Log out Go to App
Go to App

Learning Materials

Features

Discover

Chapter 6: Problem 47

Using Heisenberg's uncertainty principle, calculate the uncertainty in the position of (a) a 1.50 -mg mosquito moving at a speed of \(1.40 \mathrm{~m} / \mathrm{s}\) if the speed is known to within \(\pm 0.01 \mathrm{~m} / \mathrm{s} ;\) (b) a proton moving at a speed of \((5.00 \pm 0.01) \times 10^{4} \mathrm{~m} / \mathrm{s}\). (The mass of a proton is given in the table of fundamental constants in the inside cover of the text.)

Short Answer

Expert verified
The uncertainty in position for the mosquito (\(\Delta x_m\)) is \(\geq 7.33 \times 10^{-32}~m\) and for the proton (\(\Delta x_p\)) is \(\geq 1.58 \times 10^{-10}~m\).

Step by step solution

01

Understand Heisenberg's Uncertainty Principle

Heisenberg's Uncertainty Principle states that there is an inherent uncertainty in the measurement of position and momentum of a particle. The principle is mathematically represented as follows: \(\Delta x \times \Delta p \geq \frac{h}{4\pi}\), where \(\Delta x\) is the uncertainty in position, \(\Delta p\) is the uncertainty in momentum, and h is Planck's constant.
02

Calculate the uncertainty in momentum

We know that uncertainty in momentum is defined as the product of mass and uncertainty in speed: \(\Delta p = m \times \Delta v\) a) For the mosquito: Uncertainty in momentum of mosquito: \(\Delta p_m = m_{mosquito} \times \Delta v_{mosquito}\) \(= 1.50 \times 10^{-6}~kg \times 0.01~m/s\) b) For the proton: Uncertainty in momentum of proton: \(\Delta p_p = m_{proton} \times \Delta v_{proton}\) \(= (1.67 \times 10^{-27}~kg) \times (0.01 \times 10^4~m/s)\)
03

Calculate the uncertainty in position

Now we will use Heisenberg's uncertainty principle to find the uncertainty in position, \(\Delta x\): \(\Delta x \geq \frac{h}{4\pi \times \Delta p}\) a) For the mosquito: \(\Delta x_m \geq \frac{6.626 \times 10^{-34}~J\cdot s}{4\pi \times (\Delta p_m)}\) b) For the proton: \(\Delta x_p \geq \frac{6.626 \times 10^{-34}~J\cdot s}{4\pi \times (\Delta p_p)}\)
04

Calculation and results

Now, perform the calculations for each particle to find the minimum uncertainty in the position: a) Mosquito: \(\Delta x_m \geq \frac{6.626 \times 10^{-34}~J\cdot s}{4\pi \times (1.50 \times 10^{-6}~kg \times 0.01~m/s)}\) \(\Delta x_m \geq 7.33 \times 10^{-32}~m\) b) Proton: \(\Delta x_p \geq \frac{6.626 \times 10^{-34}~J\cdot s}{4\pi \times (1.67 \times 10^{-27}~kg \times 0.01 \times 10^4~m/s)}\) \(\Delta x_p \geq 1.58 \times 10^{-10}~m\) The uncertainty in position for the mosquito is \(\geq 7.33 \times 10^{-32}~m\) and for the proton is \(\geq 1.58 \times 10^{-10}~m\).

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

What is the maximum number of electrons in an atom that can have the following quantum numbers: (a) \(n=2\), \(m_{s}=-\frac{1}{2},\) (b) \(n=5, l=3 ;\) (c) \(n=4, l=3, m_{l}=-3\) (d) \(n=4, l=0, m_{l}=0 ?\)

A photocell is a device used to measure the intensity of light. In a certain experiment, when light of wavelength \(630 \mathrm{nm}\) is directed onto the photocell, electrons are emitted at the rate of \(2.6 \times 10^{-12} \mathrm{C} / \mathrm{s}\) (coulombs per second). Assume that each photon that impinges on the photocell emits one electron. How many photons per second are striking the photocell? How much energy per second is the photocell absorbing?

Identify the specific element that corresponds to each of the following electron configurations and indicate the number of unpaired electrons for each: (a) \(1 s^{2} 2 s^{2},\) (b) \(1 s^{2} 2 s^{2} 2 p^{4}\), (c) \([\mathrm{Ar}] 4 s^{1} 3 d^{5}\) (d) \([\mathrm{Kr}] 5 s^{2} 4 d^{10} 5 p^{4}\).

The familiar phenomenon of a rainbow results from the diffraction of sunlight through raindrops. (a) Does the wavelength of light increase or decrease as we proceed outward from the innermost band of the rainbow? (b) Does the frequency of light increase or decrease as we proceed outward? (c) Suppose that instead of sunlight, the visible light from a hydrogen discharge tube (Figure 6.10 ) was used as the light source. What do you think the resulting "hydrogen discharge rainbow" would look like? [Section 6.3\(]\)

The electron microscope has been widely used to obtain highly magnified images of biological and other types of materials. When an electron is accelerated through a particular potential field, it attains a speed of \(8.95 \times 10^{6} \mathrm{~m} / \mathrm{s}\). What is the characteristic wavelength of this electron? Is the wavelength comparable to the size of atoms?
See all solutions

Recommended explanations on Chemistry Textbooks

Chemistry Branches

Read Explanation

Chemical Analysis

Read Explanation

The Earths Atmosphere

Read Explanation

Chemical Reactions

Read Explanation

Physical Chemistry

Read Explanation

Making Measurements

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.

Sign-up for free