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Question #43813

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The Question

In a hypothetical model of dense nuclear matter, a nucleon of rest mass $m$ is confined within a nucleus of mass number $A$ . The nuclear radius follows the standard empirical relation $R = R_0 A^{1/3}$ , where $R_0$ is a constant. Due to strong short-range correlations, the nucleon is localized to a small sub-region such that its position uncertainty is $\Delta x = \frac{R}{A}$ . The corresponding momentum uncertainty is given by the Heisenberg-like relation $\Delta p \approx \frac{\hbar}{\Delta x}$ . Let $K$ be the minimum kinetic energy of the nucleon associated with this confinement. Which of the following correctly describes the scaling of $K$ with $A$ in the limits of small $A$ (where the nucleon is non-relativistic) and very large $A$ (where the nucleon becomes ultra-relativistic)?

Options

A.
\text{Small } A: K \propto A^{4/3}, \quad \text{Large } A: K \propto A^{2/3}
B.
\text{Small } A: K \propto A^{4/3}, \quad \text{Large } A: K \propto A^{4/3}
C.
\text{Small } A: K \propto A^{-2/3}, \quad \text{Large } A: K \propto A^{-1/3}
D.
\text{Small } A: K \propto A^{2/3}, \quad \text{Large } A: K \propto A^{1/3}

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Explanation

How the Examiner Thinks

Multi-Concept_Bridge

Master problems that require linking physics and math concepts to solve complex competitive questions.

Concepts Behind This Question

Nuclear Density Invariance|De Broglie Wavelength Scaling|Approximations (Binomial)

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to transition between non-relativistic and relativistic kinematic regimes based on a scaling parameter.

The primary trap is the misapplication of the Newtonian kinetic energy formula ( $K = p^2/2m$ ) across all domains.

Students often mechanically apply this formula, leading them to incorrectly conclude that $K \propto A^{4/3}$ even for very large $A$ (Option B).

By forcing the momentum uncertainty to grow with $A$ ( $\Delta p \propto A^{2/3}$ ), the problem demands a model-switch to the ultra-relativistic energy-momentum relation ( $E \approx pc$ ), where kinetic energy scales linearly with momentum, yielding $K \propto A^{2/3}$ .

Additionally, it filters out students who incorrectly assume the confinement region is the entire nucleus ( $\Delta x = R$ ), which would yield a decreasing momentum uncertainty and lead to Option C.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Nuclei - Class 12
Concept	Nuclear Density Invariance
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free