The text solves for an electron starting from rest

PHYSICS · Subjective · Fermi University Physics · Condensed Matter, Materials & Soft Matter · Free electrons, energy bands and electronic structure · Free electrons, energy bands and electronic structure

Condensed Matter Physics, Ch The free-electron gas: what Drude gets right and what only Fermi fixes

Problem

The text solves m dv/dt=−eEx−mv/τm\,\mathrm{d}v/\mathrm{d}t=-eE_x-mv/\tau for an electron starting from rest. Suppose instead the electron already has velocity v0v_0 along xx at the instant the field is switched on. Solve the equation for v(t)v(t) with this general initial condition, and show that however large or small v0v_0 is, v(t)v(t) approaches the same steady drift velocity vd=−eτEx/mv_d=-e\tau E_x/m as t→∞t\to\infty. Hence explain why the conductivity σ=ne2τ/m\sigma=ne^2\tau/m does not depend on the speeds electrons happen to have when the field is switched on.

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