The problem of the position operator in periodic systems has been analyzed by Resta
and others [78].
The position operator within the Schrödinger representation acts multiplying the wave
function by the space coordinate. This applies only to the bound eigenstates of a finite system
which belong to the class of square-integrable wave functions. However, if one
considers a large system within periodic boundary conditions (PBC) the position
operator becomes meaningless. Let's take the Hilbert space of the single-particle
wave functions defined by the condition
(for the sake of
simplicity a one-dimensional system is assumed), where
is the imposed
periodicity, chosen to be large with respect to atomic dimensions. An operator
maps any vector of the given space into another vector belonging to the same
space: the multiplicative position operator
is not
a legitimate operator when PBC are adopted for the state vectors, since
is not a periodic function whenever
is such.
Since the position operator is ill defined, so is its expectation value, whose
observable effects in condensed matter are related to macroscopic polarization.
For the crystalline case, the problem of the dielectric polarization has been
solved [79,80]: polarization is a manifestation of the Berry phase [81],
i.e. it is an observable which cannot be cast as the expectation value of any
operator, being instead a gauge-invariant phase of the wave function.
The most relevant features are that the expectation value is defined modulo
,
and the operator is no longer one body; it acts as a genuine many-body operator
on the periodic wave function of N electrons.
The position expectation value of a wavefunction using PBC is