QuantumPhysics.dvi

(Wang) #1

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)

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