Question

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

(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

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, \( \Delta x = 1.75 \ m \).

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).

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:\[ \Delta v \geq \frac{1.0545718 \times 10^{-34}}{2 \cdot 0.01 \cdot 1.75} \approx 3.01 \times 10^{-34} \: m/s \]

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 \( t \):\[ t \approx 5.81 \times 10^{33} \: seconds \]

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 (\( 4.41 \times 10^{17} \text{ seconds} \)).

Key concepts

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.