What is the minimum uncertainty in position, in nm, of an electron whose velocity is known to be between 3×10⁵ m/s and 4 ×10⁵ m/s? Give your answer to one significant figure.

Verified step by step guidance

1. Start by identifying the uncertainty in velocity (Δv). Since the velocity is known to be between 3×10⁵ m/s and 4×10⁵ m/s, the uncertainty in velocity is Δv = 4×10⁵ m/s - 3×10⁵ m/s = 1×10⁵ m/s.
2. Use Heisenberg's Uncertainty Principle, which states Δx * Δp ≥ ℏ / 2, where Δx is the uncertainty in position, Δp is the uncertainty in momentum, and ℏ is the reduced Planck's constant (ℏ ≈ 1.055×10⁻³⁴ J·s).
3. Express the uncertainty in momentum (Δp) in terms of the uncertainty in velocity (Δv) using the relationship Δp = m * Δv, where m is the mass of the electron (m ≈ 9.11×10⁻³¹ kg).
4. Substitute Δp = m * Δv into the uncertainty principle equation to find Δx: Δx ≥ ℏ / (2 * m * Δv).
5. Plug in the known values (ℏ = 1.055×10⁻³⁴ J·s, m = 9.11×10⁻³¹ kg, and Δv = 1×10⁵ m/s) into the equation Δx ≥ ℏ / (2 * m * Δv) to calculate the minimum uncertainty in position, and convert the result to nanometers (1 nm = 1×10⁻⁹ m).

Key Concepts

Heisenberg's Uncertainty Principle

Heisenberg's Uncertainty Principle states that it is impossible to simultaneously know both the exact position and exact momentum of a particle. This principle implies that the more accurately we know a particle's velocity (momentum), the less accurately we can know its position. This relationship is fundamental in quantum mechanics and highlights the inherent limitations in measuring quantum systems.

Momentum

Momentum is defined as the product of an object's mass and its velocity. In the context of quantum mechanics, momentum is a crucial variable that relates to the uncertainty in position. For an electron, knowing its velocity allows us to calculate its momentum, which is essential for applying the uncertainty principle to determine the minimum uncertainty in its position.

Calculating Uncertainty

To calculate the uncertainty in position, we can use the formula derived from Heisenberg's principle: Δx * Δp ≥ ħ/2, where Δx is the uncertainty in position, Δp is the uncertainty in momentum, and ħ is the reduced Planck's constant. By determining the range of velocities, we can find the uncertainty in momentum and subsequently calculate the minimum uncertainty in position for the electron.