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arXiv:2609.38204 · American Journal of Physics

Bloch's theorem: An operator-based derivation

Celal Sirin

Published Jul 23, 2026• 0 citations
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Abstract

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.

DOI
10.1119/5.0310014

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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.
Bloch’s theorem, which is one of the cornerstones of condensed matter physics, has played a fundamental role in the development of many theories, such as the band theory, Fermi liquid theory, and BCS theory. It states that when the system has discrete translational symmetry, the corresponding eigenstate can be labeled by a conserved quantity, or a good quantum number — Crystal momentum. Based on Bloch’s theorem, many important physical quantities can be expressed as the integral over the entire Brillouin zone or the Fermi surface. In realistic macroscopic materials, the translational symmetry is explicitly broken down due to the existence of boundaries. It is natural to ask why we can still use Bloch’s theorem to understand the physical properties of real materials? In traditional textbooks of solid-state physics, a thermodynamic limit argument is used to explain the above question. Since the lattice size of the macroscopic material is very large, e.g. N≈1023 , its asymptotic behaviors can be described by the thermodynamic limit N→∞ . Therefore, if we fix the lattice constant, i.e., a, the system length will extend to infinity, i.e. L=aN→∞ . Since wave functions, e.g. ψ(x)∝eikx , should be bounded at the infinity, e.g. |ψ(x→±∞)|<∞ , the corresponding momentum is restricted to be real numbers. This revives Bloch’s theorem. Physically, for a finite-size system with open boundary conditions, the boundary only acts the role of scattering potential and can be regarded as a perturbation to the Bloch Hamiltonian. Therefore, the eigenstate of the OBC Hamiltonian is a superposition of all the scattered Bloch waves with the same energy. In this sense, one can say that, although the existence of boundary definitely breaks the translational symmetry, Bloch’s theorem is perturbatively broken down, and one can still use Bloch’s theorem to understand the physical properties in real materials. Even for interacting systems, Bloch’s theorem is also approximately preserved. Indeed, in a many-body system, the elementary excitations (or quasiparticles) are referred to the eigenmodes with well-defined energy, momentum, and dispersion relation. This means the excitations of the system are extended Bloch-like states in the bulk, which is consistent with the physical intuition in a disorder-free system. In this paper, we show that all the above arguments are challenged in some Hermitian bosonic systems, in which all the eigenstates, or quasiparticles are localized at the boundary, indicating that Bloch’s theorem is nonperturbatively broken down. This phenomenon generalizes the concept of the non-Hermitian skin effect from dissipative systems to non-dissipative systems, and can be understood as the Hermitian skin effect in general. Here the physical origin of the Hermitian skin effect is that although the bosonic Hamiltonian is Hermitian, its elementary excitations are determined by a non-Hermitian matrix. In order to understand the quasiparticles in such systems, the so-called generalized Brillouin zone theory is necessary. Finally, based on the Bogoliubov theory, the quench dynamic is also studied to illustrate the non-perturbative breakdown of Bloch’s theorem.
A general Hamiltonian H of electrons in finite concentration, interacting via any two-body coupling inside a crystal of arbitrary dimension, is considered. For simplicity and without loss of generality, a one-band model is used to account for the electron-crystal interaction. The electron motion is described in the Hilbert space ${\mathit{S}}_{\mathrm{\ensuremath{\varphi}}}$, spanned by a basis of Slater determinants of one-electron Bloch wave functions. Electron pairs of total momentum K and projected spin \ensuremath{\zeta}=0,\ifmmode\pm\else\textpm\fi{}1 are considered in this work. The Hamiltonian then reads H=HD{\mathit{H}}_{\mathit{D}}HD​+${\ensuremath{\sum}}_{\mathit{K},\mathrm{\ensuremath{\zeta}}}$${\mathit{H}}_{\mathit{K},\mathrm{\ensuremath{\zeta}}}$, where HD{\mathit{H}}_{\mathit{D}}HD​ consists of the diagonal part of H in the Slater determinant basis. ${\mathit{H}}_{\mathit{K},\mathrm{\ensuremath{\zeta}}}$ describes the off-diagonal part of the two-electron scattering process which conserves K and \ensuremath{\zeta}. This Hamiltonian operates in a subspace of ${\mathit{S}}_{\mathrm{\ensuremath{\varphi}}}$, where the Slater determinants consist of pairs characterized by the same K and \ensuremath{\zeta}. It is shown that the whole set of eigensolutions \ensuremath{\psi},\ensuremath{\epsilon} of the time-independent Schr\"odinger equation (H-\ensuremath{\epsilon})\ensuremath{\psi}=0 divides into two classes, ${\mathrm{\ensuremath{\psi}}}_{1}$,${\mathrm{\ensuremath{\epsilon}}}_{1}$ and ${\mathrm{\ensuremath{\psi}}}_{2}$,${\mathrm{\ensuremath{\epsilon}}}_{2}$. The eigensolutions of class 1 are characterized by the property that for each solution ${\mathrm{\ensuremath{\psi}}}_{1}$,${\mathrm{\ensuremath{\epsilon}}}_{1}$ there is a single K and \ensuremath{\zeta} such that (HD{\mathit{H}}_{\mathit{D}}HD​+${\mathit{H}}_{\mathit{K},\mathrm{\ensuremath{\zeta}}}$-${\mathrm{\ensuremath{\epsilon}}}_{1}$)${\mathrm{\ensuremath{\psi}}}_{\mathit{K},\mathrm{\ensuremath{\zeta}}}$=0 where, in general, ${\mathrm{\ensuremath{\psi}}}_{1}$\ensuremath{\ne}${\mathrm{\ensuremath{\psi}}}_{\mathit{K},\mathrm{\ensuremath{\zeta}}}$, whereas each solution ${\mathrm{\ensuremath{\psi}}}_{2}$,${\mathrm{\ensuremath{\epsilon}}}_{2}$ of class 2 fulfills (HD{\mathit{H}}_{\mathit{D}}HD​-${\mathrm{\ensuremath{\epsilon}}}_{2}$)${\mathrm{\ensuremath{\psi}}}_{2}$=0. We prove also that the eigenvectors of class 1 have off-diagonal long-range order, whereas those of class 2 do not. Finally, our result shows that off-diagonal long-range order is not a sufficient condition for superconductivity. \textcopyright{} 1996 The American Physical Society.
Abstract Bootstrap methods, initially developed for solving statistical and quantum field theories, have recently been shown to capture the discrete spectrum of quantum mechanical problems, such as the single particle Schrödinger equation with an anharmonic potential. The core of bootstrap methods builds on exact recursion relations of arbitrary moments of some quantum operator and the use of an adequate set of positivity criteria. We extend this methodology to models with continuous Bloch band spectra, by considering a single quantum particle in a periodic cosine potential. We find that the band structure can be obtained accurately provided the bootstrap uses moments involving both position and momentum variables. We also introduce several new techniques that can apply generally to other bootstrap studies. First, we devise a trick to reduce by one unit the dimensionality of the search space for the variables parametrizing the bootstrap. Second, we employ statistical techniques to reconstruct the distribution probability allowing to compute observables that are analytic functions of the canonical variables. This method is used to extract the Bloch momentum, a quantity that is not readily available from the bootstrap recursion itself.
A translation operator is introduced to describe the quantum dynamics of a position-dependent mass particle in a null or constant potential. From this operator, we obtain a generalized form of the momentum operator as well as a unique commutation relation for $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{x}$ and ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{p}}_{\ensuremath{\gamma}}$. Such a formalism naturally leads to a Schr\"odinger-like equation that is reminiscent of wave equations typically used to model electrons with position-dependent (effective) masses propagating through abrupt interfaces in semiconductor heterostructures. The distinctive features of our approach are demonstrated through analytical solutions calculated for particles under null and constant potentials like infinite wells in one and two dimensions and potential barriers.
It is sometimes convenient to emphasize the local aspects of a part of a crystalline system, and use the corresponding localized states to build a basis of the states of the whole system. In many cases it is only a subspace of these local states that is relevant, and the Hubbard operators provide a fairly simple way to write the corresponding projected Hamiltonian. Two examples of this type of treatment are presented int this work. The first is a Co2+ impurity in a MgO crystal interacting through a Jahn-Teller term with the crystal phonons, and it is shown how this interaction affects the electronic Raman scattering. The second is the Anderson lattice when the local electrons have an infinite Coulomb repulsion, and a diagrammatic expansion with cumulants is discussed. We propose a method to obtain approximate Green's functions for the Anderson lattice that employs the exact solutions of an atomic problem, and the corresponding spectral density of the local electrons is calculated.