A one-dimensional chain has lattice constant

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

Solid State Physics, Ch Bloch's theorem and the nearly-free-electron method

Problem

a=4.0\ \mathrm{\AA}

. At the zone boundary

k=\pi/a

two degenerate free-electron plane waves

e^{i\pi x/a}

and

e^{-i\pi x/a}

combine into the real standing wave

\psi(x)=\cos(\pi x/a)

. ((a)) State the crystal momentum label

\hbar k

of this state, in

\mathrm{kg\,m\,s^{-1}}

. ((b)) Evaluate

\langle p\rangle = \dfrac{\int_0^{2a}\psi^*(x)\big(-i\hbar\,d/dx\big)\psi(x)\,dx}{\int_0^{2a}|\psi(x)|^2\,dx}

and compare it with

\hbar k

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