Suppose Fuzzy, a quantum mechanical duck, lives in a world in which h = 2 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.
I don't recognize the symbol before J.s
it is π (pi)
To solve this problem, we will use the Heisenberg uncertainty principle, which states that the uncertainty in position (Δx) and the uncertainty in momentum (Δp) of a particle are related by the equation:
Δx * Δp ≥ h/4π
Where h is the Planck's constant and is given as h = 2π J.s.
a) To find the minimum uncertainty in Fuzzy's speed, we need to find the uncertainty in momentum Δp. Since we know that the mass of Fuzzy is 2.0 kg and the speed (v) is related to momentum (p) as p = m * v, where m is the mass and v is the speed, we can rearrange the equation to find the uncertainty in momentum:
Δp = m * Δv
We are given the mass of Fuzzy (m = 2.0 kg) and the width of the region where Fuzzy is initially known to be (Δx = 1.0 m).
Now, we can plug these values into the Heisenberg uncertainty principle equation to find Δv:
(Δx) * (Δp) ≥ h/4π
(1.0 m) * (2.0 kg * Δv) ≥ (2π J.s) / (4π)
2.0 kg * (1.0 m) * Δv ≥ 1/2 J.s
2.0 kg * Δv ≥ 1/2 J.s
Δv ≥ 1/4 kg.m/s
Therefore, the minimum uncertainty in Fuzzy's speed is Δv ≥ 1/4 kg.m/s.
b) Now, let's determine the uncertainty in position after 5.0 s, assuming the uncertainty in speed Δv is maintained.
To find the uncertainty in position (Δx) after time t, we can use the formula Δx = Δv * t.
Substituting the given values, we have:
Δx = (1/4 kg.m/s) * (5.0 s)
Δx = 5/4 kg.m
Therefore, the uncertainty in position after 5.0 s is Δx = 5/4 kg.m.