Spatial Symmetries

Spatial symmetries compare a quantum system with translated, rotated, inverted, or uniformly moving descriptions of itself. They determine whether momentum or angular momentum is conserved, which spatial labels organize the spectrum, how wavefunctions transform between frames, and what symmetry remains when a lattice or boundary selects preferred locations and directions.

The central discipline is to separate three questions:

1. What is the spatial transformation?
2. How is it represented on states and observables?
3. Does the Hamiltonian, its domain, and its external environment share it?

An operator may represent a perfectly valid transformation without being a symmetry of the Hamiltonian. Momentum exists in a trapped system, but it is not conserved there. Parity exists as an inversion operator, but an asymmetric potential need not commute with it. Lattice translations may survive even after continuous translations are lost.

Canonical Boundaries

This page owns the taxonomy, shared workflow, and conceptual transitions among the chapter’s pages. Detailed derivations remain at their canonical homes.

Question	Canonical home	Role here
why does momentum generate translations?	Translations and Momentum	connects finite translations, $P$, and $[X,P]$
why is $P=-i\hbar\partial_x$?	Momentum Operator as Generator	owns the wavefunction derivation and domain cautions
when is momentum conserved?	Translation-Invariant Hamiltonians	tests potentials, boundaries, and many-body systems
how do inertial frames act quantum mechanically?	Galilean Boosts	treats the mass-dependent phase and boost generator
how do rotations enter?	Rotations Preview	routes to the full angular-momentum chapter
why is inversion discrete?	Parity as Spatial Inversion	compares parity with proper rotations
what survives in a crystal?	Crystalline Symmetry Preview	introduces lattice translations and Bloch phases

The detailed angular-momentum algebra begins with Rotations in Three Dimensions. The full discrete-symmetry treatment belongs to Parity. Bloch theorem, band theory, and material applications belong to the planned quantum-matter treatment rather than this preview.

A Taxonomy of Spatial Operations

Operation	Parameter structure	Typical representative	Generator or label	Connected to identity?
translation	$\mathbf a\in\mathbb R^3$	$T(\mathbf a)$	momentum $\mathbf P$	yes
proper rotation	$\mathcal R\in SO(3)$	$U(\mathcal R)$	angular momentum $\mathbf J$	yes
Galilean boost	$\mathbf v\in\mathbb R^3$	$B(\mathbf v,t)$	boost $\mathbf K(t)$	yes
spatial inversion	$\mathbf r\mapsto-\mathbf r$	parity $\Pi$	eigenvalue $\pi=\pm1$	no in three dimensions
lattice translation	$\mathbf R\in\Lambda$	$T(\mathbf R)$	crystal momentum $\mathbf k$	discrete
point-group operation	finite rotations, reflections, inversion	$U(g)$	irreducible-representation label	usually discrete

Translations and rotations are spatial transformations at a fixed time. A Galilean boost is a spacetime transformation relating inertial frames. Parity is an orientation-reversing operation. Crystal symmetries are subgroups selected by a periodic environment. Putting all of them under one heading is useful only if these differences remain visible.

Translations and the Active Convention

In one dimension, the active translation that moves a state to the right by $a$ is

$$T(a)=e^{-iaP/\hbar}.$$

Its position-space action is

$$(T(a)\psi)(x)=\psi(x-a).$$

The minus sign in the argument is geometrically necessary: a wavepacket formerly centered at $x_0$ is centered at $x_0+a$ after the transformation. Expanding both forms for small $a$ gives

$$\begin{aligned} \psi(x-a) &= \psi(x)-a\psi'(x)+O(a^2), \\ T(a) &= I-\frac{iaP}{\hbar}+O(a^2). \end{aligned}$$

Comparison yields the familiar position-space expression

$$P=-i\hbar\frac{d}{dx}.$$

This is not an independent postulate. It is the differential expression for the self-adjoint generator of translations on the full line, with an appropriate domain. On an interval, half-line, ring, or space with nontrivial boundary conditions, the differential expression alone does not settle the operator’s self-adjointness or spectrum.

Translating the Position Observable

The same convention gives

$$T^\dagger(a)XT(a)=X+aI.$$

Expanding the left side,

$$T^\dagger(a)XT(a) = X+\frac{ia}{\hbar}[P,X] +O(a^2).$$

Matching the first-order displacement gives

$$[X,P]=i\hbar I.$$

The canonical commutator is therefore the infinitesimal statement that momentum generates translations of position. This relation has domain subtleties in infinite dimensions and cannot be represented exactly by finite-dimensional matrices, because the trace of a finite-dimensional commutator vanishes.

In three dimensions,

$$T(\mathbf a) = \exp\!\left( -\frac{i}{\hbar}\mathbf a\cdot\mathbf P \right),$$

with

$$[X_i,P_j]=i\hbar\delta_{ij}I, \qquad [P_i,P_j]=0$$

for ordinary translations without gauge-field modifications. The commuting momentum components reflect the abelian translation group.

Momentum Eigenstates and Translation Phases

If

$$P|p\rangle=p|p\rangle,$$

then

$$T(a)|p\rangle = e^{-iap/\hbar}|p\rangle.$$

Momentum is thus the label of the one-dimensional irreducible unitary representations of the translation group. In position representation, the generalized eigenfunctions are plane waves,

$$\langle x|p\rangle \propto e^{ipx/\hbar},$$

where the proportionality depends on normalization convention. Plane waves are not normalizable vectors in $L^2(\mathbb R)$; they are generalized eigenstates. Normalizable wavepackets are superpositions of these translation eigenstates.

When Translation Is a Symmetry

A Hamiltonian is invariant under a translation when

$$T(\mathbf a)H T^\dagger(\mathbf a)=H$$

for every allowed $\mathbf a$. For continuous translations, differentiating at the identity gives

$$[H,P_i]=0$$

for each invariant direction. Only those components are conserved. A system can be invariant along $y$ and $z$ while a potential varies along $x$.

For one particle,

$$H=\frac{P^2}{2m}+V(X),$$

and

$$[H,P]=i\hbar V'(X).$$

Continuous translation symmetry therefore requires a constant potential on the modeled region. The nonzero derivative is the operator form of force:

$$\frac{d}{dt}\langle P\rangle = -\langle V'(X)\rangle.$$

The existence of the translation operator does not imply that it commutes with this Hamiltonian. Symmetry is a property of the full model, including its potential, boundaries, domain, and fixed external fields.

Boundaries, Many-Body Systems, and Gauge Fields

Boundary conditions can break translation symmetry even when the differential expression for $H$ looks translation invariant. A particle in a box has a kinetic Hamiltonian in the interior, but the walls select locations. On a ring, continuous translations can survive because periodic boundary conditions are themselves translation invariant.

For many particles with pair interactions depending only on relative positions,

$$H = \sum_i\frac{\mathbf P_i^2}{2m_i} + \sum_{i<j}V(\mathbf X_i-\mathbf X_j),$$

simultaneous translation of every particle leaves the relative coordinates unchanged. The conserved generator is total momentum,

$$\mathbf P_{\mathrm{tot}} = \sum_i\mathbf P_i.$$

In electromagnetic backgrounds, distinguish canonical momentum from kinetic momentum. The Gauge, Phase, and Magnetic Geometry chapter explains why ordinary translations may have to be combined with gauge transformations, producing magnetic translations. The statement $\mathbf P=-i\hbar\boldsymbol\nabla$ identifies the canonical translation generator in a chosen representation; it is not automatically the mechanical momentum $m\dot{\mathbf X}$.

Galilean Boosts

A Galilean boost relates inertial frames moving with constant relative velocity. In a passive convention,

$$t'=t, \qquad \mathbf x'=\mathbf x-\mathbf v t,$$

and a particle’s momentum transforms as

$$\mathbf p'=\mathbf p-m\mathbf v.$$

Quantum wavefunctions require more than a coordinate substitution. They acquire a mass-dependent phase so that momentum and energy transform correctly. At fixed time, the boost generator can be written

$$\mathbf K(t)=m\mathbf X-t\mathbf P.$$

For a free particle,

$$H=\frac{\mathbf P^2}{2m},$$

the explicit time dependence of $\mathbf K(t)$ cancels its commutator evolution:

$$\frac{\partial\mathbf K}{\partial t} + \frac{i}{\hbar}[H,\mathbf K] =0.$$

Thus the boost generator is a constant of motion in the appropriate sense even though it depends explicitly on time.

The Galilei algebra contains

$$[K_i,P_j]=i\hbar m\delta_{ij}I.$$

The mass appears as a central charge. This is tied to projective quantum representations and to the phase acquired under boosts. Galilean boosts are not low-speed Lorentz boosts with every relativistic feature retained; they belong to a different spacetime symmetry group and preserve absolute time.

Proper Rotations

An ordinary three-dimensional rotation $\mathcal R\in SO(3)$ acts on a spinless scalar wavefunction by

$$(U(\mathcal R)\psi)(\mathbf r) = \psi(\mathcal R^{-1}\mathbf r).$$

The inverse appears for the same reason as the shifted argument in a translated wavefunction. For a rotation through angle $\theta$ about $\hat{\mathbf n}$,

$$U(\hat{\mathbf n},\theta) = \exp\!\left( -\frac{i\theta}{\hbar} \hat{\mathbf n}\cdot\mathbf J \right).$$

For a spinless particle, $\mathbf J=\mathbf L=\mathbf X\times\mathbf P$. With spin,

$$\mathbf J=\mathbf L+\mathbf S.$$

A scalar Hamiltonian is rotationally invariant when

$$U(\mathcal R)HU^\dagger(\mathcal R)=H$$

for every rotation, equivalently when $[H,J_i]=0$ for all components under the usual assumptions. The generators themselves obey noncommuting relations, so conservation of all three components does not mean that all three can be simultaneously sharp.

The Rotations Preview fixes conventions and routes onward. The detailed $SO(3)$/$SU(2)$ distinction, angular-momentum algebra, ladder operators, spherical harmonics, and central potentials belong to the next chapter.

Parity as Spatial Inversion

Parity sends

$$\mathbf r\longmapsto-\mathbf r.$$

For a spinless scalar wavefunction,

$$(\Pi\psi)(\mathbf r)=\psi(-\mathbf r),$$

with

$$\Pi^2=I, \qquad \pi=\pm1.$$

Position and momentum are polar vectors and change sign,

$$\begin{aligned} \Pi\mathbf X\Pi^{-1}&=-\mathbf X, \\ \Pi\mathbf P\Pi^{-1}&=-\mathbf P. \end{aligned}$$

Orbital angular momentum is an axial vector and does not:

$$\Pi\mathbf L\Pi^{-1}=\mathbf L.$$

In three dimensions, full inversion has determinant $-1$ and is not a proper rotation. It lies in $O(3)$ but not $SO(3)$ and is not connected continuously to the identity. This is why parity has no infinitesimal generator analogous to momentum or angular momentum.

Parity is a symmetry only when

$$[H,\Pi]=0.$$

For $H=P^2/(2m)+V(X)$ in one dimension, this requires $V(-x)=V(x)$ together with parity-invariant boundary conditions. In that case stationary states can be chosen even or odd, and matrix elements of parity-odd operators obey simple selection rules.

Crystalline Symmetry

A periodic potential preserves only translations by lattice vectors:

$$V(\mathbf r+\mathbf R)=V(\mathbf r), \qquad \mathbf R\in\Lambda.$$

The symmetry condition is

$$[H,T(\mathbf R)]=0 \qquad \text{for every }\mathbf R\in\Lambda.$$

This does not imply $[H,\mathbf P]=0$. Continuous momentum conservation is generally lost, while the commuting lattice translations still supply simultaneous labels. Their eigenstates satisfy

$$T(\mathbf R)|\psi_{\mathbf k}\rangle = e^{-i\mathbf k\cdot\mathbf R} |\psi_{\mathbf k}\rangle.$$

Reciprocal-lattice vectors $\mathbf G$ obey

$$e^{i\mathbf G\cdot\mathbf R}=1,$$

so

$$\mathbf k\sim\mathbf k+\mathbf G.$$

Crystal momentum is therefore a label modulo reciprocal-lattice vectors, not ordinary momentum with an unrestricted unique value. A crystal also retains a finite point group of rotations, reflections, or inversions compatible with the lattice. The combination of translations and point operations forms a space group, with additional nonsymmorphic possibilities beyond this preview.

A Spatial-Symmetry Workflow

1. Specify the operation. Translation, proper rotation, inversion, boost, or lattice operation are different transformations.
2. State active or passive convention. Record how coordinates, states, and operators change.
3. Write the representative. Use the finite unitary or discrete operator before taking derivatives.
4. Identify the generator or eigenvalue label. Momentum and angular momentum generate continuous operations; parity has discrete eigenvalues.
5. Transform every external structure. Decide whether fields, walls, sources, and apparatus are part of the transformed system or fixed background.
6. Test the full Hamiltonian and domain. A symmetric differential expression with asymmetric boundaries is not a symmetric quantum problem.
7. Determine the surviving subgroup. Partial translations, axial rotations, or lattice translations may remain after a larger symmetry is reduced.
8. Extract consequences. State conserved quantities, good labels, degeneracies, and selection rules with their assumptions.
9. Distinguish exact from approximate statements. Weak trapping, anisotropy, disorder, or fields can make old labels only approximate.

This procedure is more reliable than inferring symmetry from the visual shape of one term in the Hamiltonian.

Chapter Map

Page	Central question	Best use
Translations and Momentum	how does $T(a)$ connect displacement, momentum, and the canonical commutator?	concise first derivation
Momentum Operator as Generator	why does translation give $-i\hbar\partial_x$?	detailed derivation, signs, and domains
Translation-Invariant Hamiltonians	what features of a model preserve momentum?	potentials, boundaries, periodicity, and many particles
Galilean Boosts	how do quantum states transform between inertial frames?	phases, generators, and mass central charge
Rotations Preview	how do rotations act before the full angular-momentum algebra?	conventions and transition to the next chapter
Parity as Spatial Inversion	why is inversion discrete and how does it act on vectors?	even/odd states and parity selection rules
Crystalline Symmetry Preview	what replaces continuous symmetry in a periodic system?	Bloch phases, reciprocal equivalence, and point groups

Reading Paths

Wave mechanics and momentum

1. Translations and Momentum
2. Momentum Operator as Generator
3. Translation-Invariant Hamiltonians
4. Free Particle

Toward angular momentum and spin

1. Rotations Preview
2. Rotations in Three Dimensions
3. Angular Momentum Operators
4. What Spin Is

Discrete and lattice structure

1. Parity as Spatial Inversion
2. Crystalline Symmetry Preview
3. Parity
4. Math Needed for Quantum Matter

Frame changes and projective structure

1. Active and Passive Transformations
2. Projective Representations
3. Galilean Boosts

Distinctions Worth Keeping

Do not identify	With	Reason
momentum operator	momentum conservation	conservation requires translation symmetry of $H$
canonical momentum	kinetic momentum	gauge fields can separate the two
symmetric local formula	symmetric quantum problem	domains and boundaries can break symmetry
proper rotation	parity	inversion reverses orientation in three dimensions
orbital angular momentum	total angular momentum	spin contributes to the rotation generator
boost	spatial translation	a boost changes inertial frame and mixes position with time
crystal momentum	ordinary momentum	$\mathbf k$ is defined modulo reciprocal vectors
discrete lattice translation	continuous translation	the former need not imply $[H,\mathbf P]=0$
transformed background	fixed background	the physical symmetry test depends on which is intended

Common Mistakes

• Using $(T(a)\psi)(x)=\psi(x+a)$ while also claiming the state moves right by $a$.
• Memorizing $P=-i\hbar\partial_x$ without specifying the Hilbert space and domain.
• Inferring momentum conservation merely from the existence of $P$.
• Ignoring walls, defects, fields, or boundary conditions in a symmetry test.
• Treating the vector potential as invariant without specifying its gauge transformation.
• Applying an active sign convention to a passive Galilean boost formula.
• Calling parity a rotation in ordinary three-dimensional quantum mechanics.
• Using orbital $\mathbf L$ when spin makes total $\mathbf J$ the rotation generator.
• Confusing lattice momentum conservation modulo $\mathbf G$ with continuous momentum conservation.
• Assuming every visual symmetry of a potential survives the operator domain.

Cross-Links

• Rotations and Orbital Angular Momentum
• Continuous Symmetries and Conservation Laws
• Active and Passive Transformations
• Canonical Commutation Relations
• Position and Momentum Representations
• Heisenberg Group
• Translation Operator
• Momentum Operator
• Magnetic Translations
• Gauge, Phase, and Magnetic Geometry

References

• H. Weyl, The Theory of Groups and Quantum Mechanics, Dover, 1950.
• E. P. Wigner, Group Theory and Its Application to the Quantum Mechanics of Atomic Spectra, Academic Press, 1959.
• L. D. Landau and E. M. Lifshitz, Quantum Mechanics: Non-Relativistic Theory, 3rd ed., Butterworth-Heinemann, 1977.
• R. Shankar, Principles of Quantum Mechanics, 2nd ed., Springer, 1994.
• J. J. Sakurai and J. Napolitano, Modern Quantum Mechanics, 3rd ed., Cambridge University Press, 2020.
• V. Bargmann, “On Unitary Ray Representations of Continuous Groups,” Annals of Mathematics 59, 1–46, 1954.
• N. W. Ashcroft and N. D. Mermin, Solid State Physics, Holt, Rinehart and Winston, 1976.
• M. Tinkham, Group Theory and Quantum Mechanics, Dover, 2003.

Exercises

1. Starting from $T^\dagger(a)XT(a)=X+aI$ and $T(a)=I-iaP/\hbar+O(a^2)$, derive the canonical commutator.

Solution

The adjoint expands as

$$T^\dagger(a) = I+\frac{iaP}{\hbar}+O(a^2).$$

Therefore

$$\begin{aligned} T^\dagger XT &= \left(I+\frac{iaP}{\hbar}\right) X \\ &\quad\times \left(I-\frac{iaP}{\hbar}\right) +O(a^2) \\ &= X+\frac{ia}{\hbar}[P,X] +O(a^2). \end{aligned}$$

Comparing the first-order term with $X+aI$ gives

$$\frac{i}{\hbar}[P,X]=I.$$

Hence $[P,X]=-i\hbar I$, or

$$[X,P]=i\hbar I.$$

2. Let $V(x+a_0)=V(x)$ for one nonzero period $a_0$. Explain why $T(a_0)$ can commute with $H=P^2/(2m)+V(X)$ even though $P$ need not commute with $H$.

Solution

Translation by the lattice period sends the potential to itself:

$$\begin{aligned} T(a_0)V(X)T^\dagger(a_0) &= V(X-a_0), \\ &= V(X). \end{aligned}$$

up to the stated active-convention placement of the shift. The kinetic term is also invariant, so $[H,T(a_0)]=0$.

Continuous invariance would require the same statement for every real $a$. Differentiating that stronger condition would give $[H,P]=0$. A nonconstant periodic potential obeys

$$[H,P]=i\hbar V'(X)\ne0$$

in general. Thus discrete lattice translation symmetry survives while continuous translation symmetry and ordinary momentum conservation do not.

3. Let $H=P^2/(2m)+V(X)$ with $V(-x)=V(x)$. If $|e\rangle$ and $|o\rangle$ have even and odd parity, respectively, determine whether $\langle e|X|e\rangle$, $\langle o|X|o\rangle$, and $\langle e|X|o\rangle$ are forbidden by parity.

Solution

Position is parity odd:

$$\Pi X\Pi^{-1}=-X.$$

For a parity eigenstate $|\pi\rangle$,

$$\begin{aligned} \langle\pi|X|\pi\rangle &= \langle\pi|\Pi^{-1}(\Pi X\Pi^{-1})\Pi|\pi\rangle \\ &=-\langle\pi|X|\pi\rangle. \end{aligned}$$

Therefore both diagonal matrix elements vanish. Between opposite-parity states, the two state parities and the odd operator multiply to an even overall factor, so $\langle e|X|o\rangle$ is allowed by parity. It may still vanish for another reason.

4. For a one-dimensional free particle, define $K(t)=mX-tP$. Show that it is a constant of motion despite its explicit time dependence.

Solution

The explicit derivative is

$$\frac{\partial K}{\partial t}=-P.$$

Using $H=P^2/(2m)$ and $[P^2,X]=-2i\hbar P$,

$$\begin{aligned} \frac{i}{\hbar}[H,K] &= \frac{i}{\hbar} \left[\frac{P^2}{2m},mX\right] \\ &=P. \end{aligned}$$

Hence

$$\frac{\partial K}{\partial t} + \frac{i}{\hbar}[H,K] =0.$$

The explicit and dynamical changes cancel, so the Heisenberg-picture boost generator is constant.

5. A lattice translation eigenstate satisfies $T(\mathbf R)|\psi_{\mathbf k}\rangle=e^{-i\mathbf k\cdot\mathbf R}|\psi_{\mathbf k}\rangle$. Show why $\mathbf k$ and $\mathbf k+\mathbf G$ label the same translation eigenvalues when $\mathbf G$ is reciprocal.

Solution

By definition of a reciprocal-lattice vector,

$$e^{i\mathbf G\cdot\mathbf R}=1$$

for every lattice vector $\mathbf R$. Therefore

$$\begin{aligned} e^{-i(\mathbf k+\mathbf G)\cdot\mathbf R} &= e^{-i\mathbf k\cdot\mathbf R} e^{-i\mathbf G\cdot\mathbf R} \\ &= e^{-i\mathbf k\cdot\mathbf R}. \end{aligned}$$

All lattice translations act with the same phases, so the two labels represent the same character of the discrete translation group. Crystal momentum is defined modulo reciprocal-lattice vectors.