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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

High-speed electrons are used to probe the interior structure of the atomic nucleus. For such electrons the expression \(\lambda = h/p\) still holds, but we must use the relativistic expression for momentum, \(p = mv/\sqrt{1 - v^2/c^2}\). (a) Show that the speed of an electron that has de Broglie wavelength \(\lambda\) is $$v = \frac{c}{\sqrt1+(mc\lambda/h)^2} $$ (b) The quantity \(h/mc\) equals 2.426 \(\times\) 10\(^{-12}\) m. (As we saw in Section 38.3, this same quantity appears in Eq. (38.7), the expression for Compton scattering of photons by electrons.) If \(\lambda\) is small compared to \(h/mc\), the denominator in the expression found in part (a) is close to unity and the speed \(v\) is very close to c. In this case it is convenient to write \(v = (1 - \Delta)c\) and express the speed of the electron in terms of rather than v. Find an expression for \(\delta\) valid when \(\lambda \ll h mc\). [\(Hint:\) Use the binomial expansion (1 + \(z)^n = 1 + nz + [n(n - 1)z^2/2] + \cdots\) g, valid for the case 0 z 0 6 1.4 (c) How fast must an electron move for its de Broglie wavelength to be 1.00 \(\times\) 10\(^{-15}\) m, comparable to the size of a proton? Express your answer in the form \(v =(1 - \Delta)c\), and state the value of \(\Delta\)

For your work in a mass spectrometry lab, you are investigating the absorption spectrum of one-electron ions. To maintain the atoms in an ionized state, you hold them at low density in an ion trap, a device that uses a configuration of electric fields to confine ions. The majority of the ions are in their ground state, so that is the initial state for the absorption transitions that you observe. (a) If the longest wavelength that you observe in the absorption spectrum is 13.56 nm, what is the atomic number Z for the ions? (b) What is the next shorter wavelength that the ions will absorb? (c) When one of the ions absorbs a photon of wavelength 6.78 nm, a free electron is produced. What is the kinetic energy (in electron volts) of the electron?

Why is it easier to use helium ions rather than neutral helium atoms in such a microscope? (a) Helium atoms are not electrically charged, and only electrically charged particles have wave properties. (b) Helium atoms form molecules, which are too large to have wave properties. (c) Neutral helium atoms are more difficult to focus with electric and magnetic fields. (d) Helium atoms have much larger mass than helium ions do and thus are more difficult to accelerate.

How does the wavelength of a helium ion compare to that of an electron accelerated through the same potential difference? (a) The helium ion has a longer wavelength, because it has greater mass. (b) The helium ion has a shorter wavelength, because it has greater mass. (c) The wavelengths are the same, because the kinetic energy is the same. (d) The wavelengths are the same, because the electric charge is the same.

What is the de Broglie wavelength of a red blood cell, with mass 1.00 \(\times\) 10\(^{-11}\) g, that is moving with a speed of 0.400 cm/s? Do we need to be concerned with the wave nature of the blood cells when we describe the flow of blood in the body?

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