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
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.
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.
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.
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 Fall '08
 Mccall
 Angular Momentum, Fundamental physics concepts, Noether's theorem, Nancy Makri

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