Phy5645/Uncertainty Relations Problem 2

This a problem from the book of W.Greiner named Quantum Mechanics.

Consider there is a box with a particle (a nucleon) in it. Width of the box is L. Determine the magnitude of kinetic energy of the particle.

a picture of the box taken from the same book

According to Heisenberg uncertanity principle: $\Delta p\Delta x\cong \hbar$ and so $\Delta p\cong {\frac {\hbar }{\Delta x}}$. On the other hand, as we know that $E={\frac {\ P^{2}}{2m}}$. Therefore,$\Delta E={\frac {(\Delta P)^{2}}{2m}}$

If we plug $\Delta p\$ into the energy equation; it will look $\Delta E\cong {\frac {\hbar ^{2}}{2m(\Delta x)^{2}}}$

Let the side of the box shrink $L=\Delta x\rightarrow \ 0$

Knowing that Nucleons have size of $\approx 10^{12}cm$, Nucleon mass $mc^{2}\cong 938MeV$, and $\hbar c\cong 197\times 10^{-13}cmMev$, then we can calculate kinetic energy.

$\Delta E\cong {\frac {\hbar ^{2}}{2m(\Delta x)^{2}}}={\frac {\hbar ^{2}c^{2}}{2mc^{2}(\Delta x)^{2}}}={\frac {(197\times 10^{-13}cmMev)^{2}}{(2)(938MeV)(10^{-12})^{2}}}\approx 0.2MeV$