Chapter 6, Problem 53QP

Interpretation Introduction

Interpretation:

The minimum uncertainty in the velocity of a ping-pong ball is to be determined and a comment is to be made on the magnitude of the uncertainty in the speed.

Concept introduction:

According to the Heisenberg uncertainty principle, the product of the uncertainty in the position and the momentum of a particle cannot be less than $\frac{\text{h}}{4\pi }$. It is expressed as follows:

$\Delta x\cdot \Delta p\ge \frac{\text{h}}{4\pi }$

Here, $\Delta x$ is the uncertainty in position, $\Delta p$ is the uncertainty in momentum, and $\text{h}$ is Planck’s constant $\left(6.63\times {10}^{-34}\text{ J}\cdot \text{s}\right)$.

As momentum is the product of mass and velocity, the equation of the Heisenberg uncertainty principle can also be written as follows:

$\Delta x\cdot m\Delta u\ge \frac{\text{h}}{4\pi }$

Here, $\Delta x$ is the uncertainty in the position of a particle, $m$ is the mass of the particle, $\Delta u$ is the uncertainty in velocity, and $\text{h}$ is Planck’s constant $\left(6.63\times {10}^{-34}\text{ J}\cdot \text{s}\right)$.

$\text{Conversion factor for nm to meter is}\frac{1.0\text{ nm}}{1.0\times {10}^{-9}\text{ m}}$.

$\text{Conversion factor for gram to kilogram is:}\frac{1000\text{ g}}{1.0\text{ kg}}$.