Computational Physics

Exercises 165

not necessarily equal toq, and that these solutions can be written as a periodic

function times eiqx ̃. In our case this implies that

(

An+ 1

Bn+ 1

)

=eiqa ̃

(

An

Bn

)

and therefore

T

(

An

Bn

)

=ei ̃qa

(

An

Bn

)

.

This equation defines the band spectrum of the system. It is now easier to find the
vectorq ̃as a function of energy than vice versa: the above-mentioned eigenvalues
(which depend on energy viaqandκ) must be equal to eiqa ̃.
(b) [C] Write a simple computer program to determine the spectrum. In an APW
approach, the wave function outside the barriers is written as eiqmx, where
qm=k+ 2 πm/aand−π/a<k<π/a(mis integer). It is now convenient to
confine ourselves to the unit cell[−a/2,a/ 2 ]and to use Bloch boundary
conditions on that cell. For a Bloch stateχm:
χm(−a/ 2 )=χ(a/ 2 )e−ika.
For eachqr, the value of the wave function outside and inside the barrier can be
matched at the boundaries of the barrier. Show thatCmandDmare given by

Cm=

sin[(κ+qm)/ 2 ]

sin(κ)

,

Dm=

sin[(κ−qm)/ 2 ]

sin(κ)

.

In the APW method, the coefficientsbmof the expansion

ψ(x)=

∑

m

bmχm(x)

are found by solving the generalised eigenvalue problem

Hb=ESb

in which the matrixSis given by

Sml=

∫−/ 2

−a/ 2

e−iqmxeiqlxdx+

∫a/ 2

/ 2

e−iqmxeiqlxdx

+

∫/ 2

−/ 2

[Cm∗e−iκmx+D∗meiκmx][Cleiκnx+Dle−iκlx]dx

=Sintml+Smlext

where we have split the expression forSinto an integration over the interior of the

barriers and the part outside.