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Ch.7 - Quantum-Mechanical Model of the Atom
Chapter 7, Problem 56

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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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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.
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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.
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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.
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