= Solution
An initially localized escaping <electron> occupies a superposition of lead <momentum> states. In a uniform free one-dimensional lead,
$$
\psi(x,t)=\int a(k)e^{ikx-i\hbar k^2t/(2m^*)}\,dk.
$$
A packet centered at $k_0$ moves with <group velocity> $\hbar k_0/m^*$ and generally spreads because its components have different velocities. For an initially unchirped minimum-uncertainty <Gaussian wave packet>,
$$
\sigma_x(t)=\sigma_x(0)
\sqrt{1+\left[\frac{\hbar t}{2m^*\sigma_x^2(0)}\right]^2}.
$$
This illustrates dispersive evolution; the actual emission contact determines $a(k)$ and whether only outward-propagating components are appreciable. It does not instantaneously become a single lead <plane wave>.
A \b[<single-electron transistor> used as a sensitive electrometer] exploits <Coulomb blockade>: a nearby charge shifts the offset $Q_{\rm ext}$ and therefore the conductance peaks. This requires stable calibrated <capacitances> and low charge noise, rather than merely a small conductor.
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