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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 beam of alpha particles is incident on a target of lead. A particular alpha particle comes in "head-on" to a particular lead nucleus and stops 6.50 \(\times\) 10\(^{-14}\) m away from the center of the nucleus. (This point is well outside the nucleus.) Assume that the lead nucleus, which has 82 protons, remains at rest. The mass of the alpha particle is 6.64 \(\times\) 10\(^{-27}\) kg. (a) Calculate the electrostatic potential energy at the instant that the alpha particle stops. Express your result in joules and in MeV. (b) What initial kinetic energy (in joules and in MeV) did the alpha particle have? (c) What was the initial speed of the alpha particle?

Consider a particle with mass m moving in a potential \(U = {1\over2} kx^2\), as in a mass-spring system. The total energy of the particle is \(E = (p^2/2m) + 12 kx^2\). Assume that \(p\) and \(x\) are approximately related by the Heisenberg uncertainty principle, so \(px \approx h\). (a) Calculate the minimum possible value of the energy \(E\), and the value of \(x\) that gives this minimum E. This lowest possible energy, which is not zero, is called the \(zero-point \space energy\). (b) For the \(x\) calculated in part (a), what is the ratio of the kinetic to the potential energy of the particle?

Radiation has been detected from space that is characteristic of an ideal radiator at \(T\) = 2.728 K. (This radiation is a relic of the Big Bang at the beginning of the universe.) For this temperature, at what wavelength does the Planck distribution peak? In what part of the electromagnetic spectrum is this wavelength?

(a) A particle with mass \(m\) has kinetic energy equal to three times its rest energy. What is the de Broglie wavelength of this particle? (\(Hint\): You must use the relativistic expressions for momentum and kinetic energy: \(E^2 = (pc^2) + (mc^2)^2\) and \(K = E - mc^2\).) (b) Determine the numerical value of the kinetic energy (in MeV) and the wavelength (in meters) if the particle in part (a) is (i) an electron and (ii) a proton.

(a) An electron moves with a speed of 4.70 \(\times\) 10\(^6\) m/s. What is its de Broglie wavelength? (b) A proton moves with the same speed. Determine its de Broglie wavelength.

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