Using Heisenberg’s uncertainty principle, calculate the uncertainty in the position of b. a proton moving at a speed of (5.00±0.01) × 104 m/s. The mass of a proton is 1.673×10−27 kg.

Verified step by step guidance

1. Identify the given values: speed of the proton \( v = 5.00 \times 10^4 \text{ m/s} \) with an uncertainty \( \Delta v = 0.01 \times 10^4 \text{ m/s} \), and the mass of the proton \( m = 1.673 \times 10^{-27} \text{ kg} \).
2. Use Heisenberg's uncertainty principle formula: \( \Delta x \cdot \Delta p \geq \frac{h}{4\pi} \), where \( \Delta x \) is the uncertainty in position and \( \Delta p \) is the uncertainty in momentum.
3. Calculate the uncertainty in momentum \( \Delta p \) using \( \Delta p = m \cdot \Delta v \).
4. Substitute \( \Delta p \) into the uncertainty principle formula to solve for \( \Delta x \): \( \Delta x \geq \frac{h}{4\pi \cdot \Delta p} \).
5. Use the value of Planck's constant \( h = 6.626 \times 10^{-34} \text{ Js} \) to calculate \( \Delta x \).

Key Concepts

Heisenberg's Uncertainty Principle

Heisenberg's uncertainty principle states that it is impossible to simultaneously know both the exact position and momentum of a particle. The more accurately we know one of these values, the less accurately we can know the other. This principle is fundamental in quantum mechanics and highlights the inherent limitations in measuring subatomic particles.

Momentum

Momentum is defined as the product of an object's mass and its velocity. In the context of the uncertainty principle, momentum is a crucial variable because it relates to the particle's motion. For a proton, its momentum can be calculated using the formula p = mv, where p is momentum, m is mass, and v is velocity.

Calculating Uncertainty

To calculate the uncertainty in position (Δx) using the uncertainty principle, we use the formula Δx * Δp ≥ ħ/2, where Δp is the uncertainty in momentum and ħ is the reduced Planck's constant. The uncertainty in momentum can be derived from the uncertainty in velocity (Δv) and the mass of the particle, allowing us to find the corresponding uncertainty in position.