Applying uncertainty principle and the difference in $\Delta x$

Question

These two questions seem to be very similar, but the textbook uses a bit different methods for calculating $\Delta x$ of uncertainty principle.

Question A) Suppose that there is a room with the same length of its side (x-axis, y-axis, z-axis). A ball is 100g.

In this case, the solution says that I should put $\Delta x$ as 15m.

Question B) Find the kinetic energy of an electron bound by nucleus using uncertainty principle. Assume that the nucleus has the side of $1.0 \times 10^{-14} m$.

In this case, however, the solution says that I should put $\Delta x$ as the half of $1.0 \times 10^{-14}m$.

What is the difference between these two cases? (I know that the second case is only approximation.)

Answer 1

For a particle in a cubic box the solution is fairly simple because the wavefunction separates into functions of $x$, $y$ and $z$, and you can calculate $\Delta x$ exactly. The Wikipedia article on the particle in a box gives:

$$ \Delta x^2 = \frac{L^2}{12} \left( 1 - \frac{6}{n^2\pi^2} \right) $$

though I get $\Delta x \approx 18m$ not $15m$. The calculation for a spherical box is a lot harder, and in any case a spherical box is only an approximation to reality. This is presumably why you've been told to just assume $\Delta x$ is half the size of the nucleus rather than trying to calculate it.