The uncertainty involved in the measurement of velocity within a distance of [tex]$0.1 A^0$[/tex] is:
a) [tex]$5.79 \times 10^6 m / s$[/tex]
b) [tex]$5.79 \times 10^7 m / s$[/tex]
c) [tex]$5.79 \times 10^8 m / s$[/tex]
d) [tex]$5.79 \times 10^5 m / s$[/tex]
Answer (2)
Apply the Heisenberg uncertainty principle: Δ x Δ p g e 4 π h .
Substitute Δ p = m Δ v into the uncertainty principle.
Solve for Δ v : Δ vg e 4 πm Δ x h .
Plug in the given values to find Δ v ≈ 5.79 × 1 0 6 m / s , so the answer is 5.79 × 1 0 6 m / s .
Explanation
Problem Setup We are given the uncertainty in position, Δ x = 0.1 A 0 , and we need to find the uncertainty in velocity, Δ v . We will use the Heisenberg uncertainty principle to relate these quantities.
Heisenberg Uncertainty Principle The Heisenberg uncertainty principle states that Δ x Δ p ≥ 4 π h , where Δ x is the uncertainty in position, Δ p is the uncertainty in momentum, and h is Planck's constant.
Momentum and Uncertainty Since momentum p = m v , the uncertainty in momentum is Δ p = m Δ v , where m is the mass of the particle. We assume the particle is an electron, so m = 9.11 × 1 0 − 31 k g .
Substitution Substitute Δ p = m Δ v into the Heisenberg uncertainty principle: Δ x ( m Δ v ) ≥ 4 π h .
Solving for Velocity Uncertainty Solve for the uncertainty in velocity: Δ v ≥ 4 πm Δ x h .
Calculation Plug in the values: h = 6.626 × 1 0 − 34 J s , π ≈ 3.14 , m = 9.11 × 1 0 − 31 k g , and Δ x = 0.1 × 1 0 − 10 m . Therefore, Δ v ≥ 4 × π × 9.11 × 1 0 − 31 × 0.1 × 1 0 − 10 6.626 × 1 0 − 34 Δ v ≥ 4 × 3.14159 × 9.11 × 1 0 − 31 × 1 0 − 11 6.626 × 1 0 − 34 Δ v ≥ 1.1447 × 1 0 − 40 6.626 × 1 0 − 34 Δ v ≥ 5.7879 × 1 0 6 m / s
Final Answer The uncertainty in velocity is approximately 5.79 × 1 0 6 m / s . Comparing this to the given options, we see that option (a) is the correct answer.
Examples
The Heisenberg Uncertainty Principle is not just a theoretical concept; it has practical implications in fields like microscopy and materials science. For instance, when trying to observe extremely small objects using electron microscopes, the very act of measuring the electron's position introduces uncertainty in its velocity. This limits the precision with which we can simultaneously know both the position and momentum of the electron, affecting the resolution and accuracy of the microscope. Understanding this principle helps scientists optimize their experimental setups to minimize these uncertainties and obtain the most accurate measurements possible. In quantum computing, controlling the uncertainty is crucial for maintaining the coherence of qubits, which are the fundamental units of quantum information.
The uncertainty in velocity, according to the Heisenberg Uncertainty Principle, is approximately 5.79 × 1 0 6 m/s . Therefore, the correct answer is option (a).
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