Question

The mean momentum of a nucleon in a nucleus with mass number A varies as

• A. $A^3$
• B. $A^2$
• C. $A^{-2/3}$
• D. $A^{-1/3}$
• E. $A^{1/3}$

Hint:

Think of this as a particle in a box problem: as the box size ($A^{1/3}$) gets bigger, the energy levels and corresponding momentum get smaller.

Answer 1

The Correct Option is D

Concept:
According to the uncertainty principle ($\Delta p \Delta x \approx \hbar$), the momentum of a nucleon is inversely proportional to the dimensions of the region it is confined in ($p \propto 1/R$).

Step 1: Relate nuclear radius to mass number A.
The radius $R$ of a nucleus is given by $R = R_0 A^{1/3}$.

Step 2: Determine the proportionality of momentum.
\[ p \propto \frac{1}{R} \propto \frac{1}{A^{1/3}} = A^{-1/3} \] This means as the mass number increases, the nucleons are confined to a larger volume, and their mean momentum (and kinetic energy) decreases.