/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 41 The minimum uncertainty \(\Delta... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 29: Problem 41

The minimum uncertainty \(\Delta y\) in the position \(y\) of a particle is equal to its de Broglie wavelength. Determine the minimum uncertainty in the speed of the particle, where this minimum uncertainty \(\Delta v_{y}\) is expressed as a percentage of the particle's speed \(v_{y}\left(\text { Percentage }=\frac{\Delta v_{y}}{v_{y}} \times 100 \%\right)\) Assume that relativistic effects can be ignored.

Short Answer

Expert verified
The minimum uncertainty in speed \( \Delta v_y \) is about 7.96% of the speed \( v_y \).

Step by step solution

01

Understand the Problem

According to the problem, the uncertainty in position, \( \Delta y \), is equal to the de Broglie wavelength of the particle. This means we can set \( \Delta y = \frac{h}{mv_y} \), with \( h \) as Planck's constant, \( m \) as the particle's mass, and \( v_y \) as its speed. We'll need to find the minimum uncertainty in speed, \( \Delta v_y \), as a percentage of \( v_y \).
02

Heisenberg Uncertainty Principle

The Heisenberg uncertainty principle states \( \Delta y \Delta p_y \geq \frac{\hbar}{2} \), where \( \Delta p_y = m \Delta v_y \) and \( \hbar = \frac{h}{2\pi} \). Since \( \Delta y = \frac{h}{mv_y} \), substitute \( \Delta y \) in the uncertainty relation to find \( \Delta v_y \).
03

Substitute and Simplify

By substituting \( \Delta y = \frac{h}{mv_y} \), the uncertainty principle becomes \( \frac{h}{mv_y} \cdot m \Delta v_y \geq \frac{h}{4 \pi} \). Simplify to get \( \Delta v_y \geq \frac{v_y}{4\pi} \).
04

Calculate Percentage

To find the percentage, use the formula \( \text{Percentage} = \frac{\Delta v_y}{v_y} \times 100\% \). Substituting \( \Delta v_y \geq \frac{v_y}{4\pi} \), we get \( \text{Percentage} \geq \frac{1}{4\pi} \times 100\% \).
05

Final Calculation

Calculate \( \frac{1}{4\pi} \times 100\% \) to find the minimum percentage. It yields approximately 7.96%.

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.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

de Broglie wavelength
The de Broglie wavelength relates the wave-like properties of particles to their momentum. It is an essential concept in quantum mechanics, connecting the behavior of particles to classical wave theory. Defined by the equation \( \lambda = \frac{h}{p} \), where \( \lambda \) is the wavelength, \( h \) is Planck's constant, and \( p \) is the momentum. This discovery shows that every moving particle or object has an associated wave.
  • For particles with larger momentum, the de Broglie wavelength is shorter.
  • This concept is crucial when discussing the limits of classical mechanics and the wave-particle duality of matter.
  • It helps in understanding phenomena like electron diffraction and the behavior of subatomic particles within an atom.
Understanding the de Broglie wavelength allows us to comprehend the transition from classical physics, which deals with macroscopic objects, to quantum mechanics, where we explore the microscopic world of particles.
Quantum Mechanics
Quantum mechanics is a fundamental theory in physics that describes the physical properties of nature at the scale of atoms and subatomic particles. It departs from classical mechanics primarily in its inability to account for the particle-wave duality and indeterministic nature of microscopic phenomena.
  • One of the essential features is the Heisenberg Uncertainty Principle, which introduces intrinsic limits to the precision with which pairs of physical properties such as position and momentum can be simultaneously known.
  • Unlike classical mechanics, where an object's trajectory can be predicted with certainty, quantum mechanics provides probabilities of finding a particle in a certain position.
  • It offers a framework for understanding the behavior of electrons in atoms, leading to the modern understanding of chemical bonding and molecular structure.
Quantum mechanics doesn't just change our understanding of small particles; it also impacts the technology we use, including semiconductors and various types of imaging.
Planck's constant
Planck's constant is a fundamental constant in physics, denoted by \( h \). It is a central element in the field of quantum mechanics and plays a pivotal role in quantifying the relationship between the energy of a photon and the frequency of its electromagnetic wave.
  • Defined by the equation \( E = hu \), where \( E \) is energy, and \( u \) is frequency, it highlights the quantized nature of energy transfer at microscopic scales.
  • With a value of approximately \( 6.62607015 \times 10^{-34} \text{ J}\cdot\text{s} \), Planck's constant is exceedingly small, which is why quantum effects are typically only noticeable at atomic or subatomic levels.
  • In the context of de Broglie wavelength, Planck's constant illustrates how particles can exhibit both wave-like and particle-like properties.
Understanding Planck's constant allows scientists to make sense of the quantum mechanical systems and bridge the gap between classical and quantum physics.
Particle Physics
Particle physics is the branch of physics that studies the fundamental particles of the universe and the forces with which they interact. These particles are the building blocks of matter, and understanding their interactions helps in explaining the universe at the most basic and fundamental level.
  • It investigates particles such as electrons, protons, neutrons, and the more exotic quarks and leptons which form the basis for atoms.
  • The standard model of particle physics provides the most comprehensive explanation of these fundamental particles and their interactions via fundamental forces, except for gravity.
  • This field is responsible for breakthroughs like the discovery of the Higgs boson, a particle associated with imparting mass to other particles.
Overall, particle physics extends quantum mechanics and helps to explore questions about the universe's origins, structure, and the underlying laws of nature. It is a critical area of study that contributes to both theoretical understanding and practical technological advancements.

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

A proton is located at a distance of 0.420 m from a point charge of \(+8.30 \mu C\) . The repulsive electric force moves the proton until it is at a distance of 1.58 \(m\) from the charge. Suppose that the electric potential energy lost by the system were carried off by a photon. What would be its wavelength?

The kinetic energy of a particle is equal to the energy of a photon. The particle moves at 5.0% of the speed of light. Find the ratio of the photon wavelength to the de Broglie wavelength of the particle.

The work function of a metal surface is \(4.80 \times 10^{-19} \mathrm{J}\) . The maximum speed of the electrons emitted from the surface is \(v_{\mathrm{A}}=7.30 \times 10^{5} m/s\) when the wavelength of the light is \(\lambda_{\mathrm{A}} .\) However, a maximum speed of \(v_{\mathrm{B}}=5.00 \times 10^{5} m / s\)is observed when the wavelength is \(\lambda_{\mathrm{B}}\) . Find the wavelengths \(\lambda_{\mathrm{A}}\) and \(\lambda_{\mathrm{B}}\) .

How fast does a proton have to be moving in order to have the same de Broglie wavelength as an electron that is moving with a speed of \(4.50 \times 10^{6} \mathrm{m} / \mathrm{s} ?\)

In a Young's double-slit experiment that uses electrons, the angle that locates the first-order bright fringes is \(\theta_{\mathrm{A}}=1.6 \times 10^{-4}\) degrees when the magnitude of the electron momentum is \(p_{\mathrm{A}}=1.2 \times 10^{-22} \mathrm{kg} \cdot \mathrm{m} / \mathrm{s}\) . With the same double slit, what momentum magnitude \(p_{\mathrm{B}}\) is necessary so that an angle of \(\theta_{\mathrm{B}}=4.0 \times 10^{-4}\) degrees locates the first-order bright fringes?
See all solutions

Recommended explanations on Physics Textbooks

Famous Physicists

Read Explanation

Conservation of Energy and Momentum

Read Explanation

Medical Physics

Read Explanation

Space Physics

Read Explanation

Electrostatics

Read Explanation

Atoms and Radioactivity

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.