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 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.
a) The minimum uncertainty in Fuzzy's speed can be calculated using the uncertainty principle, which states that the product of the uncertainties in position and momentum should be greater than or equal to h/4π.
Given:
Mass of Fuzzy (m) = 2.0 kg
Width of the region (Δx) = 1.0 m
Planck's constant (h) = 2π J.s
To find the minimum uncertainty in speed (Δv), we first need to calculate the uncertainty in momentum (Δp). The uncertainty in momentum can be calculated using the equation:
Δp = m * Δv
Therefore, rearranging the equation, we can find the uncertainty in speed:
Δv = Δp / m
Using the uncertainty principle, we have:
(Δx * Δp) >= h / (4π)
Simplifying and substituting Δp = m * Δv:
(Δx * m * Δv) >= h / (4π)
Substituting the given values:
(1.0 m) * (2.0 kg) * (Δv) >= (2π J.s) / (4π)
2.0 kg * (Δv) >= 0.5 J.s
Dividing both sides by 2.0 kg:
Δv >= 0.5 J.s / 2.0 kg
Δv >= 0.25 m/s
So, the minimum uncertainty in Fuzzy's speed is 0.25 m/s.
b) To determine the uncertainty in position after a given time, we need to use the equation for displacement:
Δx = Δv * t
where Δx is the uncertainty in position, Δv is the uncertainty in speed, and t is the time period.
Given:
Δv = 0.25 m/s
t = 5.0 s
Substituting the values:
Δx = (0.25 m/s) * (5.0 s)
Δx = 1.25 m
Therefore, the uncertainty in position after 5.0 s is 1.25 m.
To solve this problem, we need to use the Heisenberg Uncertainty Principle, which states that the product of the uncertainties in position and momentum of a particle is greater than or equal to Planck's constant divided by 2π.
a) To find the minimum uncertainty in Fuzzy's speed, we need to find the uncertainty in his momentum. The uncertainty in momentum can be calculated by taking the product of the mass of Fuzzy and the uncertainty in his velocity.
Given:
Mass of Fuzzy (m) = 2.0 kg
Width of the region (Δx) = 1.0 m (uncertainty in position)
To find the uncertainty in velocity (V), we'll use the formula:
V = Δx / t
Where t is the time interval, which we'll take as an infinitesimally small time period. Assuming t to be very small, we can approximate V as Δx / Δt.
Δt is the time taken to traverse Δx, and in this case, since the region is 1.0 m, Δx is also the displacement. We'll take the Δt to be the time taken to travel this displacement.
Using the formula: Displacement (Δx) = initial velocity (v₀) * time (Δt)
We can rearrange the formula to find Δt:
Δt = Δx / v₀
Substituting the values:
Δt = 1.0 m / v₀
To find v₀, we need to consider the initial situation, where the uncertainty in velocity is minimum. The minimum uncertainty in the velocity occurs when particles are described by an equal probability of being anywhere within the region. In this case, we can assume the initial velocity (v₀) is half the maximum possible velocity.
The maximum possible velocity is the velocity of light (c). However, since we are dealing with a macroscopic object, we'll consider a much smaller maximum velocity, such as the velocity of sound in air (v).
Assuming v = 343 m/s (speed of sound in air), we'll take v₀ = v/2.
Substituting the value of v₀ in Δt = 1.0 m / v₀:
Δt = 1.0 m / (v/2) = 2.0 m / v
Now, the uncertainty in velocity (Δv) is given by:
Δv = v - v₀
Substituting v₀ = v/2:
Δv = v - (v/2) = v/2
The uncertainty in momentum (Δp) is given by:
Δp = m * Δv
Substituting the value of mass (m) = 2.0 kg and Δv = v/2:
Δp = 2.0 kg * (v/2) = v kg·m/s
Finally, we can use the Heisenberg Uncertainty Principle to find the uncertainty in speed:
Δp * Δv ≥ h / (2π)
Substituting the value of h = 2π J.s and Δp = Δv = v/2:
(v/2) * (v/2) ≥ (2π J.s) / (2π)
Simplifying the equation:
(v²/4) ≥ 1 J.s
To find the minimum uncertainty in speed, we'll solve for v:
v² ≥ 4 J.s
v ≥ √(4 J.s)
v ≥ 2 m/s
Therefore, the minimum uncertainty in Fuzzy's speed is 2 m/s.
b) Using the uncertainty principle, we know that the product of the uncertainties in position and momentum is greater than or equal to h / (2π). We can rearrange this equation to solve for the uncertainty in position (Δx):
Δx * Δp ≥ h / (2π)
Substituting the values, we have:
Δx * (m * Δv) ≥ h / (2π)
Substituting the value of mass (m) = 2.0 kg and Δv = v/2:
Δx * (2.0 kg * (v/2)) ≥ (2π J.s) / (2π)
Simplifying the equation:
Δx * (v kg·m/s) ≥ 1 J.s
Since we have already found the minimum uncertainty in speed to be 2 m/s, we can substitute this value:
Δx * (2 kg·m/s) ≥ 1 J.s
Simplifying further:
2Δx ≥ 1 kg·m/s
To find the uncertainty in position (Δx), we divide both sides of the equation by 2:
Δx ≥ 0.5 kg·m/s
Now, we can multiply the minimum uncertainty in speed (2 m/s) by the time interval (5.0 s) to find the uncertainty in position:
Δx = Δv * t
Δx = 0.5 kg·m/s * 5.0 s
Δx = 2.5 kg·m
Therefore, the uncertainty in position after 5.0 seconds is 2.5 kg·m.