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conservation-laws

# conservation-laws - Chem 540 Nancy Makri Symmetries...

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Chem. 540 Nancy Makri Symmetries, Commutators and Conservation Laws In class we proved a theorem that relates the time derivative of an observable to the commutator between the corresponding operator and the system’s Hamiltonian. Thus, if an operator (which is not explicitly time-dependent) commutes with the Hamiltonian, the corresponding observable is a conserved quantity. The classical version of this involves the classical analogue of the commutator, which is called Poisson bracket. If a dynamical variable Poisson commutes with the Hamiltonian, the dynamical variable is a constant of the motion. Noether’s theorem relates this to symmetries. It states that any differentiable symmetry of the action in- tegral in a physical system has a corresponding conservation law. For example, the Lagrangian (and thus the action) of a system with orientational symmetry is rotationally invariant. This symmetry leads to con- servation of the total angular momentum. Perhaps the simplest way to understand this is through Hamilton’s equations of motion: A symmetry manifests itself as the absence of dependence of the system’s Lagrangian (by extension, the Hamiltonian) on a particular coordinate (in a judiciously chosen coordinate system of course!) If the Hamiltonian does not contain explicitly a particular coordinate, the corresponding canonical momentum is conserved. Hamilton-Jacobi theory takes advantage of this in the most elegant (yet formal, often impractical) way. I note in passing that a classical system of N degrees of freedom can have up to N (independent) constants of the motion. Most systems in the real world have between 1 and N constants of the motion (e.g., the en- ergy, often the angular momentum, and sometimes some non-obvious quantities, which may be related to hidden symmetries). We also proved that when two operators commute, they share common eigenstates. We should be a bit careful here: if there are degeneracies, not every eigenstate of either operator will be an eigenstate of the other.

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