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\$$times10^{-25} kg-m/s. $$Use the Heisenberg uncertainty principle to evaluate the validity of this claim.

Ch 39: Particles Behaving as Waves

Young & Freedman Calc - University Physics 14th Edition

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Young & Freedman Calc 14th Edition

Ch 39: Particles Behaving as Waves

문제 47

39장, 문제 47

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-m/s. Use the Heisenberg uncertainty principle to evaluate the validity of this claim.

검증된 단계별 안내

Step 1: Recall the Heisenberg uncertainty principle, which states that the product of the uncertainties in position (Δx) and momentum (Δp) must satisfy the inequality Δx * Δp ≥ ℏ/2, where ℏ is the reduced Planck's constant (ℏ = 1.0545718 × 10^-34 J·s).

Step 2: Identify the given values in the problem: the standard deviation of the position (Δx) is 0.12 nm (convert this to meters: 0.12 × 10^-9 m), and the standard deviation of the momentum (Δp) is 3.0 × 10^-25 kg·m/s.

Step 3: Calculate the product of the uncertainties Δx * Δp using the given values. Multiply the converted position uncertainty (in meters) by the momentum uncertainty (in kg·m/s).

Step 4: Compare the calculated product Δx * Δp to the minimum value required by the Heisenberg uncertainty principle, ℏ/2. Compute ℏ/2 using ℏ = 1.0545718 × 10^-34 J·s and divide it by 2.

Step 5: Evaluate whether the scientist's claim is valid by checking if the calculated product Δx * Δp meets or exceeds ℏ/2. If it does not, the claim violates the Heisenberg uncertainty principle and is invalid.

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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 of a particle. Mathematically, it is expressed as Δx * Δp ≥ ħ/2, where Δx is the uncertainty in position, Δp is the uncertainty in momentum, and ħ is the reduced Planck's constant. This principle highlights the fundamental limits of measurement in quantum mechanics.

Standard Deviation in Measurement

Standard deviation is a statistical measure that quantifies the amount of variation or dispersion in a set of values. In the context of the Heisenberg Uncertainty Principle, the standard deviations of position (Δx) and momentum (Δp) represent the uncertainties in these measurements. A smaller standard deviation indicates a more precise measurement, but according to the uncertainty principle, this precision in one measurement leads to greater uncertainty in the other.

Planck's Constant

Planck's constant (ħ) is a fundamental constant in quantum mechanics that relates the energy of a photon to its frequency. It has a value of approximately 1.055 x 10^-34 J·s. In the context of the Heisenberg Uncertainty Principle, it sets the scale for the limits of precision in measurements of position and momentum, emphasizing that at quantum scales, these uncertainties are not just theoretical but have real implications for particle behavior.