Problem 45 - College Physics

Question

A pesky \(1.5 \mathrm{mg}\) mosquito is annoying you as you attempt to study physics in your room, which is \(5.0 \mathrm{~m}\) wide and \(2.5 \mathrm{~m}\) high. You decide to swat the bothersome insect as it flies toward you, but you need to estimate its speed to make a successful hit. (a) What is the maximum uncertainty in the horizontal position of the mosquito? (b) What limit does the Heisenberg uncertainty principle place on your ability to know the horizontal velocity of this mosquito? Is this limitation a serious impediment to your attempt to swat it?

Step-by-Step Solution

Answer

(a) Maximum uncertainty in position is 5.0 m. (b) Velocity uncertainty is \(7 \times 10^{-30}\, \text{m/s}\); not a serious impediment.

Step 1: Understand the Problem

We need to find the maximum uncertainty in the horizontal position of the mosquito and the implications of the Heisenberg Uncertainty Principle on the mosquito's horizontal velocity. The mosquito's mass and the room dimensions are given.

Step 2: Determine the Maximum Uncertainty in Position

The maximum uncertainty in the mosquito's horizontal position \(\Delta x\) can be assumed as the width of the room, given as \(5.0\, \text{m}\).

Step 3: Apply the Heisenberg Uncertainty Principle

The Heisenberg Uncertainty Principle states that \(\Delta x \cdot \Delta p \geq \frac{\hbar}{2}\), where \(\Delta p\) is the uncertainty in linear momentum and \(\hbar = \frac{h}{2\pi}\) is the reduced Planck's constant \(1.05 \times 10^{-34}\, \text{J}\cdot\text{s}\).

Step 4: Calculate the Uncertainty in Momentum

Since \(\Delta x = 5.0\, \text{m}\), we set up the inequality: \(5.0\, \text{m} \cdot \Delta p \geq 5.25 \times 10^{-35}\, \text{J}\cdot\text{s}\). Solving for \(\Delta p\), we find \(\Delta p \geq 1.05 \times 10^{-35}\, \text{kg}\cdot\text{m/s}\).

Step 5: Calculate the Uncertainty in Velocity

Using the formula \(\Delta p = m \cdot \Delta v\), where the mosquito's mass \(m = 1.5 \times 10^{-6}\, \text{kg}\), solve for \(\Delta v\): \(\Delta v \geq \frac{1.05 \times 10^{-35}}{1.5 \times 10^{-6}} = 7 \times 10^{-30}\, \text{m/s}\).

Step 6: Evaluate the Limitation's Impact

The result \(\Delta v = 7 \times 10^{-30}\, \text{m/s}\) is extremely small, indicating that the uncertainty principle does not pose a significant limitation to swatting the mosquito.

Key Concepts

Uncertainty in Position and Momentum

In the realm of quantum mechanics, one fundamental concept is the Heisenberg Uncertainty Principle. This principle states that there is a fundamental limit to how precisely we can know both the position and momentum of a particle simultaneously. The equation is characterized by \[ \Delta x \cdot \Delta p \geq \frac{\hbar}{2} \],where \( \Delta x \) represents the uncertainty in position, and \( \Delta p \) is the uncertainty in momentum. Here, \( \hbar \), the reduced Planck's constant, equals approximately \( 1.05 \times 10^{-34} \text{ J}\cdot\text{s} \).

• For the pesky mosquito problem, the maximum horizontal position uncertainty \( \Delta x \) is assumed to be equal to the room's width: 5 meters.
• The resulting uncertainty in the mosquito's momentum \( \Delta p \) defined by \( \Delta x \geq 5 \text{ m} \) yields a minimal \( \Delta p \).

The conclusion is that while the position of the mosquito can vary within these constraints, the uncertainty in its velocity is extremely small due to the minute scale of our Planck's constant, which has a negligible effect in this context.

Quantum Mechanics in Everyday Life

Many people might think that quantum mechanics only applies to atoms, molecules, or phenomena on a microscopic scale. However, its principles can also relate to seemingly everyday scenarios. In the mosquito swatting exercise, though the uncertainty principle is not significantly impactful on perception or actions due to the enormous differences in scales, it does remind us of the elegance of nature's rules at the fundamental level.

• This principle governs the underlying physics, even when dealing with larger objects like bugs in our everyday surroundings.
• Conceptualizing these abstracts helps us appreciate their applications across various domains, from quantum computing developments to understanding universal mechanics.

Physics Problem-Solving

Solving physics problems often requires a good grasp of theoretical principles and the ability to apply them to practical situations. Let's break down the mosquito problem step-by-step:

• Identify the Problem: Understand what you're trying to find. Here, it involves estimating the mosquito's position and velocity uncertainty.
• Use the Given Information: Utilize given data such as mass, room dimensions, and Heisenberg's formula to set up the problem.
• Apply the Principle: Use the Heisenberg Uncertainty Principle to link position and velocity uncertainty.
• Carry Out Calculations: Perform calculations with careful attention to maintaining units and accuracy.
• Evaluate Relevance: Determine if the uncertainty affects the objective — here, if it’s significant for swatting the mosquito, which it isn’t!

Tackling these problems not only involves plugging numbers but understanding the wider picture of how principles like quantum mechanics play out in practical daily scenarios.