Chapter 3.6, Problem 3.6.2SR

Interpretation Introduction

Interpretation:

The minimum uncertainty in the position of a proton with the mass of $\text{1}{\text{.67 × 10}}^{-\text{27}}\text{ kg}$ moving at a speed of ${\text{4 × 10}}^{\text{6}}\text{ m/s ± 1 percent}$ should be calculated using the concept of the Heisenberg uncertainty principle.

Concept Introduction:

The Heisenberg uncertainty principle says that it is impossible to know simultaneously both the momentum ($\text{p}$) and the position ($\text{x}$) of a particle with certainty.

Mathematically,

$\text{Δx}\cdot \text{Δp }\ge \frac{\text{h}}{\text{4π}}$

For a particle having the mass $\text{m}$,

$\text{Δx}\cdot \text{mΔu }\ge \frac{\text{h}}{\text{4π}}$

Where $\text{Δx and Δu}$ are uncertainty in measuring the position and velocity of the particle respectively; $\text{h}$ is Planck’s constant ($\text{6}{\text{.63 × 10}}^{-\text{34}}\text{ Js}$).