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Ch 39: Particles Behaving as Waves
Young & Freedman Calc - University Physics 15th Edition
Young & Freedman Calc15th EditionUniversity PhysicsISBN: 9780135159552당신이 사용하는 게 아니라요?교과서 변경
모든 교과서Young & Freedman Calc 15th EditionCh 39: Particles Behaving as Waves문제 444
38장, 문제 444

A pesky 1.51.5-mg mosquito is annoying you as you attempt to study physics in your room, which is 5.05.0 m wide and 2.52.5 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?

검증된 단계별 안내
1
Step 1: Understand the problem and identify the key concepts. This problem involves the Heisenberg Uncertainty Principle, which states that the uncertainty in position (Δx) and the uncertainty in momentum (Δp) are related by the inequality Δx * Δp ≥ ℏ / 2, where ℏ is the reduced Planck's constant (ℏ ≈ 1.055 × 10⁻³⁴ J·s). Momentum is related to velocity by p = m * v, where m is the mass and v is the velocity.
Step 2: Calculate the maximum uncertainty in the horizontal position of the mosquito. Since the mosquito is flying in a room that is 5.0 m wide, the maximum uncertainty in its horizontal position (Δx) can be taken as the width of the room, Δx = 5.0 m.
Step 3: Use the Heisenberg Uncertainty Principle to find the minimum uncertainty in the horizontal momentum (Δp). Rearrange the inequality to solve for Δp: Δp ≥ ℏ / (2 * Δx). Substitute the known values of ℏ and Δx into this equation to calculate Δp.
Step 4: Relate the uncertainty in momentum (Δp) to the uncertainty in velocity (Δv). Since momentum is given by p = m * v, the uncertainty in velocity can be expressed as Δv = Δp / m. Use the mosquito's mass (m = 1.5 mg = 1.5 × 10⁻⁶ kg) and the previously calculated Δp to find Δv.
Step 5: Analyze whether the uncertainty in velocity (Δv) is significant. Compare the calculated Δv to the typical speed of a mosquito (which is on the order of 1 m/s). If Δv is much smaller than the mosquito's speed, the uncertainty is negligible, and the Heisenberg Uncertainty Principle does not pose a serious impediment to swatting the mosquito.

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.
영상 길이:
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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Heisenberg Uncertainty Principle

The Heisenberg Uncertainty Principle states that it is impossible to simultaneously know both the exact position and exact momentum (or velocity) of a particle. This principle highlights a fundamental limit in quantum mechanics, where the more precisely one property is known, the less precisely the other can be known. For small particles like a mosquito, this principle becomes significant when estimating their position and velocity.
추천 영상:
가이드 코스
14:47
Diffraction with Huygen's Principle

Momentum

Momentum is a vector quantity defined as the product of an object's mass and its velocity. In physics, momentum is crucial because it is conserved in isolated systems, meaning the total momentum before and after an event remains constant. Understanding momentum helps in analyzing the motion of objects, including the mosquito in this scenario, and is essential for calculating its velocity.
추천 영상:
가이드 코스
05:17
Intro to Momentum

Measurement Uncertainty

Measurement uncertainty refers to the doubt that exists about the result of any measurement. In the context of the mosquito's position and velocity, it quantifies the limits of precision in determining these values. This uncertainty can arise from various factors, including the limitations of measuring instruments and the inherent variability in the system being measured, which is particularly relevant when applying the Heisenberg Uncertainty Principle.
추천 영상:
가이드 코스
07:49
Proper Frames and Measurements
관련 실천
교과서 질문

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 0.120.12 nm and its momentum component along this axis with a standard deviation of 3.0×10−253.0\(\times\)10^{-25} kg-m/s. Use the Heisenberg uncertainty principle to evaluate the validity of this claim.

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교과서 질문

The uncertainty in the y-component of a proton's position is 2.0×10−122.0\(\times\)10^{-12} m. What is the minimum uncertainty in a simultaneous measurement of the yy-component of the proton's velocity?

1705
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교과서 질문

10.010.0-g marble is gently placed on a horizontal tabletop that is 1.751.75 m wide.

(a) What is the maximum uncertainty in the horizontal position of the marble?

(b) According to the Heisenberg uncertainty principle, what is the minimum uncertainty in the horizontal velocity of the marble?

(c) In light of your answer to part (b), what is the longest time the marble could remain on the table? Compare this time to the age of the universe, which is approximately 1414 billion years. (Hint: Can you know that the horizontal velocity of the marble is exactly zero?)

1748
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