QuantumPhysics.dvi

This relation must hold for all states|k′;T〉, and it means that

〈k;T|k′;T〉= 2πδ(k′−k) k,k′∈[−kc,kc] (5.26)

whereδ(k−k′) is theDiracδ-function. It is generally defined by the relation

f(x) =

∫

dy δ(x−y)f(y) (5.27)

for any dense set of infinitely differentiable functionsf(x) (strictly speaking with the extra

technical assumption of compact support). It is instructive to verify this relation directly

from the expression of|k;T〉in terms of|n〉,

〈k′;T|k;T〉=a

+∑∞

n=−∞

eina(k−k

′)

(5.28)

This gives us a convenient representation of theδ-function,

2 πδ(k−k′) =a

+∑∞

n=−∞

eina(k−k

′)

k,k′∈[−kc,kc] (5.29)

We conclude by taking the limit of the energy eigenvalues of the latticeHamiltonian in the

N→∞limit. It is given by

Ek=A 0 − 2 A 1 cos(ka)− 2 A 2 cos(2ka) k∈[−kc,kc] (5.30)

This is a very general result for the propagation of free waves on alattice with spacinga.

5.3 Propagation on a circle

Another way of taking the limitN→∞is obtained by keeping the physical size of the system

Na= 2πLfinite. For fixedL, the lattice spacing must then tend to 0 as follows,a= 2πL/N.

This time, the momentum operator retains a discrete spectrum, and the position operator

acquires a continuous spectrum. We choosex 0 =−π(N+ 1)L, such that

an+x 0 =x −πL≤x≤+πL (5.31)

It is clear from the conjugation relation ofXandTthat asa→0, the operatorTapproaches

the identity operator linearly ina. We define the momentum operatorP by

T=I−i

a

h ̄

P+O(a^2 ) (5.32)