$10.0$-g marble is gently placed on a horizontal tabletop that is $1.75$ m wide.
(a) What is the maximum uncertainty in the horizontal position of the marble?
(b) According to the Heisenberg uncertainty principle, what is the minimum uncertainty in the horizontal velocity of the marble?
(c) In light of your answer to part (b), what is the longest time the marble could remain on the table? Compare this time to the age of the universe, which is approximately $14$ billion years. (Hint: Can you know that the horizontal velocity of the marble is exactly zero?)

Verified step by step guidance

1. Step 1: Begin by addressing part (a). The maximum uncertainty in the horizontal position of the marble can be considered as the width of the tabletop, which is given as 1.75 m. This is because the marble could be anywhere along the tabletop, and the uncertainty in its position spans the entire width.
2. Step 2: For part (b), use the Heisenberg uncertainty principle, which states: Δx * Δp ≥ ℏ / 2, where Δx is the uncertainty in position, Δp is the uncertainty in momentum, and ℏ is the reduced Planck's constant (ℏ ≈ 1.054 × 10⁻³⁴ Js). Since Δp = m * Δv (momentum is mass times velocity), rearrange the equation to find the minimum uncertainty in velocity: Δv ≥ ℏ / (2 * m * Δx). Substitute the values: m = 10.0 g = 0.010 kg, Δx = 1.75 m, and ℏ = 1.054 × 10⁻³⁴ Js.
3. Step 3: For part (c), calculate the longest time the marble could remain on the table. The marble's horizontal velocity uncertainty (Δv) from part (b) implies that it could move off the table over time. The time it takes to traverse the width of the table can be estimated as t = Δx / Δv. Substitute Δx = 1.75 m and the value of Δv obtained in part (b) into this equation.
4. Step 4: Compare the calculated time from part (c) to the age of the universe, which is approximately 14 billion years (1.4 × 10¹⁰ years). Convert the calculated time into years if necessary and determine whether the marble's time on the table is significant in comparison to the universe's age.
5. Step 5: Reflect on the hint provided in the problem. The Heisenberg uncertainty principle implies that the horizontal velocity of the marble cannot be exactly zero, as there would then be infinite uncertainty in its position. This reinforces the idea that the marble's position and velocity are inherently uncertain, even in a seemingly static scenario.

Key Concepts

Heisenberg Uncertainty Principle

The Heisenberg Uncertainty Principle states that it is impossible to simultaneously know both the exact position and exact momentum (or velocity) of a particle. This principle highlights a fundamental limit to measurement in quantum mechanics, where the more precisely one property is measured, the less precisely the other can be known. This concept is crucial for understanding the behavior of particles at the quantum level, such as the marble in this question.

Quantum Mechanics

Quantum mechanics is the branch of physics that deals with the behavior of matter and energy at very small scales, typically at the level of atoms and subatomic particles. It introduces concepts such as wave-particle duality and quantization of energy, which differ significantly from classical physics. Understanding quantum mechanics is essential for interpreting phenomena like the uncertainty principle and the behavior of particles like the marble.

Measurement and Observables

In physics, measurement refers to the process of determining the value of a physical quantity, which is often represented as an observable. Observables in quantum mechanics, such as position and momentum, are subject to uncertainty due to the nature of quantum states. This concept is important for the question as it relates to how accurately we can know the marble's position and velocity, impacting our calculations of time and comparisons to cosmic scales.