Using Heisenberg's uncertainty principle, calculate the uncertainty in the position of (b) a proton moving at a speed of 15.00 { 0.012 * 104 m/s. (The mass of a proton is given in the table of fundamental constants in the inside cover of the text.)

Verified step by step guidance

1. Identify the given values: the speed of the proton is 15.00 \( \pm \) 0.012 \times 10^4 \text{ m/s} \), and the mass of a proton is approximately \( 1.67 \times 10^{-27} \text{ kg} \).
2. Calculate the uncertainty in velocity (\( \Delta v \)) using the given speed and its uncertainty: \( \Delta v = 0.012 \times 10^4 \text{ m/s} \).
3. Use Heisenberg's uncertainty principle formula: \( \Delta x \cdot \Delta p \geq \frac{h}{4\pi} \), where \( \Delta p \) is the uncertainty in momentum and \( h \) is Planck's constant (\( 6.626 \times 10^{-34} \text{ Js} \)).
4. Calculate the uncertainty in momentum (\( \Delta p \)) using \( \Delta p = m \cdot \Delta v \), where \( m \) is the mass of the proton.
5. Solve for the uncertainty in position (\( \Delta x \)) using \( \Delta x \geq \frac{h}{4\pi \cdot \Delta p} \).

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 limitations of measurement at the microscopic scale.

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 key variable because it relates to the motion of the particle. For a proton moving at a given speed, its momentum can be calculated, which is essential for determining the uncertainty in its position.

Calculating Uncertainty

To calculate the uncertainty in position (Δx) using Heisenberg's principle, the formula Δx * Δp ≥ ħ/2 is used, where Δp is the uncertainty in momentum and ħ is the reduced Planck's constant. By determining the uncertainty in momentum based on the speed of the proton, one can rearrange the equation to find the corresponding uncertainty in position.