Warning: foreach() argument must be of type array|object, bool given in /var/www/html/web/app/themes/studypress-core-theme/template-parts/header/mobile-offcanvas.php on line 20

A 10.0-g marble is gently placed on a horizontal tabletop that is 1.75 m wide. (a) What is the maximum uncertainty in the horizontal position of the marble? (b) According to the Heisenberg uncertainty principle, what is the minimum uncertainty in the horizontal velocity of the marble? (c) In light of your answer to part (b), what is the longest time the marble could remain on the table? Compare this time to the age of the universe, which is approximately 14 billion years. (\(Hint\): Can you know that the horizontal velocity of the marble is \(exactly\) zero?)

Short Answer

Expert verified
(a) \( \Delta x = 1.75 \ m \); (b) \( \Delta v \geq 3.01 \times 10^{-34} \ m/s \); (c) Time is about \( 5.81 \times 10^{33} \) seconds, much greater than the age of the universe.

Step by step solution

01

Determine Position Uncertainty

Since the marble can be anywhere on the tabletop, the maximum uncertainty in its position, \( \Delta x \), is equal to the width of the table. Thus, \( \Delta x = 1.75 \ m \).
02

Apply Heisenberg's Uncertainty Principle

According to Heisenberg's uncertainty principle, the uncertainty in position (\( \Delta x \)) and the uncertainty in momentum (\( \Delta p \)) are related by: \[ \Delta x \cdot \Delta p \geq \frac{\hbar}{2} \]where \( \hbar = 1.0545718 \times 10^{-34} \: J\cdot s \) is the reduced Planck's constant. We need to find \( \Delta p \), which is \( m \cdot \Delta v \) (mass times velocity uncertainty).
03

Calculate Minimum Uncertainty in Velocity

For the marble:\[ \Delta x \cdot m \cdot \Delta v \geq \frac{\hbar}{2} \]So, solving for \( \Delta v \):\[ \Delta v \geq \frac{\hbar}{2m \cdot \Delta x} \]Substitute \( m = 0.01 \ kg \) and \( \Delta x = 1.75 \ m \) to find:\[ \Delta v \geq \frac{1.0545718 \times 10^{-34}}{2 \cdot 0.01 \cdot 1.75} \approx 3.01 \times 10^{-34} \: m/s \]
04

Calculate Maximum Time on Table

The marble's maximum time on the table can be estimated by dividing the width of the table by the minimum velocity:\[ t = \frac{1.75 \ m}{3.01 \times 10^{-34} \ m/s} \]Calculate \( t \):\[ t \approx 5.81 \times 10^{33} \: seconds \]
05

Compare Time with Universe Age

Convert 14 billion years to seconds:\[ 14 \times 10^9 \text{ years} \times 365.25 \times 24 \times 60 \times 60 \approx 4.41 \times 10^{17} \text{ seconds} \]Compare:The time the marble could theoretically remain on the table (\( 5.81 \times 10^{33} \text{ seconds} \)) is much larger than the age of the universe (\( 4.41 \times 10^{17} \text{ seconds} \)).

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 Vaia!

Key Concepts

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

Quantum Mechanics
Quantum mechanics is the branch of physics that studies the behavior of very small particles, such as electrons and photons, which cannot be described by classical physics. In this realm, phenomena such as superposition and entanglement highlight the unique nature of particles.
Heisenberg's uncertainty principle plays a crucial role in quantum mechanics by introducing limits to the precision with which certain pairs of physical properties, like position and momentum, can be simultaneously known. Despite its counterintuitive nature, this principle forms an integral part of quantum theory, providing a foundational understanding of particle behavior.
In this exercise, we use quantum mechanics to explore how uncertainties in measuring the position and velocity of a marble relate to fundamental quantum principles. While the marble is commonplace and much larger than typical quantum objects, applying quantum mechanics helps explain precision limits in real-world scenarios.
Position Uncertainty
Position uncertainty refers to the lack of precise knowledge about where an object is located at any given time. According to Heisenberg's uncertainty principle, determining a particle's accurate position inherently affects our ability to know its momentum, and vice versa.
In the exercise given, the position uncertainty of the marble (\( \Delta x \)) is equal to the width of the table since it can be located anywhere along that width. Thus, \( \Delta x = 1.75 \, m \).
This uncertainty showcases our inherent limitation in knowing a particle's exact location, illustrating how quantum mechanics affects measurement precision, even for something as simple as a marble on a table. It highlights a fundamental principle: the more precisely one property is known, the less precisely the other can be known.
Velocity Uncertainty
Velocity uncertainty refers to the inability to exactly determine how fast a particle is moving in a specific direction. In quantum mechanics, when position uncertainty is high, velocity uncertainty cannot be minimal.
The formulation for velocity uncertainty involves the relationship established by the Heisenberg uncertainty principle, \( \Delta x \cdot m \cdot \Delta v \geq \frac{\hbar}{2} \). By rearranging this formula, we can solve for the minimum velocity uncertainty. For the marble:
  • \( \Delta v \geq \frac{\hbar}{2m \cdot \Delta x} \)
  • Using a mass (\( m \)) of 0.01 \, kg and a position uncertainty (\( \Delta x \)) of 1.75 \, m, we calculate \( \Delta v \geq 3.01 \times 10^{-34} \, m/s \).
This tiny velocity value underscores quantum mechanics' influence on measurement limits, reflecting the reality that absolute certainty in location or velocity is unattainable.
Planck's Constant
Planck's constant (\( h \)) is a fundamental constant in quantum physics, pivotal in the quantification of energy levels and action-like quantities in atomic scale interactions. Reduced Planck’s constant (\( \hbar = \frac{h}{2\pi} \)) is often used in the context of Heisenberg's uncertainty principle.
In our exercise, \( \hbar \) is critical for calculating the relationship between position and momentum uncertainties. Its value, \( 1.0545718 \times 10^{-34} \, J\cdot s \), is applied in quantum formulas to find limits on the precision of simultaneous measurements.
Planck's constant bridges quantum and classical physics, explaining phenomena that lack classical explanation, such as particle-wave duality and the quantization of light. In the analysis of the marble, \( \hbar \) embodies the constraints dictated by quantum principles, highlighting the universality of quantum effects across various scales.

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

Why is it easier to use helium ions rather than neutral helium atoms in such a microscope? (a) Helium atoms are not electrically charged, and only electrically charged particles have wave properties. (b) Helium atoms form molecules, which are too large to have wave properties. (c) Neutral helium atoms are more difficult to focus with electric and magnetic fields. (d) Helium atoms have much larger mass than helium ions do and thus are more difficult to accelerate.

What is the de Broglie wavelength of a red blood cell, with mass 1.00 \(\times\) 10\(^{-11}\) g, that is moving with a speed of 0.400 cm/s? Do we need to be concerned with the wave nature of the blood cells when we describe the flow of blood in the body?

The negative muon has a charge equal to that of an electron but a mass that is 207 times as great. Consider a hydrogenlike atom consisting of a proton and a muon. (a) What is the reduced mass of the atom? (b) What is the ground-level energy (in electron volts)? (c) What is the wavelength of the radiation emitted in the transition from the \(n\) = 2 level to the \(n\) = 1 level?

How many photons per second are emitted by a 7.50-mW CO\(_2\) laser that has a wavelength of 10.6 \(\mu\)m?

If your wavelength were 1.0 m, you would undergo considerable diffraction in moving through a doorway. (a) What must your speed be for you to have this wavelength? (Assume that your mass is 60.0 kg.) (b) At the speed calculated in part (a), how many years would it take you to move 0.80 m (one step)? Will you notice diffraction effects as you walk through doorways?

See all solutions

Recommended explanations on Physics Textbooks

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