# Translational symmetry

**Translational symmetry** is the invariance of a physical system, its laws, or its equations under a shift of position: moving the whole system, or the coordinate origin, by some distance leaves the physics unchanged. In its continuous form, invariance under arbitrary shifts in space underlies conservation of momentum through [Noether's theorem](https://www.edgepedia.org/noethers-theorem); in its discrete form, invariance under shifts by lattice vectors governs the quantum mechanics of crystals through [Bloch's theorem](https://www.edgepedia.org/blochs-theorem).<sup>[1](https://interactivetextbooks.tudelft.nl/nb1140/content/Lagrangianmechanics.html)</sup><sup> • </sup><sup>[2](https://www.sp.phy.cam.ac.uk/files/media/qcmp_supplementary_notes_2020-2021.pdf)</sup>

| Key fact | Detail |
|---|---|
| Definition | A system has translational symmetry when shifting positions by a vector leaves the Lagrangian, Hamiltonian, or laws of physics unchanged<sup>[1](https://interactivetextbooks.tudelft.nl/nb1140/content/Lagrangianmechanics.html)</sup> |
| Continuous case | Invariance under space translation implies conservation of momentum (Noether's theorem)<sup>[1](https://interactivetextbooks.tudelft.nl/nb1140/content/Lagrangianmechanics.html)</sup> |
| Discrete case | A lattice is symmetric under translation by Bravais lattice vectors; this discrete symmetry underlies Bloch's theorem<sup>[2](https://www.sp.phy.cam.ac.uk/files/media/qcmp_supplementary_notes_2020-2021.pdf)</sup> |
| Bloch form | Eigenstates of a periodic-potential Hamiltonian can be written as a plane wave \( e^{i\mathbf{q}\cdot\mathbf{r}} \) times a function with the periodicity of the lattice<sup>[2](https://www.sp.phy.cam.ac.uk/files/media/qcmp_supplementary_notes_2020-2021.pdf)</sup> |
| Quantum statement | If the Hamiltonian commutes with a translation operator, that operator maps eigenstates to eigenstates of the same energy<sup>[2](https://www.sp.phy.cam.ac.uk/files/media/qcmp_supplementary_notes_2020-2021.pdf)</sup> |
| Related conservation laws | Time translation symmetry gives energy conservation; rotation symmetry gives angular momentum conservation<sup>[1](https://interactivetextbooks.tudelft.nl/nb1140/content/Lagrangianmechanics.html)</sup> |

## Continuous translations and momentum conservation

In classical mechanics, translational symmetry is expressed through the Lagrangian. A system of \( N \) particles with masses \( m_I \), position vectors \( \mathbf{r}_I \), and a potential \( V \) that depends only on the distances between particles, \( V(\{|\mathbf{r}_I - \mathbf{r}_J|\}) \), has a Lagrangian invariant under the spatial translations \( \mathbf{r}_I \to \mathbf{r}_I + \epsilon\,\mathbf{k} \) for any \( \epsilon \) and fixed vector \( \mathbf{k} \). Noether's theorem then gives that \( \sum_I \mathbf{p}_I \cdot \mathbf{k} \) is conserved, and since this holds for all \( \mathbf{k} \), the total momentum \( \mathbf{P} = \sum_I \mathbf{p}_I \) is conserved.<sup>[3](https://users.ox.ac.uk/~math0391/CMlectures.pdf)</sup>

The same result appears in a simpler coordinate form. If the Lagrangian of a system does not depend on a coordinate \( q_i \), the associated generalized momentum \( p_i = \partial \mathcal{L}/\partial \dot{q}_i \) is conserved; such a coordinate is called cyclic. For Cartesian coordinates, independence of \( x \), \( y \), or \( z \) means the corresponding linear momentum is conserved. A thrown ball illustrates the distinction: its Lagrangian \( \mathcal{L} = \tfrac{1}{2}m(\dot{x}^2 + \dot{y}^2 + \dot{z}^2) - mgz \) is independent of \( x \) and \( y \), so horizontal momentum is conserved, while gravity breaks the symmetry in the vertical direction.<sup>[1](https://interactivetextbooks.tudelft.nl/nb1140/content/Lagrangianmechanics.html)</sup>

**Homogeneity of space.** The physical content is that no point in space is special. If space is homogeneous along a direction, the potential is invariant under translation along that direction and the corresponding momentum component is conserved; homogeneity along each of the \( X \), \( Y \), and \( Z \) directions conserves \( p_x \), \( p_y \), and \( p_z \) respectively.<sup>[4](https://www.iitg.ac.in/physics/fac/padmakumarp/Courses/PH101/Lecture12.pdf)</sup> In quantum-mechanical language, an operator that commutes with the Hamiltonian corresponds to a conserved observable, and energy, momentum, and angular momentum conservation correspond to invariance under time translations, space translations, and space rotations.<sup>[5](https://faraday.physics.utoronto.ca/GeneralInterest/DBailey/SubAtomic/Lectures/LectF13/Lect13.htm)</sup>

Noether's theorem applies to continuous symmetries, those connected smoothly to the identity. Translation is continuous: any finite shift can be built from infinitesimal steps.<sup>[5](https://faraday.physics.utoronto.ca/GeneralInterest/DBailey/SubAtomic/Lectures/LectF13/Lect13.htm)</sup> Discrete symmetries, such as the parity reflection \( x \to -x \), which cannot be reached by an infinitesimal amount, lead instead to multiplicative conserved quantities rather than additive ones.<sup>[6](https://faraday.physics.utoronto.ca/GeneralInterest/DBailey/SubAtomic/Lectures/LectF09/Lect09.htm)</sup>

## Discrete translational symmetry and Bloch's theorem

A crystal lattice is not invariant under arbitrary shifts, only under translation by lattice vectors. This discrete symmetry has consequences of a different kind. In quantum mechanics, if an operator such as the Hamiltonian is unchanged under a symmetry operation, applying that operation to an eigenstate produces another eigenstate with the same eigenvalue. Either the two states are the same up to a complex prefactor, or they are degenerate, in which case a basis of the degenerate subspace can be chosen whose members are also eigenstates of the symmetry operation.<sup>[2](https://www.sp.phy.cam.ac.uk/files/media/qcmp_supplementary_notes_2020-2021.pdf)</sup>

For a one-electron Hamiltonian with a periodic lattice potential, \( V(\mathbf{r} + \mathbf{T}) = V(\mathbf{r}) \) for any [Bravais lattice](https://www.edgepedia.org/bravais-lattice) vector \( \mathbf{T} \), the eigenstates can therefore be chosen to be eigenstates of the discrete lattice translation operation. Expanding such a state in reciprocal lattice vectors \( \mathbf{G} \) and factoring out a plane wave gives

\[
\psi_{\mathbf{q}}(\mathbf{r}) = e^{i\mathbf{q}\cdot\mathbf{r}} \sum_{\mathbf{G}} c_{\mathbf{q}-\mathbf{G}}\, e^{-i\mathbf{G}\cdot\mathbf{r}} = e^{i\mathbf{q}\cdot\mathbf{r}}\, u_{j,\mathbf{q}}(\mathbf{r}),
\]

where \( u_{j,\mathbf{q}}(\mathbf{r}) \) is built from functions \( e^{-i\mathbf{G}\cdot\mathbf{r}} \) and must have the same periodicity as the lattice. This result is known as Bloch's theorem: the eigenstates of the one-electron Hamiltonian may be selected so that each takes the form of a plane wave times a function that repeats with the Bravais lattice's periodicity.<sup>[2](https://www.sp.phy.cam.ac.uk/files/media/qcmp_supplementary_notes_2020-2021.pdf)</sup> Equivalently, a Bloch state is a simultaneous eigenstate of all lattice translation operators.<sup>[7](https://cmp.df.unipi.it/roddaro/SSPnotes/ch03-bands.html)</sup>

The wavevector \( \mathbf{k} \) in the Bloch form identifies the irreducible representation of the group of lattice translation operations, while the index \( n \) labels any other distinguishing features of the wavefunction, often arising from rotational symmetry.<sup>[8](https://chemicalgrouptheory.pubpub.org/pub/q0geofda/release/1)</sup>

## What the symmetry explains

Bloch's theorem is the organizing result of band theory. The continuous energy spectrum of a free particle fractures into bands \( E_n(\mathbf{k}) \) once a periodic potential is present, and phenomena from conduction and semiconductor gaps to effective masses and optical selection rules follow from the geometry of those bands.<sup>[9](https://www.physics2u.com/vault/blochs-theorem-from-translation-symmetry.html)</sup> The theorem's reach extends beyond electrons: the same structure reappears for phonons, photonic crystals, and any wave equation with discrete translational invariance.<sup>[9](https://www.physics2u.com/vault/blochs-theorem-from-translation-symmetry.html)</sup>

The symmetry argument also shows what breaks it. If interactions depend on relative coordinates between particles, as genuine two-body forces do, translating one particle alone is not a symmetry and the single-particle Bloch construction does not apply directly; the theorem holds for the single electron in a fixed periodic potential.<sup>[9](https://www.physics2u.com/vault/blochs-theorem-from-translation-symmetry.html)</sup>

## Continuous versus discrete symmetry

The two forms of translational symmetry lead to different kinds of conserved structure. A continuous translation symmetry yields an additive conserved quantity, momentum, via Noether's theorem, which requires the action to be invariant under the symmetry transformation.<sup>[5](https://faraday.physics.utoronto.ca/GeneralInterest/DBailey/SubAtomic/Lectures/LectF13/Lect13.htm)</sup><sup> • </sup><sup>[10](https://www.physics.utoronto.ca/~poppitz/poppitz/PHY354.html)</sup> A discrete translation symmetry, such as a lattice period \( a \), permits only translations by integral multiples of \( a \) to give the same physics.<sup>[11](https://phys.libretexts.org/Bookshelves/Waves_and_Acoustics/The_Physics_of_Waves_%28Goergi%29/05%3A_Waves/5.01%3A_New_Page)</sup> In the quantum problem, translating among a finite set of states returns a discrete number of energy levels, and the wavefunction splits into a plane wave times a periodic function, the statement of Bloch's theorem.<sup>[12](https://etneil.github.io/grad_qm_lec_notes/translation.html)</sup> Discrete time-translation symmetry in periodically driven (Floquet) systems can even be spontaneously broken, producing discrete time crystals, whereas spontaneous breaking of continuous time-translation symmetry is proved impossible for equilibrium states.<sup>[13](https://arxiv.org/abs/2005.03138)</sup>

## References

1. [Lagrangian mechanics, Introduction to Particle and Continuum Physics, TU Delft](https://interactivetextbooks.tudelft.nl/nb1140/content/Lagrangianmechanics.html)
2. [Quantum Condensed Matter Physics, supplementary notes, University of Cambridge](https://www.sp.phy.cam.ac.uk/files/media/qcmp_supplementary_notes_2020-2021.pdf)
3. [Classical Mechanics lecture notes, University of Oxford](https://users.ox.ac.uk/~math0391/CMlectures.pdf)
4. [PH101 Lecture 12: Translational symmetry and homogeneity of space, IIT Guwahati](https://www.iitg.ac.in/physics/fac/padmakumarp/Courses/PH101/Lecture12.pdf)
5. [PHY357 Lectures 13: Conservation Laws, University of Toronto](https://faraday.physics.utoronto.ca/GeneralInterest/DBailey/SubAtomic/Lectures/LectF13/Lect13.htm)
6. [PHY357 Lectures 9–12: Conservation Laws, University of Toronto](https://faraday.physics.utoronto.ca/GeneralInterest/DBailey/SubAtomic/Lectures/LectF09/Lect09.htm)
7. [Intro to bands, Solid State Physics notes, University of Pisa](https://cmp.df.unipi.it/roddaro/SSPnotes/ch03-bands.html)
8. [Bloch's Theorem, Chemical Group Theory (Gordon J. Miller, CHEM 583 notes)](https://chemicalgrouptheory.pubpub.org/pub/q0geofda/release/1)
9. [Bloch's Theorem from Translational Symmetry, physics2u](https://www.physics2u.com/vault/blochs-theorem-from-translation-symmetry.html)
10. [PHY354 Advanced Classical Mechanics, University of Toronto](https://www.physics.utoronto.ca/~poppitz/poppitz/PHY354.html)
11. [Space Translation Invariance, Physics LibreTexts](https://phys.libretexts.org/Bookshelves/Waves_and_Acoustics/The_Physics_of_Waves_%28Goergi%29/05%3A_Waves/5.01%3A_New_Page)
12. [Translation symmetry, Graduate Quantum Mechanics Lecture Notes](https://etneil.github.io/grad_qm_lec_notes/translation.html)
13. [Condensed Matter Physics in Time Crystals, arXiv:2005.03138](https://arxiv.org/abs/2005.03138)

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*Topic: Encyclopedia › Physical world and mathematics › Physics › Classical physics › Mechanics › Motion, forces, and dynamics › Dynamics (mechanics)*

*Initially written Sep 17, 2026 · Reviewed: Oct 15, 2026 · Edited: Oct 15, 2026 · Last review: Oct 15, 2026*

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