An electron traveling at 3.7×10⁵ m/s has an uncertainty in its velocity of 1.88×10⁵ m/s. What is the uncertainty in its position?

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Identify the relevant principle: This problem involves the Heisenberg Uncertainty Principle, which relates the uncertainty in position (\( \Delta x \)) and the uncertainty in momentum (\( \Delta p \)).

Recall the Heisenberg Uncertainty Principle formula: \( \Delta x \cdot \Delta p \geq \frac{h}{4\pi} \), where \( h \) is Planck's constant (\( 6.626 \times 10^{-34} \text{ Js} \)).

Determine the uncertainty in momentum (\( \Delta p \)): Since momentum \( p = m \cdot v \), the uncertainty in momentum \( \Delta p = m \cdot \Delta v \), where \( m \) is the mass of the electron (\( 9.11 \times 10^{-31} \text{ kg} \)) and \( \Delta v \) is the uncertainty in velocity.

Substitute the known values into the formula for \( \Delta p \): \( \Delta p = 9.11 \times 10^{-31} \text{ kg} \times 1.88 \times 10^{5} \text{ m/s} \).

Use the Heisenberg Uncertainty Principle to solve for \( \Delta x \): Rearrange the formula to \( \Delta x \geq \frac{h}{4\pi \cdot \Delta p} \) and substitute the calculated \( \Delta p \) and known \( h \) to find \( \Delta x \).

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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 to measurement in quantum mechanics, indicating that the more precisely one property is measured, the less precisely the other can be controlled or known.

Momentum

Momentum is a physical quantity defined as the product of an object's mass and its velocity. In the context of quantum mechanics, momentum is often represented as a vector quantity, and its uncertainty is directly related to the uncertainty in the particle's position, as described by the Heisenberg Uncertainty Principle.

Calculating Uncertainty

To calculate the uncertainty in position (Δx) when given the uncertainty in velocity (Δv), one can use the formula Δx * Δp ≥ ħ/2, where Δp is the uncertainty in momentum and ħ is the reduced Planck's constant. Since momentum (p) is mass (m) times velocity (v), the uncertainty in momentum can be expressed as Δp = m * Δv, allowing for the determination of Δx when Δv is known.

An electron traveling at 3.7×105 m/s has an uncertainty in its velocity of 1.88×105 m/s. What is the uncertainty in its position?