Translation Operator

Operator

The active translation operator by displacement $\mathbf a$ is commonly written

$$U(\mathbf a) = \exp \left( -\frac{i}{\hbar}\mathbf a\cdot\mathbf P \right).$$

On scalar wavefunctions,

$$(U(\mathbf a)\psi)(\mathbf r) = \psi(\mathbf r-\mathbf a).$$

The generator relation is

$$U(\mathbf a)^\dagger \mathbf R U(\mathbf a) = \mathbf R+\mathbf a.$$

Assumptions

• The convention is active translation of the state by $\mathbf a$.
• $\mathbf P$ is the canonical momentum generator.
• The system has a translation group acting on the relevant configuration space.
• Boundary conditions may make translations discrete or may break translation symmetry.

Common Mistakes

• Mixing active state translations with passive coordinate changes.
• Dropping the sign in the exponential after changing conventions.
• Assuming continuous translation symmetry in a finite box, lattice, or with boundaries.
• Confusing ordinary translations with magnetic translations in a vector potential.

Canonical Links

• Translations and Momentum
• Momentum Operator as Generator
• Momentum Operator
• Unitary Symmetries
• Unitary Representations

References

• H. Weyl, The Theory of Groups and Quantum Mechanics, Dover, 1950.
• J. J. Sakurai and J. Napolitano, Modern Quantum Mechanics, 3rd ed., Cambridge University Press, 2020.