Problem 48 - College Physics

Question

A pesky 1.5 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) 5.0 m; (b) \( 2.8 \times 10^{-29} \) m/s; Heisenberg limit is negligible.

Step 1: Understanding the Maximum Uncertainty in Position

To determine the maximum uncertainty in the mosquito's horizontal position, we can assume it is the entire width of the room since the mosquito could be anywhere from the left to the right wall. Thus, the uncertainty in position, \( \Delta x \), is 5.0 meters.

Step 2: Applying the Heisenberg Uncertainty Principle

The Heisenberg Uncertainty Principle states that \( \Delta x \cdot \Delta p \geq \frac{h}{4\pi} \), where \( \Delta p \) is the uncertainty in momentum and \( h \) is the Planck's constant \( (6.626 \times 10^{-34} \, \mathrm{m^2 \, kg \, s^{-1}}) \). The momentum \( p = m \cdot v \), so \( \Delta p = m \cdot \Delta v \). We need to solve for \( \Delta v \).

Step 3: Calculating \( \Delta v \), the Velocity Uncertainty

Rearrange the Heisenberg equation to find \( \Delta v \): \[ \Delta v \geq \frac{h}{4 \pi \cdot m \cdot \Delta x} \] Substituting \( m = 1.5 \times 10^{-6} \, \mathrm{kg} \) and \( \Delta x = 5.0 \, \mathrm{m} \), we get: \[ \Delta v \geq \frac{6.626 \times 10^{-34}}{4 \pi \cdot 1.5 \times 10^{-6} \cdot 5.0} \approx 2.8 \times 10^{-29} \, \mathrm{m/s} \]

Step 4: Assessing the Effect of Velocity Uncertainty on Hitting the Mosquito

The uncertainty limitation results in an extremely small \( \Delta v \) of about \( 2.8 \times 10^{-29} \, \mathrm{m/s} \), which is negligible. Thus, this limitation does not impede your attempt to swat the mosquito, as the actual speed of a mosquito is much higher than this uncertainty.

Key Concepts

Position Uncertainty

Position uncertainty is all about understanding the limitations of knowing exactly where an object is located. In the case of our pesky mosquito, the maximum uncertainty in its horizontal position is determined by the dimensions of the room. Since the mosquito could be anywhere across the room's width, the uncertainty, denoted as \( \Delta x \), is 5.0 meters.
This value represents the range within which the mosquito might be positioned horizontally. Recognizing this range is crucial in settings where precision is essential, such as in advanced physics experiments.
Position uncertainty highlights that complete certainty about an object's location is fundamentally limited, especially in quantum mechanics where precise measurement changes the situation.

Momentum Uncertainty

Momentum uncertainty revolves around the uncertainty in measuring an object's momentum. Momentum, represented by \( p \), is calculated as the product of an object's mass and velocity (\( p = m \cdot v \)).
According to the Heisenberg Uncertainty Principle, there is a trade-off between how precisely we know an object's position and its momentum. As we have an uncertainty in position \( \Delta x \), there is an accompanying uncertainty in momentum \( \Delta p \).
The expression \( \Delta x \cdot \Delta p \geq \frac{h}{4\pi} \) defines this relationship, indicating that the more accurately we know the position, the less accurately we can know the momentum, and vice versa. This principle has profound implications in quantum mechanics, where momentum uncertainty plays a pivotal role in understanding the behavior of particles.

Velocity Measurement

Velocity measurement is a key component when considering how fast an object moves and the precision of such measurements. The mosquito's velocity is a significant factor when attempting to swat it. The Heisenberg Uncertainty Principle shows that we can't know the velocity exactly without a trade-off in the certainty of its position.
By rearranging the principle's equation, \( \Delta v \geq \frac{h}{4 \pi \cdot m \cdot \Delta x} \), we find the minimum uncertainty in velocity, \( \Delta v \). For our mosquito, this comes out to be approximately \( 2.8 \times 10^{-29} \, \mathrm{m/s} \).

• This tiny number tells us that the uncertainty in measuring its velocity is practically negligible.
• It means that the mosquito's actual speed far exceeds this uncertainty, making it realistically possible to aim for it successfully.

In practical terms, this lets us understand why such a small uncertainty doesn't affect day-to-day tasks, like swatting a mosquito.

Quantum Mechanics

Quantum mechanics is the branch of physics that deals with the behavior of very small particles, such as electrons and photons, which behave differently than larger everyday objects. It teaches us that at the microscopic scale, many classical concepts like precise velocity or position don't apply exactly as they do in the macroscopic world.
Key principles like the Heisenberg Uncertainty Principle stem from quantum mechanics, illustrating that on a tiny scale, there's a fundamental limit to how much we can know about a particle's position and momentum at the same time. This principle underlines the probabilistic nature of quantum mechanics, where outcomes can only be predicted in terms of probabilities.
Quantum mechanics is crucial for understanding advanced technology and various natural phenomena, providing a framework that governs the microscopic universe.