  Inside a medium, showers originating from a very high-energy particle may
develop via medium-induced splitting processes such as hard bremsstrahlung or
pair production. During shower development, two consecutive splittings
sometimes overlap quantum mechanically, so that they cannot be treated
independently. Some of these effects can be absorbed into an effective value of
a medium parameter known as $\hat q$. Previous calculations (with certain
simplifying assumptions) have found that, after adjusting the value of $\hat
q$, the leftover effect of overlapping splittings is quite small for purely
gluonic large-$N_c$ showers but is very much larger for large-$N_f$ QED
showers, at comparable values of $N\alpha$. Those works did not quite make for
apples-to-apples comparisons: the gluon shower work investigated energy
deposition from a gluon-initiated shower, whereas the QED work investigated
charge-deposition from an electron-initiated shower. As a first step to tighten
up the comparison, this paper investigates energy deposition in the QED case.
Along the way, we develop a framework that should be useful in the future to
explore whether the very small effect of overlapping splitting in purely
gluonic showers is an artifact of having ignored quarks.
