A beam of electrons is fired in free space over a distance of $10^{4} \mathrm{~km}$. If the transversal width of the initial packet is $1 \mu \mathrm{m}$, what will be the minimum spread upon arrival if the kinetic energy of the electrons is $13.6 \mathrm{eV}$ ?
The initial wave packet of an electron in the beam has uncertainties $\Delta p_{x}, \Delta x$ in a transverse direction related by the uncertainty relation

$$
\Delta p_{x} \cdot \Delta x \geq \frac{\hbar}{2} .
$$

If it has momentum $p$, the angle of divergence is

$$
\theta=\frac{\Delta p_{x}}{p} \gtrsim \frac{1}{2 \Delta x} \frac{\hbar}{p}=\frac{1}{2 \Delta x} \frac{\hbar c}{p c} .
$$

As

$$
\begin{gathered}
p=\sqrt{2 m T}=\frac{1}{c} \sqrt{2 m c^{2} T}=\frac{1}{c} \sqrt{2 \times 0.51 \times 10^{6} \times 13.6}=3.72 \times 10^{3} \mathrm{eV} / \mathrm{c}, \\
\Delta x \approx 10^{-6} \mathrm{~m},
\end{gathered}
$$

we have

$$
\theta \gtrsim \frac{6.6 \times 10^{-16} \times 3 \times 10^{8}}{2 \times 10^{-6} \times 3.72 \times 10^{3}}=2.66 \times 10^{-5} \mathrm{radian}
$$

and the minimum transverse spread is

$$
L \theta_{\min }=266 \mathrm{~m} .
$$
