Suppose Fuzzy, a quantum mechanical duck, lives in a world in which h = 2π J.s. Fuzzy has a mass
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.
To solve this problem, we need to use the principles of Heisenberg's uncertainty principle, which states that the product of the uncertainty in position (Δx) and the uncertainty in momentum (Δp) of a quantum particle is always greater than or equal to Planck's constant divided by two (h/2π).
a) To find the minimum uncertainty in Fuzzy's speed, we need to calculate the uncertainty in momentum using the formula Δp = mΔv, where m is the mass and Δv is the uncertainty in velocity. Uncertainty in velocity can be approximated as the uncertainty in speed, so we will use Δv = Δx.
Given:
h = 2π J.s
m = 2.0 kg
Δx = 1.0 m
Using the uncertainty principle, we have: Δp x Δx ≥ h/2π
Δp x 1.0 ≥ 2π/2π
Δp ≥ 2π/2π
Δp ≥ 1
Therefore, the minimum uncertainty in Fuzzy's momentum is 1 kg.m/s.
Now, to find the minimum uncertainty in speed, we can use the formula Δv = Δp / m:
Δv = 1 / 2.0
Δv = 0.5 m/s
So, the minimum uncertainty in Fuzzy's speed is 0.5 m/s.
b) To determine the uncertainty in position after 5.0 s, we need to use the uncertainty in velocity (0.5 m/s) and the time period (5.0 s).
Δx = Δv x t
Δx = 0.5 x 5.0
Δx = 2.5 m
Therefore, the uncertainty in Fuzzy's position after 5.0 s will be 2.5 meters.