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Unformatted text preview: 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 systems Hamiltonian. Thus, if an operator (which is not explicitly timedependent) 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. Noethers 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 Hamiltons equations of motion: A symmetry manifests itself as the absence of dependence of the systems 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. HamiltonJacobi 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 nonobvious 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. What we proved is that if [ , ]...
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This note was uploaded on 02/08/2012 for the course CHEM 540 taught by Professor Mccall during the Fall '08 term at University of Illinois, Urbana Champaign.
 Fall '08
 Mccall

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