Position Operator in Periodic Systems

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 $\Psi(x+L) = \Psi(x)$ (for the sake of simplicity a one-dimensional system is assumed), where $L$ 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 $x$ is not a legitimate operator when PBC are adopted for the state vectors, since $x \Psi(x)$ is not a periodic function whenever $\Psi(x)$ 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 $L$ , 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

$$\langle X \rangle = {L \over 2\pi} \; \; \langle \Psi \mid e^{i {2\pi \over L} \hat{X} } \mid \Psi \rangle\enspace .$$ (137)

The expectation value $\langle X \rangle$ is thus defined only modulo L. The right-hand side of Eq.( 137) is not simply the expectation value of an operator: the given form, as the imaginary part of a logarithm, is indeed essential. Furthermore, its main ingredient is the expectation value of the multiplicative operator $e^{i {2\pi \over L} \hat{X} }$ , a genuine many-body operator. In general, one defines an operator to be one body whenever it is the sum of N identical operators, acting on each electronic coordinate separately.