QuantumPhysics.dvi

Here,ais the lattice spacing of the crystal. As a result, the translation operator by a finite shifta,
denoted byT(a), and given by

T(a) = exp{iap/ ̄h} (10.68)

must commute withH. We choose a basis of eigenstates forHwhich also diagonalizesT(a). The
eigenvalues ofT(a) are phases specified by a wave-numberk, so that

T(a)|k,n〉 = e−ika|k,n〉

H|k,n〉 = Ek,n|k,n〉 (10.69)

By construction, for givena, the wave-number is periodic with periodk→k+ 2π/a, so states are
labeled uniquely whenkruns over and interval of length 2π/a, called a Brillouin zone, for example
the first Brillouin zone is specified by

−

π

a

≤k≤

π

a

(10.70)

The additional indexnis a further quantum number needed to describe all the states.

It is useful to view the problem in a Schr ̈odinger equation picture, for wave functionsψk,n(x) =
〈x|k,n〉, so thatψk,n(x+a) =eikaψk,n(x). Alternatively, the wave function may be expressed in
terms of a periodic wave functionφk,n(x) and a phase factor,

ψk,n(x) = eikxφk,n(x)

φk,n(x+a) = φk,n(x) (10.71)

where the periodic wave function now obeys ak-dependent Schr ̈odinger equations,
(
(p+ ̄hk)^2
2 m
+V(x)

)

φk,n(x) =Ek,nφk,n(x) (10.72)

The quantum numbernnow labels the discrete spectrum of this periodic equation.

For vanishing potential,V = 0, it is straightforward to solve this problem, and we have the
energy levels of a periodic free system,

pφk,n=

2 πn ̄h

a

φk,n n= 0,± 1 ,± 2 ,··· (10.73)

so that the energies are

Ek,n=

̄h^2

2 m

(

k+

2 πn

a

) 2

(10.74)

For certain values ofkwithin the Brillouin zone, energy levels are be degenerate,namely

Ek,n′=Ek,n n′ 6 =n (10.75)