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Chapter 36: Problem 50

Suppose that Fuzzy, a quantum-mechanical duck, lives in a world in which Planck's constant \(\hbar=1.00 \mathrm{~J}\) s. Fuzzy has a mass of \(0.500 \mathrm{~kg}\) and initially is known to be within a \(0.750-\mathrm{m}-\) wide pond. What is the minimum uncertainty in Fuzzy's speed? Assuming that this uncertainty prevails for \(5.00 \mathrm{~s}\), how far away could Fuzzy be from the pond after 5.00 s?

Short Answer

Expert verified
Answer: The minimum uncertainty in Fuzzy's speed is approximately 1.33 m/s, and Fuzzy could be up to 6.65 m away from the pond after 5.00 seconds.

Step by step solution

01

Recall Heisenberg's Uncertainty Principle

Heisenberg's uncertainty principle states that: \(\Delta x \Delta p_x \geq \frac{\hbar}{2}\) Where \(\Delta x\) is the position uncertainty, \(\Delta p_x\) is the momentum uncertainty, and \(\hbar\) is the reduced Planck's constant.
02

Write down the given information

We are given the following information in the problem: - Reduced Planck's constant, \(\hbar = 1.00\,\text{Js}\) - Fuzzy's mass, \(m = 0.500\,\text{kg}\) - Initial position uncertainty, \(\Delta x = 0.750\,\text{m}\)
03

Calculate the minimum momentum uncertainty

Using Heisenberg's uncertainty principle, we can find the minimum uncertainty in Fuzzy's momentum: \(\Delta p_x \geq \frac{\hbar}{2\Delta x}\) Substitute the given values into the formula: \(\Delta p_x \geq \frac{1.00\,\text{Js}}{2 \times 0.750\,\text{m}}\) Calculate the minimum momentum uncertainty: \(\Delta p_x \geq \frac{1.00}{1.50} \approx 0.667\,\text{kg\,m/s}\)
04

Calculate the minimum uncertainty in Fuzzy's speed

To find the minimum speed uncertainty, we will use the formula for uncertainty in speed by dividing the momentum uncertainty by the mass of Fuzzy: \(\Delta v_x = \frac{\Delta p_x}{m}\) Substitute the calculated value and the given mass into the formula: \(\Delta v_x = \frac{0.667\,\text{kg\,m/s}}{0.500\,\text{kg}}\) Calculate the minimum speed uncertainty: \(\Delta v_x \approx 1.33\,\text{m/s}\)
05

Calculate the distance Fuzzy could be away from the pond

We are assuming that the uncertainty prevails for \(5.00\,\text{s}\). In the worst-case scenario, Fuzzy could be traveling at the highest possible speed within this uncertainty. To find the maximum distance Fuzzy can be away from the pond after \(5.00\,\text{s},\) we will use the formula: \(\text{Distance} = \text{Speed} \times \text{Time}\) Substitute the values into the formula: \(\text{Distance} = 1.33\,\text{m/s} \times 5.00\,\text{s}\) Calculate the distance: \(\text{Distance} \approx 6.65\,\text{m}\) In conclusion, the minimum uncertainty in Fuzzy's speed is \(1.33\,\text{m/s}\), and Fuzzy could be up to \(6.65\,\text{m}\) away from the pond after \(5.00\,\text{s}\).

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Most popular questions from this chapter

Consider a quantum state of energy \(E\), which can be occupied by any number \(n\) of some bosonic particles, including \(n=0\). At absolute temperature \(T\), the probability of finding \(n\) particles in the state is given by \(P_{n}=N \exp \left(-n E / k_{\mathrm{B}} T\right)\), where \(k_{\mathrm{B}}\) is Boltzmann's constant and the normalization factor \(N\) is determined by the requirement that all the probabilities sum to unity. Calculate the mean or expected value of \(n\), that is, the occupancy, of this state, given this probability distribution.

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