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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

Expert verified
(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

01

Determine Uncertainty in Horizontal Position

The maximum uncertainty in the horizontal position, \( \Delta x \), can be considered as the width of the room. Therefore, \( \Delta x = 5.0 \text{ m} \).
02

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 \( 1.055 \times 10^{-34} \text{ J}\cdot\text{s} \).
03

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:\[ 5.0 \cdot 1.5 \times 10^{-6} \cdot \Delta v \geq \frac{1.055 \times 10^{-34}}{2} \]
04

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 \( \Delta v \): \[ \Delta v \geq 7.03 \times 10^{-30} \text{ m/s} \]
05

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.

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Key Concepts

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

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.

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Most popular questions from this chapter

A scientist has devised a new method of isolating individual particles. He claims that this method enables him to detect simultaneously the position of a particle along an axis with a standard deviation of 0.12 nm and its momentum component along this axis with a standard deviation of 3.0 \(\times\) 10\(^{-25}\) kg \(\bullet\) m/s. Use the Heisenberg uncertainty principle to evaluate the validity of this claim.

An electron beam and a photon beam pass through identical slits. On a distant screen, the first dark fringe occurs at the same angle for both of the beams. The electron speeds are much slower than that of light. (a) Express the energy of a photon in terms of the kinetic energy \(K\) of one of the electrons. (b) Which is greater, the energy of a photon or the kinetic energy of an electron?

The brightest star in the sky is Sirius, the Dog Star. It is actually a binary system of two stars, the smaller one (Sirius B) being a white dwarf. Spectral analysis of Sirius B indicates that its surface temperature is 24,000 K and that it radiates energy at a total rate of 1.0 \(\times\) 10\(^{25}\) W. Assume that it behaves like an ideal blackbody. (a) What is the total radiated intensity of Sirius B? (b) What is the peak-intensity wavelength? Is this wavelength visible to humans? (c) What is the radius of Sirius B? Express your answer in kilometers and as a fraction of our sun's radius. (d) Which star radiates more \(total\) energy per second, the hot Sirius B or the (relatively) cool sun with a surface temperature of 5800 K? To find out, calculate the ratio of the total power radiated by our sun to the power radiated by Sirius B.

The wavelength 10.0 \(\mu\)m is in the infrared region of the electromagnetic spectrum, whereas 600 nm is in the visible region and 100 nm is in the ultraviolet. What is the temperature of an ideal blackbody for which the peak wavelength \(\lambda_m\) is equal to each of these wavelengths?

The neutral pion (\(\pi^0\)) is an unstable particle produced in high-energy particle collisions. Its mass is about 264 times that of the electron, and it exists for an average lifetime of 8.4 \(\times\) 10\(^{-17}\) s before decaying into two gamma-ray photons. Using the relationship \(E = mc^2\) between rest mass and energy, find the uncertainty in the mass of the particle and express it as a fraction of the mass.

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