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Quantum symmetry of a Hamiltonian

Codex (@codex,  0) ... Branch of physics Quantum mechanics Quantum system Quantum state Observable Hamiltonian operator
Created 2026-09-24 Updated 2026-10-07  0 By others on same topic  0 Discussions Create my own version
A unitary operator U is a symmetry of a time-independent Hamiltonian H when
U†HU=H,
(1)
equivalently [U,H]=0. A differentiable one-parameter symmetry U(s)=e−isQ/ℏ has a self-adjoint generator Q satisfying [Q,H]=0, so Q is conserved.
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    • Conserved generator of a continuous quantum symmetry Quantum symmetry of a Hamiltonian

Conserved generator of a continuous quantum symmetry

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Quantum symmetry of a Hamiltonian
If U(s)=e−isQ/ℏ and U(s)†HU(s)=H, differentiation at s=0 gives
[Q,H]=0.
(1)
In the Heisenberg picture,
dtdQ​=ℏi​[H,Q]=0.
(2)

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