Noether Theorem in Quantum Mechanics

Statement

In closed quantum mechanics, a continuous unitary symmetry generated by $G$ is represented as

$$U(\alpha) = e^{-i\alpha G/\hbar}.$$

If the Hamiltonian is invariant,

$$U(\alpha)HU^\dagger(\alpha)=H$$

for all $\alpha$, then

$$[G,H]=0.$$

If $G$ has no explicit time dependence, it is conserved:

$$\frac{d}{dt}\langle G\rangle=0.$$

Assumptions

• The symmetry is represented by a differentiable unitary family.
• $G$ is the self-adjoint generator of that family.
• The Hamiltonian is invariant under the transformation.
• $G$ has no explicit time dependence, or the explicit term is included.
• Operator domains are controlled when $G$ or $H$ is unbounded.

Quantum-Mechanical Meaning

The quantum version is the commutator form of the Noether pattern:

• translation symmetry gives momentum conservation;
• rotation symmetry gives angular-momentum conservation;
• time-translation symmetry gives energy conservation for time-independent Hamiltonians.

The statement is not the full field-theoretic Noether theorem. It is the operator version used in ordinary quantum dynamics.

Canonical Links

• Quantum Noether Principle
• Commutators and Conservation Laws
• Generators
• Translation-Invariant Hamiltonians
• From Quantum Generators to Noether Currents
• Conservation Laws
• Unitary Symmetries
• Stone Theorem

Common Mistakes

• Calling a transformation a symmetry before checking the Hamiltonian.
• Forgetting explicit time dependence in the conserved operator.
• Treating one stationary expectation value as an operator conservation law.
• Ignoring projective phases or antiunitary symmetries.
• Applying a classical Noether theorem slogan without identifying the quantum generator.

Quick Check

Why does translation invariance imply momentum conservation for a time-independent Hamiltonian?

Solution

Translations are generated by momentum: $U(a)=e^{-iaP/\hbar}$. If $U(a)HU^\dagger(a)=H$ for all $a$, differentiating at $a=0$ gives $[P,H]=0$. With no explicit time dependence in $P$, the expectation value and spectral distribution of momentum are conserved.

References

• E. Noether, “Invariante Variationsprobleme,” Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, 235-257, 1918.
• E. P. Wigner, Group Theory and Its Application to the Quantum Mechanics of Atomic Spectra, Academic Press, 1959.
• 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.