QuantumPhysics.dvi

5.2 Propagation in an infinite 1-dimensional lattice

Keeping the lattice spacingafixed, we may let the physical extent of the lattice become

infinite by lettingN → ∞, so that alsoL=Na→ ∞. To do this symmetrically about

the origin, we use the results of the last subsection, but we shall drop the tildes onn. As

N → ∞, the spectrum of the position operator remains discrete, with eigenstates|n〉, and

linearly spaced by the lattice spacinga. Its only modification asN→∞is that the spectrum

ofXextends to larger and larger values.

The translation operator, however, now acquires a continuous spectrum, since its eigen-

valueseiφm=e^2 πim/Nbecome increasingly dense on the unit circle. We represent the eigen-

values in terms of the physical quantity of momentump, or equivalently, of wave numberk,

which is related to momentum byp= ̄hk,

φm

a

→k −

π

a

≤k≤+

π

a

=kc (5.20)

The range of the momentum or wave number is characteristic of propagation on a lattice,

and is referred to as theBrillouin zone. The eigenstates|k;T〉of the translation operator

become labeled by a continuous parameter k. To take the limit properly, however, the

discrete normalization of the states|km;T〉must be changed to a continuum normalization,

|km;T〉

√

Na=|k;T〉 ka=φm (5.21)

With this new normalization, the states|k;T〉are now given by an infinite sum,

|k;T〉=

√

a

+∑∞

n=−∞

einak|n〉 (5.22)

The completeness relation on the momentum states now involves an integral rather than a

discrete sum,

I=

∫+kc

−kc

dk

2 π

|k;T〉〈k;T| (5.23)

It may be verified explicitly by using the expression for|k;T〉in terms of|n〉, and the formula,

∫+kc

−kc

dk

2 π

a eiak(n−n

′)

=δnn′ (5.24)

The normalization of the momentum states must now also be carried out in the continuum,

and may be deduced directly from the completeness relation itself. Applying the complete-

ness relation to an arbitrary state|k′;T〉, we have

|k′;T〉=

∫+kc

−kc

dk

2 π

|k;T〉 〈k;T|k′;T〉 (5.25)