Introduction: The Deepest Connection in Physics
In classical mechanics, Noether's theorem links symmetries to conservation laws. In quantum mechanics, this becomes: if \([\hat{H}, \hat{G}] = 0\), then \(\hat{G}\) is conserved. The generator of a symmetry is the conserved quantity.
The Quantum Noether Theorem
If operator \(\hat{G}\) commutes with the Hamiltonian:
\[[\hat{G}, \hat{H}] = 0\]then \(\langle\hat{G}\rangle\) is constant in time, and \(\hat{G}\) generates a symmetry of \(\hat{H}\).
Symmetry-Conservation Pairs
| Symmetry | Generator/Conserved Quantity |
|---|---|
| Time translation | Energy \(\hat{H}\) |
| Space translation | Momentum \(\hat{p}\) |
| Rotation | Angular momentum \(\hat{L}\) |
| Phase (global U(1)) | Particle number/charge |
Worked Example: Central Potential
For \(V = V(r)\) (depends only on distance from origin):
\([\hat{H}, \hat{L}^2] = 0\) and \([\hat{H}, \hat{L}_z] = 0\)
Angular momentum is conserved because the system is rotationally symmetric!
The Quantum Connection
Symmetries are the organizing principle of quantum physics. When \([\hat{G}, \hat{H}] = 0\), eigenstates of \(\hat{H}\) can be labeled by eigenvalues of \(\hat{G}\). This is why hydrogen states are labeled by \(l\) and \(m\)—angular momentum is conserved in the spherically symmetric Coulomb potential.