# Bloch's theorem **Bloch's theorem** states that every energy eigenfunction of a particle in a perfectly periodic potential can be written so that it repeats from one cell to the next up to a phase: ψ(x + a) = ψ(x)·exp(i·q·a), where a is the lattice period and q is a real number called the quasimomentum. Equivalently, the [[Wave_function|wave function]] is a plane wave modulated by a function with the periodicity of the lattice. The theorem does not solve the [[Schrödinger_equation|Schrödinger equation]]; it reduces it, by replacing an infinite crystal with a single cell plus a label, and every observable in the problem is periodic in q with period 2π/a.[^likharev-bloch] The bands, gaps and effective masses of [[Solid-state_physics|solid-state physics]] are consequences of that reduction. In the microsim below the reader raises μ = m·a·W/ħ², the strength of a one-dimensional chain of delta barriers, from 0 to 30, and watches a free-particle parabola break into bands separated by gaps. The equation that answers is the master dispersion relation of the Dirac comb, `cos(q·a) = cos(k·a) + (mu/(k·a))·sin(k·a)`, in which k is the particle's own wavenumber and q is the Bloch quasimomentum: wherever the right-hand side lies between −1 and +1 a real q exists and the energy (ħk)²/2m is allowed, and wherever it does not, the state is forbidden and a gap is shaded.[^likharev-comb][^spec-p38] The readouts are the band widths, the gaps and the effective mass `1/m_ef = (1/hbar^2)·d2E/dq2`, which turns negative at a band top.[^likharev-meff] On the [[Physics]] flagship this article is the *Electrons in a periodic potential* section of Part II — Core theories, and a sibling of the [[Schrödinger_equation|Schrödinger equation]] root. It is the quantum-side statement only. The band-structure root itself belongs to [[Materials_science|materials science]], where [[Electronic_band_structure|electronic band structure]] carries the classification of metals, semiconductors and insulators; the [[Semiconductor|semiconductor]] and [[P–n_junction|p–n junction]] pages are placed at row P67 of this spine and are linked from here rather than rebuilt. ## Applications and consequences The theorem's usefulness is that it converts an intractable problem — one electron in a potential with 10²³ wells — into a tractable one, a single cell with a phase condition on its boundary, solved once for each value of one continuous label. Everything below follows from that move: which systems it is allowed for, what the label means, and what a worked example looks like. ### Applicability Bloch's theorem needs one thing and assumes several. It needs the potential to be invariant under a discrete translation, so that the translation operator commutes with the Hamiltonian and the two can share eigenfunctions. It assumes, as written, a single particle moving in a fixed potential: the ions do not move, the electrons do not see one another individually, and the [[Crystal_structure|crystal]] is infinite and perfect. Each of those assumptions fails somewhere. [[Phonon|Lattice vibrations]] break the strict periodicity in time, [[Crystallographic_defect|defects]] and surfaces break it in space, and sufficiently strong electron–electron repulsion invalidates the one-electron picture altogether. The theorem nevertheless survives in a useful form in all three cases, because the corrections can be treated as scattering between Bloch states rather than as a reason to abandon them — which is why band language is still the working language for real, imperfect materials. A [[Quasicrystal|quasicrystal]] is the interesting exception: it has long-range order and sharp diffraction peaks but no translation period at all, so no Bloch label exists. ### Wave vector The quasimomentum q is not momentum. A Bloch state is not an eigenstate of the momentum operator, because the periodic part of the wave function contains many Fourier components; q is the eigenvalue of a translation, not of a derivative. Two properties follow. First, q is defined only modulo 2π/a, since exp(i·q·a) is unchanged when q shifts by that amount, so every distinct state can be labelled inside one interval of width 2π/a — the first Brillouin zone, from −π/a to π/a in one dimension.[^likharev-bloch] Reduced-zone band diagrams are drawn that way for this reason, and the sim folds its k-sampling back into that interval, so a single free parabola reappears as a stack of folded arcs.[^spec-p38] Second, ħq behaves like a [[Momentum|momentum]] in the crystal's dynamics without being one: an external force F drives the label rather than the velocity, through `hbar·dq/dt = F`, which is the equation of motion of the whole semiclassical theory of transport.[^likharev-bloch-osc] ### Detailed example The Dirac comb — delta barriers of weight W at spacing a, also called the Kronig–Penney model when the barriers have finite width — is the cheapest worked example that shows everything.[^kronig-penney] Applying the theorem to one cell and matching at the barrier gives `cos(q·a) = cos(k·a) + (mu/(k·a))·sin(k·a)` with `mu = m·a·W/hbar^2`.[^likharev-comb] The right-hand side is a function of k alone; it oscillates, and its excursions beyond ±1 are the gaps. Three features are worth watching on the dial. Every band top stays pinned at ka = nπ whatever μ, because sin(ka) vanishes there and the right-hand side is exactly cos(nπ).[^derived-bt] The band bottoms move up as μ grows — ka = 1.3065 at μ = 1, 2.2845 at μ = 5, 2.6277 at μ = 10 and 2.9458 at μ = 30 — so the bands narrow while the gaps widen.[^derived-bt] In units of ħ²/(2ma²), band 1 at μ = 5 is 4.651 wide under a gap of 12.800; at μ = 30 it has narrowed to 1.192 under a gap of 24.883, on its way to the sharp level of an isolated well.[^derived-bt] With a = 0.5 nm that energy unit is 0.152 [[Electronvolt|eV]], so μ = 5 describes a first band 0.71 eV wide beneath a 1.95 eV gap.[^derived-bt] The sim does not solve the relation for k; it samples ka at a few thousand points and plots ±acos of the right-hand side wherever that is defined, which costs under a tenth of a millisecond and needs no stored table.[^manual04-comb] ## Statement In one dimension, with U(x + a) = U(x), the theorem says that the eigenfunctions of the Hamiltonian may be chosen in the form ψ_q(x) = exp(i·q·x)·u_q(x), where u_q(x + a) = u_q(x), and equivalently ψ_q(x + a) = ψ_q(x)·exp(i·q·a).[^likharev-bloch] Two words in that sentence carry weight. *May be chosen*: where two states are degenerate the Bloch form is a choice of basis, not a property forced on every eigenfunction. And *the eigenfunctions of the Hamiltonian*: the theorem is about stationary states, so a wave packet built from several q values obeys it only through its components. The physical content is that a periodic potential cannot localise a particle. Whatever the amplitude of the potential, and however deep the wells, the probability density |ψ|² of a Bloch state is periodic and therefore spread over the whole crystal: a particle in a perfect lattice is never bound to one cell, only made slower. What a strong potential does instead is narrow the range of energies over which propagation is possible, which is exactly what the sim's μ dial shows. In three dimensions the same statement runs over the [[Reciprocal_lattice|reciprocal lattice]] of a [[Bravais_lattice|Bravais lattice]], with ψ(r + R) = ψ(r)·exp(i·q·R) for every lattice vector R, and the label q lives in a zone that is a polyhedron rather than an interval. ## Proof Three standard routes reach the same result, and they are worth knowing separately because each generalises in a different direction: the first to arbitrary lattices, the second to any symmetry described by a commuting operator, and the third to the labelling of degenerate states. ### Using lattice periodicity Define the translation operator T that sends x to x + a. Because U is periodic and the kinetic term is translation-invariant, T commutes with the Hamiltonian, so eigenfunctions of H can be chosen to be eigenfunctions of T as well. If ψ is such a function, Tψ = λψ for some complex λ. Conservation of probability requires |λ| = 1 — otherwise the amplitude would grow or decay without bound across the crystal and the state could not be normalised even in the Born–von Kármán sense — so λ can be written exp(i·q·a) with q real. That is the theorem.[^likharev-bloch] The argument uses nothing about the shape of U, which is why the same statement covers a smooth sinusoidal potential and the sim's array of delta spikes. ### Using operators The operator route makes the same point in the language of simultaneous eigenfunctions. Successive translations compose, T(a)T(a′) = T(a + a′), so the eigenvalues must satisfy λ(a)λ(a′) = λ(a + a′); the only continuous solutions of that functional equation are exponentials, which forces λ = exp(i·q·a) without any appeal to normalisation. Writing ψ = exp(i·q·x)·u and substituting into the Schrödinger equation then yields an equation for u on a single cell with periodic boundary conditions, in which q appears as a parameter. Each such problem has a discrete spectrum, E₁(q), E₂(q), …, and those functions are the bands. ### Using group theory The translations of a lattice form an abelian group, and the states of a system with a symmetry group are labelled by its irreducible representations. For an abelian group every irreducible representation is one-dimensional, and for the translation group they are precisely the phase factors exp(i·q·R). Bloch's theorem is therefore the statement that an energy eigenstate carries one label per generator of the translation group — nothing more. The value of this framing is that it extends: adding the point-group operations of the [[Crystal_structure|crystal]] to the translations explains the degeneracies at symmetry points of the zone, which the elementary [[Group_theory|group-theoretic]] argument predicts without any calculation. ## Velocity and effective mass The reason band diagrams are drawn at all is that their slope and curvature are the dynamics. The group velocity of a packet built around q is `v = (1/hbar)·dE/dq`, so a flat band means slow carriers and a state at a band extremum does not move. Differentiating again with an external force gives an acceleration proportional to the curvature, which is written as an effective mass through `1/m_ef = (1/hbar^2)·d2E/dq2`.[^likharev-meff] A sharply curved band is light and a flat one heavy; the Portal Book's reference values are silicon at 0.26 electron masses in the conduction band and 0.39 in the valence band, with indium antimonide down to 0.0145.[^likharev-meff] The comb reproduces the structure of that statement but not its magnitudes, and the gap is instructive. At the bottom of band 1 the derived effective mass is 1.019 electron masses at μ = 1, 1.275 at μ = 5, 1.719 at μ = 10 and 3.685 at μ = 30 — always heavier than a free electron, never approaching silicon's 0.26.[^derived-bt] At the band top the curvature is negative, and the closed form is exactly −μ/π²: −0.101, −0.507, −1.013 and −3.040 for the same four values of μ.[^derived-bt] A negative mass is not a paradox but a bookkeeping convention: an electron near a band top responds to a force in the direction opposite to a free particle, and the standard repair is to describe the empty states instead, as positively charged holes with positive mass. At large μ the two numbers converge — 3.685 at the bottom against 3.040 in magnitude at the top for μ = 30 — which is the tight-binding limit asserting itself, since a pure cosine band `E = E_n + 2·hbar·eta_n·cos(q·a)` of width 4ħ|η_n| has equal and opposite curvature at its two ends.[^likharev-tb][^derived-bt] At the opposite end of the dial the [[Free_electron_model|weak-potential]] picture applies: gaps of width 2|U_n| open at q = πm/a, where the nearly free electron is Bragg-reflected.[^likharev-nfe] Between them lies the last consequence worth naming, the divergence of the density of states at every band edge, which follows from dN = (L/2π)dq the moment dE/dq goes to zero.[^likharev-dos] ## Mathematical caveat Three caveats belong with the theorem. The Bloch functions of an infinite crystal are not square-integrable — |ψ|² is periodic, so its integral over all space diverges — and the usual repair is Born–von Kármán boundary conditions, which wrap a crystal of N cells into a ring and quantise q into N allowed values per zone. Nothing physical depends on N, but statements about counting states do. Second, where two bands are degenerate at a point of the zone the Bloch form fixes only the subspace, not the individual functions, and any unitary mixture within the degenerate subspace is equally a Bloch state; the choice matters when perturbations are added, which is where the nearly free electron calculation gets its 2|U_n| splitting.[^likharev-nfe] The third caveat is practical and is recorded in the Wikitube sub-manual. The extraction of the Portal Book prints the comb's dispersion relation with a minus sign in front of the sine term; a direct derivation for a repulsive barrier gives the plus sign used here and on the [[Electronic_band_structure|band structure]] page.[^manual04-sign] The discrepancy is testable rather than a matter of taste: as ka → 0 the right-hand side tends to 1 + μ, so a repulsive comb must have a gap at the very bottom of the spectrum. If a band reaches k = 0, the attractive comb has been coded by mistake.[^manual04-sign] ## History and related equations The mathematics predates the physics. Solutions of a linear differential equation with periodic coefficients were shown in the nineteenth century to take the form of a periodic function times an exponential, the result now known as Floquet theory.[^floquet] Felix Bloch rediscovered the same structure in 1929 in the quantum problem of an [[Electron|electron]] in a crystal lattice, which is the form it takes here.[^bloch1929] Two years later the Kronig–Penney model made a solvable one-dimensional example of it, and that model, in its delta-barrier limit, is the sim's engine.[^kronig-penney][^openstax-v3-ch9] The related equations are mostly what happens when a field is switched on. Because a constant force drives q rather than v, a [[Bloch_oscillation|Bloch oscillation]] results: q sweeps across the zone, wraps, and the packet oscillates in real space with period `t_B = 2·pi·hbar/(F·a)` and amplitude ΔE/F, rather than accelerating.[^likharev-bloch-osc] The same tilt produces the Wannier–Stark ladder, a set of levels spaced by exactly F·a.[^likharev-bloch-osc] The numbers explain why the effect is a superlattice phenomenon: an electric field of 10⁶ V/m gives F·a = 10 meV and t_B = 0.41 ps in a 10 nm superlattice, but only 0.5 meV and 8.3 ps at a bulk spacing of 0.5 nm, where scattering intervenes long before one period is complete.[^derived-bt][^likharev-bloch-osc] Pushed harder still, the packet leaks into the next band by Landau–Zener tunnelling, a formula the sub-manual flags as damaged in the extract and not to be used until the printed page is checked.[^manual04-osc] ## See also - [[Particle_in_a_one-dimensional_lattice]] — the sim's own model, treated as its own pair - [[Bloch_oscillation]] — what a constant force does to a Bloch state - [[Electronic_band_structure]] — the materials-side root this page feeds - [[Free_electron_model]] — the μ → 0 end of the dial - [[Reciprocal_lattice]] — where the quasimomentum lives in three dimensions - [[Quantum_tunnelling]] — the single-barrier problem the comb repeats - [[Band_gap]] — the width of the forbidden range - [[Fermi_level]] — where the filling of the bands stops ## References [^likharev-bloch]: Likharev, Konstantin (2013). *Essential Graduate Physics, Part QM: Quantum Mechanics*. Chapter 2, Bloch's theorem `psi(x + a) = psi(x)*exp(i*q*a)`, the quasimomentum q, and the 2π/a periodicity of all observables in q (pp. 74–75). https://open.umn.edu/opentextbooks/textbooks/part-qm-quantum-mechanics [^likharev-comb]: Likharev, Konstantin (2013). *Essential Graduate Physics, Part QM: Quantum Mechanics*. Chapter 2, the Dirac comb: `cos(q*a) = cos(k*a) + (mu/(k*a))*sin(k*a)` with `mu = m*a*W/hbar^2`, and the reduction of the N = 1 and N = 2 cases to the single- and double-barrier results (pp. 72–73, 83–84). https://open.umn.edu/opentextbooks/textbooks/part-qm-quantum-mechanics [^likharev-tb]: Likharev, Konstantin (2013). *Essential Graduate Physics, Part QM: Quantum Mechanics*. Chapter 2, tight binding: `E = E_n + 2*hbar*eta_n*cos(q*a)`, band width 4ħ|η_n|, valid for ħ|η_n| ≪ E_n (pp. 78–79). https://open.umn.edu/opentextbooks/textbooks/part-qm-quantum-mechanics [^likharev-nfe]: Likharev, Konstantin (2013). *Essential Graduate Physics, Part QM: Quantum Mechanics*. Chapter 2, the weak-potential limit: gaps of width Δ_n = 2|U_n| opening at q = πm/a (pp. 81–82). https://open.umn.edu/opentextbooks/textbooks/part-qm-quantum-mechanics [^likharev-dos]: Likharev, Konstantin (2013). *Essential Graduate Physics, Part QM: Quantum Mechanics*. Chapter 2, the one-dimensional density of states dN = (L/2π)dq and its divergence at band edges (p. 86). https://open.umn.edu/opentextbooks/textbooks/part-qm-quantum-mechanics [^likharev-meff]: Likharev, Konstantin (2013). *Essential Graduate Physics, Part QM: Quantum Mechanics*. Chapter 2, effective mass `1/m_ef = (1/hbar^2)*d2E/dq2`, negative at a band top; silicon 0.26 mₑ (conduction) and 0.39 mₑ (valence), InSb down to 0.0145 mₑ (pp. 87–89). https://open.umn.edu/opentextbooks/textbooks/part-qm-quantum-mechanics [^likharev-bloch-osc]: Likharev, Konstantin (2013). *Essential Graduate Physics, Part QM: Quantum Mechanics*. Chapter 2, Bloch oscillations: `hbar*dq/dt = F` valid for Fa much smaller than the band and gap energies, `t_B = 2*pi*hbar/(F*a)`, `Omega_B = F*a/hbar`, spatial swing `xmax = dE_n/F`, the Wannier–Stark ladder of step Fa, and the observation that the effect is seen in superlattices with a ≈ 10 nm but is masked by scattering in bulk (pp. 87–94). https://open.umn.edu/opentextbooks/textbooks/part-qm-quantum-mechanics [^manual04-comb]: Wikitube MICROSIM_GUIDE sub-manual 04, *Atomic, Quantum, Statistical and Electromagnetic Physics*, §3.5 "Energy bands of a Dirac comb": sample ka at 4,096 points and plot ±acos of the right-hand side wherever |RHS| ≤ 1 rather than root-finding (under 0.1 ms per change); no bake is needed; and the sampling caveat that qa ∝ √(distance to the edge) leaves points sparse in q near band edges. [^manual04-sign]: Wikitube MICROSIM_GUIDE sub-manual 04, §3.5 and Appendix A: the text extraction of [^likharev-comb] prints the dispersion relation with a minus sign before the sine term, while a direct derivation for a repulsive barrier gives the plus sign used here; the recorded unit test is that the right-hand side tends to 1 + μ as ka → 0, so a repulsive comb must show a gap at the bottom of the spectrum. [^manual04-osc]: Wikitube MICROSIM_GUIDE sub-manual 04, §3.6 "Bloch oscillations" and Appendix A: the Landau–Zener interband probability is damaged in the extraction of [^likharev-bloch-osc] (the printed exponent is dimensionally inconsistent) and must be checked against the printed page before use; the general form `P = exp(-2*pi*abs(H12)^2/(hbar*abs(d(E1 - E2)/dt)))` is unambiguous but its mapping onto the book's variables is not recoverable from the text. [^derived-bt]: Computed for this article from the dispersion relation of [^likharev-comb], in units of ħ²/(2ma²). Band-1 bottoms (the root of RHS = 1): ka = 1.3065 (μ = 1), 2.2845 (μ = 5), 2.6277 (μ = 10), 2.9458 (μ = 30); every band top at ka = nπ, since sin(nπ) = 0. At μ = 5 band 1 is 4.651 wide and gap 1 is 12.800, with band 2 beginning at ka = 4.7613; at μ = 30 band 1 is 1.192 and gap 1 is 24.883. With a = 0.5 nm, ħ²/(2ma²) = 0.1524 eV, giving 0.709 eV and 1.951 eV at μ = 5. Effective mass from m_ef/m = −R′(ka₀)/ka₀ with R the right-hand side: 1.0186 (μ = 1), 1.2751 (μ = 5), 1.7192 (μ = 10), 3.6853 (μ = 30) at the band-1 bottom, tending to 1 as μ → 0. At the band top ka = π the same expression is exactly −μ/π²: −0.1013, −0.5066, −1.0132 and −3.0396. Bloch-oscillation figures at an electric field of 10⁶ V/m: F·a = 10.0 meV and t_B = 0.414 ps for a = 10 nm, F·a = 0.50 meV and t_B = 8.27 ps for a = 0.5 nm. These reproduce the derived values recorded in sub-manual 04 §3.5. [^spec-p38]: Matter & Energy Cluster contract, `_registry/plans/PHYSICS_SECTIONS.md` row P38: new sibling of the Schrödinger-equation root (`wt-quantum.diracComb`, sub-manual 04 §3.5–3.6 on the framework). The reader raises μ over 0–30; the free parabola breaks into bands with gaps, band tops stay pinned at ka = nπ, the widths shrink toward the isolated-well levels, and the effective mass readout goes negative at a band top, with silicon's 0.26 mₑ and InSb's 0.0145 mₑ carried as scale values. The sim runs live by sampling ka and plotting ±acos(RHS); it needs no bake, because the sign of the extracted relation is checked by the gap at ka → 0. The row places the semiconductor and p–n junction pages at P67 rather than rebuilding them here. [^openstax-v3-ch9]: Sanny, Jeff; Ling, Samuel (2016). *University Physics Volume 3* (OpenStax). Chapter 9, "Condensed Matter Physics" (pp. 393–440), the band theory of solids at first-course level (page to pin). https://open.umn.edu/opentextbooks/textbooks/university-physics-volume-3 [^bloch1929]: Bloch, Felix (1929). "Über die Quantenmechanik der Elektronen in Kristallgittern." *Zeitschrift für Physik* 52. Pages and DOI to pin. [^floquet]: Floquet, Gaston (1883). "Sur les équations différentielles linéaires à coefficients périodiques." *Annales scientifiques de l'École Normale Supérieure*, series 2, volume 12. Pages and DOI to pin. [^kronig-penney]: Kronig, Ralph de Laer; Penney, William G. (1931). "Quantum Mechanics of Electrons in Crystal Lattices." *Proceedings of the Royal Society of London A* 130. Pages and DOI to pin. The delta-barrier limit of this model is the Dirac comb used by the sim. ## Further reading - Likharev, Konstantin (2013). *Essential Graduate Physics, Part QM: Quantum Mechanics*, Chapter 2 — Bloch's theorem, the Dirac comb, tight binding, effective mass and Bloch oscillations in closed form. https://open.umn.edu/opentextbooks/textbooks/part-qm-quantum-mechanics - Sanny, Jeff; Ling, Samuel (2016). *University Physics Volume 3*, Chapter 9 — band theory of solids at first-course level. https://open.umn.edu/opentextbooks/textbooks/university-physics-volume-3 <!-- MATTERSIM:BEGIN g33 — Matter & Energy Cluster microsim (framework build, specs/sims/Bloch's_theorem.json); do not hand-edit inside --> **Microsim — three.js (Wikitube framework):** *Bloch's theorem* <div class="wt-sim" data-src="https://wikitube-3d-microsims.netlify.app/matter/Bloch's_theorem.html" data-title="Bloch's theorem"></div> *Built from `MICROSIM_GUIDE/specs/sims/Bloch's_theorem.json`; part of the [[PORTAL_Matter|Matter portal]] spine (section sims and See-also variants).* <!-- MATTERSIM:END --> ## Wikipedia : Wikitube **Strict pair:** [Wikipedia](https://en.wikipedia.org/wiki/Bloch's_theorem) : [Wikitube](https://en.wikitube.io/wiki/Bloch's_theorem) · pinned revision [1372575844](https://en.wikipedia.org/w/index.php?oldid=1372575844) · 2026-09-11 ## Previous hub tags Hubs: `Life_Physics`. Portals: [[PORTAL_Physics]]. --- *Matter & Energy Cluster child articles, wave 1 · 2026-09-11 · drafted · Physics row P38 · sim pending (matter/Bloch's_theorem).*