Question

A pesky 1.5-mg mosquito is annoying you as you attempt to study physics in your room, which is 5.0 m wide and 2.5 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?

Short answer

(a) Maximum position uncertainty is 5.0 m. (b) Velocity uncertainty is insignificant at 7.03 x 10^-30 m/s, posing no serious impediment.

Step-by-step solution

Determine Uncertainty in Horizontal Position

The maximum uncertainty in the horizontal position, \( \Delta x \), can be considered as the width of the room. Therefore, \( \Delta x = 5.0 \text{ m} \).

Apply Heisenberg Uncertainty Principle

The Heisenberg Uncertainty Principle can be expressed as: \[ \Delta x \cdot \Delta p \geq \frac{\hbar}{2} \]where \( \Delta p \) is the uncertainty in momentum, and \( \hbar \) (the reduced Planck constant) is approximately \( 1.055 \times 10^{-34} \text{ J}\cdot\text{s} \).

Relate Momentum and Velocity

The uncertainty in momentum \( \Delta p \) is related to the uncertainty in velocity \( \Delta v \) by the equation: \[ \Delta p = m \cdot \Delta v \]where \( m = 1.5 \times 10^{-6} \text{ kg} \) is the mass of the mosquito. Substitute this into the Heisenberg equation to get:\[ 5.0 \cdot 1.5 \times 10^{-6} \cdot \Delta v \geq \frac{1.055 \times 10^{-34}}{2} \]

Solve for Uncertainty in Velocity

Rearrange the inequality to solve for \( \Delta v \): \[ \Delta v \geq \frac{1.055 \times 10^{-34}}{2 \cdot 5.0 \cdot 1.5 \times 10^{-6}} \]Calculate \( \Delta v \): \[ \Delta v \geq 7.03 \times 10^{-30} \text{ m/s} \]

Evaluate Impact on Swatting Attempt

The uncertainty in the mosquito's velocity \( \Delta v \) is extremely small, \( 7.03 \times 10^{-30} \text{ m/s} \), which is negligible compared to practical considerations when trying to swat the mosquito. Therefore, this quantum mechanical limitation is not a significant impairment to the physical task.

Key concepts

Quantum Mechanics

Quantum mechanics is a fundamental theory in physics that describes nature at the smallest scales of energy levels, such as those of subatomic particles. It is far removed from the classical mechanics that we might be more familiar with, dealing instead with probabilities and uncertainties in exactly the way the Heisenberg Uncertainty Principle does. At its heart, quantum mechanics does not predict exact results, but rather the likelihood of different outcomes.

Unlike classical mechanics, where you can know both the position and velocity of an object precisely, quantum mechanics tells us that at very small scales, such as those of electrons or the mosquito in our example, this precision is not possible. This inherent uncertainty is not due to measurement errors but is a fundamental property of the universe. The Heisenberg Uncertainty Principle highlights this, making it a cornerstone of quantum mechanics itself.

Momentum Uncertainty

Momentum uncertainty is a crucial part of the Heisenberg Uncertainty Principle, which states that the more precisely you know the position of a particle, like our mosquito, the less precisely you can know its momentum and vice versa.
In the exercise, understanding momentum, which is mass times velocity, is key.

• The mosquito has a mass of 1.5 mg, which is converted to kilograms for use in equations—so, 1.5 x 10^-6 kg.
• The uncertainty in momentum (\(\Delta p\)) plays a role in determining how well we can predict the velocity at any point.

The reduced Planck constant (\(\hbar\)), which appears in the uncertainty inequality, provides a numerical cap to how zero we can drive these uncertainties. This Principle is less noticeable in macroscopic situations but becomes critical at atomic and subatomic levels.

Velocity Uncertainty

Velocity uncertainty arises directly from trying to measure an object's velocity while having knowledge, or lack thereof, of its exact position. Our mosquito's case emphasizes this concept by demonstrating how even a very light object like this brings the Heisenberg Uncertainty to practical awareness.
The calculated uncertainty in velocity (\(\Delta v\)) turns out to be extraordinarily small, approximately 7.03 x 10^-30 m/s. This value shows just how precise our knowledge can be in certain situations, especially when translating quantum mechanical equations to real-world contexts like swatting a mosquito.
However, it's vital to remember that high precision at these scales doesn't significantly impact everyday activities due to the large scale differences with atomic or subatomic worlds:

• It shows that while quantum effects can be calculated, they don't hinder practical actions like trying to swat a mosquito.
• It mainly affects incredibly minute systems, with no noticeable influence when dealing with human-sized perceptions.

In essence, velocity uncertainty, though real, does not detract from our ability to achieve ordinary tasks.