Computational Physics

164 Solving the Schrödinger equation in periodic solids

this theorem toψand its energy derivative and use the normalisation convention to

show that ∫

core

d^3 rψ^2 (r)=−

1

2

∫

shell

d^2 aψ(a)nˆ·∇ψ( ̇ a).

6.2 [C] Consider the following periodic potential in one dimension.

na

∆

(n – 1 )a (n + 1 )a

The height of the barriers isV 0. The solution of the Schrödinger equation in

between two barriers at(n− 1 )aandnacan be written as

ψ(x)=Aneiq(x−na)+Bne−iq(x−na)

withq=

√

2 E. Assume that the energy we are interested in is higher than the barrier

heightV 0 .Onthenth barrier, the solution is written as

ψ(x)=Cneiκ(x−na)+Dne−iκ(x−na)

withκ=

√

2 (E−V 0 ).

The values ofAnandBnin neighbouring interstitial regions are connected through

the so-called ‘transfer matrix’:

(

An+ 1

Bn+ 1

)

=T(E)

(

An

Bn

)

.

Tisa2×2 matrix which depends on energy.

(a) Show that the transfer matrix is given by

T=

q

4 κ

(

T 11 T 12

T 21 T 22

)

,

with

T 11 =eiq(a−)

[

eiκ

(

1 +

κ

q

) 2

−e−iκ

(

1 −

κ

q

) 2 ]

,

T 12 =−2ieiqa

(

1 −

κ^2

q^2

)

sin(κ);

and

T 22 =T 11 ∗,

T 21 =T 12 ∗.

Show that the product of the two eigenvalues of this matrix is equal to 1. Hence

these eigenvalues can either be written as e±ik(or as e±α, realα). From Bloch’s

theorem we know that the solutions can be labelled by a wave vectorq ̃which is