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arXiv:2606.11679

Consistent Evaluation of Operators Involving the Position Operator in the Bloch Representation: Application to the Orbital Moment

Daehyeon An, Junmo Jeon, Se Kwon Kim

cond-mat.mes-hallquant-phSubmitted Jun 10, 2026• Announced Jun 11, 2026
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Abstract

The position operator plays a central role in condensed-matter observables such as velocity, orbital moment, and electric polarization. In solid-state physics, the evaluation of operators incorporating the position operator has not reached a consensus, as observed in the operator-level discrepancy between the local circulation of Wannier functions and the self-rotation of wave packets. Here, to achieve a consistent evaluation of such operators, we propose three rules for evaluating operators involving the position operator in the Bloch representation. The rules are devised to satisfy physical conditions: independence from the choice of unit cell, preservation of Hermitian conjugacy for the product of operators, and recovery of the correct intraband velocity. We further address the gauge dependence of the position operator and introduce a scheme termed gauge filtration, which systematically removes gauge-dependent contributions from the operators containing the position operator. This methodology ensures that the quantities obtained from the operator evaluation correspond to observable physical phenomena. By applying our framework, we reconcile the results concerning the self-rotation of the wave packet and the local circulation of the Wannier function. We expect our proposal to establish a consistent framework for evaluating operators involving the position operator.

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The position operator plays a central role in condensed-matter observables such as velocity, orbital moment, and electric polarization. In solid-state physics, the evaluation of operators incorporating the position operator has not reached a consensus, as observed in the operator-level discrepancy between the local circulation of Wannier functions and the self-rotation of wave packets. Here, to achieve a consistent evaluation of such operators, we propose three rules for evaluating operators involving the position operator in the Bloch representation. The rules are devised to satisfy physical conditions: independence from the choice of unit cell, preservation of Hermitian conjugacy for the product of operators, and recovery of the correct intraband velocity. We further address the gauge dependence of the position operator and introduce a scheme termed gauge filtration, which systematically removes gauge-dependent contributions from the operators containing the position operator. This methodology ensures that the quantities obtained from the operator evaluation correspond to observable physical phenomena. By applying our framework, we reconcile the results concerning the self-rotation of the wave packet and the local circulation of the Wannier function. We expect our proposal to establish a consistent framework for evaluating operators involving the position operator.
The superlattice representation is described. This representation provides occupation probabilities which form a well-defined non-negative definite analog of the phase space density. The superlattice basis functions are orthonormal wave packets characterized in general by three parameters: a band index, a wave vector, and a superlattice position vector. The crystal momentum and position are automatically coarse-grained so as to satisfy the Heisenberg uncertainty principle. The Bloch and Wannier representations are special cases of the superlattice representation for particular choices of the superlattice parameter. Joint functions of position and momentum, such as the local current density, can be represented as the expectation value of Hermitian operators through the use of this representation.
We propose a numerical method using the discrete variable representation (DVR) for constructing real-valued Wannier functions localized in a unit cell for both symmetric and asymmetric periodic potentials. We apply these results to finding Wannier functions for ultracold atoms trapped in laser-generated optical lattices. Following S. Kivelson [Phys. Rev. B 26, 4269 (1982)], for a symmetric lattice with inversion symmetry, we construct Wannier functions as eigenstates of the position operators $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{x}, \stackrel{\ifmmode \hat{}\else \^{}\fi{}}{y}$, and $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{z}$ restricted to single-particle Bloch functions belonging to one or more bands. To ensure that the Wannier functions are real-valued, we numerically obtain the band structure and real-valued eigenstates using a uniform Fourier grid DVR. We then show, by a comparison of tunneling energies, that the Wannier functions are accurate for both inversion-symmetric and asymmetric potentials to better than 10 significant digits when using double-precision arithmetic. The calculations are performed for an optical lattice with double-wells per unit cell with tunable asymmetry along the xxx axis and a single sinusoidal potential along the perpendicular directions. Localized functions at the two potential minima within each unit cell are similarly constructed, but using a superposition of single-particle solutions from the two lowest bands. We finally use these localized basis functions to determine the two-body interaction energies in the Bose-Hubbard model and show the dependence of these energies on lattice asymmetry.
A method based on expectation values of a position operator is developed for evaluating the spatial shift of a polarized Dirac electron undergoing total reflection. Longitudinal and transverse components of the shift, as well as the associated time delay, are obtained by considering the differences in the expectation values for the physically reflected wave packet and a purely geometrically reflected wave packet. For a number of position operators, this method is used to calculate the mean shift arising from a single reflection of an electron from a step-function potential. Eigenpolarization states are determined by means of a localization form-invariance argument. For these states, results obtained employing the Newton-Wigner position operator are shown to be identical to those obtained using the phase-shift analysis method. A comparison calculation giving different results is made using the current-flux method. The current-flux results are shown to be inconsistent with conservation of total angular momentum normal to the reflecting boundary.
The commutator of position and momentum operators is first considered in the coordinate and momentum representations, and then in an abstract Hilbert space. The reader is then introduced to canonical operators with the associated canonical quantization of commutation rules, which consists of maps defined on a symplectic space and taking values in the set of unitary operators on a Hilbert space. If the symplectic space is finite-dimensional, all of its irreducible representations are unitarily equivalent. The Weyl exponentiated form of the commutation relations is described and investigated. One then arrives at the definition of Weyl systems, and it is shown how to recover from them the formulation in terms of self-adjoint operators which satisfy a generalized form of commutation relations. Further topics are the Heisenberg representation for temporal evolution and the generalized uncertainty relations. Unitary operators associated with symplectic linear transformations are then considered, and, within this framework, translations, rotations and the harmonic oscillator are considered. The Weyl programme is then reassessed. Lastly, the basic postulates of modern quantum theory are eventually studied: the probabilistic nature of the predictions of results of measurements; the correspondence between observables and self-adjoint operators on Hilbert spaces; the expected result of measuring an observable quantity; and the evolution of the state vector in systems which are not affected by any external influence. The chapter ends with an outline of rigged Hilbert space formalism and its applications. The commutator in wave mechanics In the first part of this book we studied wave mechanics within the framework of the Schrödinger equation.
Spectral constraints can alter the observable manifestation of quantum geometry without changing the local eigenspace. Relativistic electron vortices provide a sharp example: the spin orbit interaction redistributes spin and orbital angular momentum, yet its magnetic contribution differs between localized three-dimensional wave packets and stationary monochromatic beams. We resolve this discrepancy by showing that fixing the energy replaces ambient momentum derivatives in position-weighted current moments by shell-tangent derivatives and derive a Hermitian selection rule for first-order observables in momentum-dependent internal subspaces. Its extension-independent form applies beyond Dirac systems, including a broad class of antisymmetric first moments of scalar-charge velocity currents. For the free positive-energy Dirac doublet, the magnetic correction is an exact non-Abelian Berry-curvature contraction with a pointwise saturable quantum-metric bound. Changing the relative phase between the two σzσ_zσz​ spinor components switches this correction from zero in a conserved transverse-helicity sector to a finite value in a radial coherent superposition. At fixed finite Lorentz factor, the radial response scales as O(θ/γ)O(θ/γ)O(θ/γ), compared with O(θ2)O(θ^2)O(θ2) for longitudinal polarization. Spectral reduction and internal coherence thus provide distinct controls of the observable imprint of quantum geometry.
We present an operator-based derivation of Bloch's theorem. In this approach, we first construct an explicit operator form of the Hamiltonian by writing the potential part in terms of an infinite sum of the momentum-translation operators. Next, we prove that it commutes with the position-translation operator corresponding to a lattice translation, using the exponential re-ordering identity. We then build on Merzbacher's treatment of simultaneous diagonalization of commuting operators to reproduce Kittel's central equation and also to derive a new formula for the simultaneous eigenket corresponding to a specific energy eigenvalue. Moreover, through the Born rule, we provide a precise account of the position and momentum probability distributions for Bloch states, and we examine the concept of crystal momentum. Finally, we apply our findings to the Kronig–Penney model, where we numerically depict the band structure and illustrate the delocalized nature of the single electron.