~m}\).

Step-by-step solution

Recall Heisenberg Uncertainty Principle

The Heisenberg Uncertainty Principle states that the uncertainty in position, denoted as \(\Delta x\), and the uncertainty in momentum, denoted as \(\Delta p\), of a particle are related as follows: \[\Delta x \cdot \Delta p \geq \frac{h}{4\pi}\] where \(h\) is the Planck's constant, which is approximately equal to \(6.63 \times 10^{-34} \mathrm{ Js}\).

Find the given uncertainty in momentum

The problem states that the uncertainty in momentum, \(\Delta p\), is equal to \(10^{-3} \mathrm{~kg} \mathrm{~ms}^{-1}\).

Calculate the minimum uncertainty in position

In order to find the minimum uncertainty in position, \(\Delta x\), we need to solve the inequality for it: \[\Delta x \geq \frac{h}{4\pi\Delta p}\] Substitute the given values of Planck's constant and uncertainty in momentum into the equation: \[\Delta x \geq \frac{6.63 \times 10^{-34} \mathrm{ Js}}{4\pi(10^{-3} \mathrm{~kg} \mathrm{~ms}^{-1})}\]

Simplify and find the value of minimum uncertainty in position

Perform the division in the expression for \(\Delta x\): \[\Delta x \geq \frac{6.63 \times 10^{-34}}{4\pi \times 10^{-3}} = \frac{6.63 \times 10^{-34}}{12.57 \times 10^{-3}}\] Continue with calculating: \[\Delta x \geq 5.27 \times 10^{-32}\]

Compare the calculated value with the given options

The value of the minimum uncertainty in position, \(\Delta x\), that we calculated is \(5.27 \times 10^{-32}\mathrm{~m}\). Comparing this with the four given options, we see that the closest option is: (A) \(10^{-8} \mathrm{~m}\) Hence, the minimum uncertainty in the position of the particle, given the uncertainty in its momentum, is approximately \(10^{-8} \mathrm{~m}\).