 Solution

ID: past-exam-of-the-mphil-in-micro-and-nanotechnology-enterprise-course-of-the-university-of-cambridge/2018/paper-2/section-b/19/iv/solution

An initially localized escaping electron occupies a superposition of lead momentum states. In a uniform free one-dimensional lead,

ψ (x, t) = \int a (k) e^{ik x - i ℏ k^{2} t / (2 m^{*})} d k .

A packet centered at

k_{0}

ℏ k_{0} / m^{*}

and generally spreads because its components have different velocities. For an initially unchirped minimum-uncertainty Gaussian wave packet,

σ_{x} (t) = σ_{x} (0) 1 + [\frac{ℏ t}{2 m ^{*} σ _{x}^{2} ( 0 )}]^{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 single-electron transistor used as a sensitive electrometer exploits Coulomb blockade: a nearby charge shifts the offset

Q_{ext}

and therefore the conductance peaks. This requires stable calibrated capacitances and low charge noise, rather than merely a small conductor.

 New to topics? Read the docs here!