QuantumPhysics.dvi

operatorT is simpler thanH, it will be advantageous to diagonalize it first. Since T is

unitary, its eigenvalues are pure phaseseiφm. We shall denote the corresponding eigenstates

by|km;T〉; we include the labelTinside the state ket, because these kets will form a basis

in whichTis diagonal and this basis is different from the basis|n〉. The labelTis included

to make this distinction clear. Thus we have,

T|km;T〉=e−iφm|km;T〉 (5.8)

FromTN=I, we haveNφm≡0 (mod 2π). The eigenvalues are all distinct, and we have

φm= 2πm/N m= 0, 1 ,···,N− 1 (5.9)

This leads toNorthogonal eigenstates (sinceTis unitary), and gives an explicit construction

of all the normalized eigenstates ofT,

|km;T〉=

1

√

N

∑N

n=1

e+inφm|n〉 m= 0, 1 ,···,N− 1 (5.10)

These states are usually referred to asBloch states.

5.1.2 Position and translation operator algebra

We may define aposition operatorXon this finite lattice by introducing a physical lattice

spacingainto the problem. The origin of position is arbitrary, which allows us to leave an

arbitrary additive constantx 0 in the eigenvalues,

X|n〉= (an+x 0 )|n〉 (5.11)

Clearly, this operator is self-adjoint and indeed corresponds to anobservable. The physical

length of the lattice is then L= aN. The translation operatorT now has the physical

interpretation of shifting the system forward by one lattice spacinga. Since the action ofT

andXis known on all states, we can compute the action ofTonX,

T†XT|n〉 = T†X|n+ 1〉=T†(a(n+ 1) +x 0 )|n+ 1〉

= (a(n+ 1) +x 0 )|n〉= (X+a)|n〉 (5.12)

This equation being valid on all states, we conclude that

T†XT=X+aI (5.13)

which makes full intuitive sense: the action of the translation generator indeed shifts the

position eigenvalues all bya. Of course, what is not well-accounted for here is the fact that

Xis really a periodic variable, defined only modaN. To account for this periodicity, a more

appropriate operator would bee^2 πiX/(Na).