Suppose Fuzzy, a quantum mechanical duck, lives in a world in which h = 2π J.s. Fuzzy has a mass

Question

of 2.0 kg and is initially known to be within a region 1.0 m wide.
a) What is the minimum uncertainty in his speed?
b) Assuming this uncertainty in speed to prevail for 5.0 s, determine the uncertainty in position after
this time.

Answer 1

To solve this problem, we can use the Heisenberg Uncertainty Principle, which states that the uncertainty in position (Δx) multiplied by the uncertainty in momentum (Δp) is greater than or equal to h/2π, where h is the Planck's constant.

Δx * Δp ≥ h/2π

a) To find the minimum uncertainty in his speed, we first need to find the uncertainty in momentum. We know the initial uncertainty in Fuzzy's position is 1.0 m (Δx = 1.0 m). Rearranging the uncertainty principle equation to solve for Δp, we get:

Δp ≥ (h/2π) / Δx

The given value of h in this world is 2π J.s. Plugging this value and the value of Δx into the formula, we get:

Δp ≥ (2π J.s) / (2π * 1.0 m)
Δp ≥ 1 kg.m/s

Now we can find the uncertainty in speed (Δv) by dividing the uncertainty in momentum by the mass of Fuzzy:

Δv = Δp / m
Δv = 1 kg.m/s / 2.0 kg
Δv = 0.5 m/s

The minimum uncertainty in Fuzzy's speed is 0.5 m/s.

b) Assuming this uncertainty in speed prevails for 5.0 s, we can find the uncertainty in position after this time by multiplying the uncertainty in speed by the time:

Δx = Δv * t
Δx = 0.5 m/s * 5.0 s
Δx = 2.5 m

The uncertainty in Fuzzy's position after 5 seconds is 2.5 meters.