QuantumPhysics.dvi

5.1.3 The spectrum and generalized Hamiltonians

We are guaranteed thatHis diagonalizable in this basis; in fact it is already diagonal, and

we readily evaluate the eigenvalues ofH:

H|km;T〉=Em|km;T〉 Em=A 0 − 2 A 1 cosφm (5.14)

Remarkably, it is very easy to include next to nearest neighbor interactions and so on. We

would do this by including inHterms which have higher powers of the translation operator.

Including next to nearest neighbor interactions would produce theHamiltonian,

H=A 0 I−A 1 T−A 1 T−^1 −A 2 T^2 −A 2 T−^2 (5.15)

By construction, this Hamiltonian is also already diagonal in the basis|km;T〉, and we may

read off the eigenvalues,

Em=A 0 − 2 A 1 cosφm− 2 A 2 cos 2φm (5.16)

5.1.4 Bilateral and reflection symmetric lattices

The periodic lattice has a natural reflection symmetry. This symmetry is often useful in

performing practical calculations. For example, in the subsequentsubsections, we shall be

led to consider the limitN→∞, which is more easily represented in symmetric form. Here,

we shall recast the periodic lattice withN sites in a manifestly reflection symmetric way.

The rearrangement depends on whetherN is even or odd; it will be convenient to define

ν≡[N/2], where [ ] stands for the integer part. To exhibit the reflection symmetry of the

periodic lattice, it suffices to choosex 0 of the preceding subsection as follows,

x 0 =−νa n ̃=n−ν (5.17)

Here, we have also shifted the labelnto a symmetric label ̃n. ForNodd, the states|ν〉and

|−ν〉are different from one another, and the range of ̃nlabeling independent states|n ̃〉is as

follows,−ν≤n ̃≤ν. ForN even, the states|−ν〉and|ν〉are to be identified|−ν〉=|ν〉,

and the range of ̃nlabeling independent states| ̃n〉is instead−ν <n ̃≤ν.

For example, whenN is odd, we haveN = 2ν+ 1, and the phasesφm may be chosen

symmetrically as follows,

φm= 2πm/N m=−ν,···, 0 ,···,ν (5.18)

The Bloch states are then given by

|km;T〉=

1

√

N

∑ν

̃n=−ν

e+inφ ̃ m|n ̃〉 m=−ν,···, 0 ,···,ν (5.19)