QuantumPhysics.dvi

equation with〈n|,

ih ̄

∂

∂t

cn(t) =〈n|H|ψ(t)〉=

∑N

m=1

〈n|H|m〉cm(t) (5.3)

To obtain the second equality, we have inserted the identity operator, represented as a sum

over the basis,I=

∑

m|m〉〈m|.

The simplest non-trivial Hamiltonian for a 1-dimensional periodic lattice retains only the

effects of nearest neighbors in the lattice. For example, given thatan electron is initially

on site|n〉with probability 1, it is natural to include only the simplest effects, namely that

there must be some probability for the electron the remain on site|n〉, and some probability

to move the electron from site|n〉to its nearest neighboring sites |n+ 1〉or|n− 1 〉. The

simplest choice is such that

〈n|H|n〉 = A 0

〈n|H|n± 1 〉 = −A 1 (5.4)

where bothA 0 andA 1 are real, and all other matrix elements are zero. It is convenient to

write the Hamiltonian as follows,

H=A 0 I−A 1 T−A 1 T† (5.5)

whereIis the identity operator, andT, andT†are given by

T=

∑N

n=1

|n+ 1〉〈n| T†=

∑N

n=1

|n〉〈n+ 1| (5.6)

and where we continue to use the periodicity|N+ 1〉=| 1 〉of the lattice.

5.1.1 Diagonalizing the translation operator

The operatorT has a remarkably simple interpretation, which may be gathered by applying

Tto an arbitrary state

T|n〉 = |n+ 1〉 n= 1,···,N− 1

T|N〉 = | 1 〉 (5.7)

Clearly,T translates the system by one lattice spacing forward, andT†=T−^1 translates

it backwards by one lattice spacing. As a result, we have TN = I. Since [H,T] = 0,

translations are a symmetry of the Hamiltonian, as the physical picture indeed suggests.

It also means that the operatorsH andT may be diagonalized simultaneously. Since the