What is the position uncertainty, in nm, of an electron...

What is the position uncertainty, in $\mathrm{nm}$, of an electron whose velocity is known to be between $3.48 \times 10^{5} \mathrm{m} / \mathrm{s}$ and $3.58 \times$ $10^{5} \mathrm{m} / \mathrm{s} ?$

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Key Concepts

Reduced Planck Constant (?)

The reduced Planck constant, denoted by ?, is a fundamental physical constant used to express the scale at which quantum effects become significant. It appears in the uncertainty relationship and sets the magnitude of the minimum product of the uncertainties of conjugate variables like position and momentum.

Relation Between Velocity and Momentum

This concept emphasizes the direct proportionality between velocity and momentum in classical mechanics, which carries over to quantum mechanics. It allows the translation of uncertainty in velocity into uncertainty in momentum, which is key to applying the Heisenberg Uncertainty Principle.

Heisenberg Uncertainty Principle

This principle is a fundamental limit in quantum mechanics stating that it is impossible to simultaneously know both the exact position and the exact momentum of a particle. The more precisely one of these properties is measured, the less precisely the other can be controlled or determined, and it is quantified by the inequality ?x · ?p ? ?/2.

Momentum Uncertainty

In quantum mechanics, momentum uncertainty is related to the uncertainty in velocity through the particle's mass. Since momentum (p) is defined as the product of mass and velocity, any uncertainty in velocity (?v) directly translates into an uncertainty in momentum (?p = m · ?v).

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