Lect31_SymmetryII

# Lect31_SymmetryII - Review A symmetry is a transformation...

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Lecture 31: Symmetry II PHY851 Quantum Mechanics I Fall, 2008 M.G. Moore Review A symmetry is a transformation which leaves the Hamiltonian unchanged Transformations are described by Unitary operators: States can transform as: Operators can transform as: We can generate continuous-symmetry operators from observables via: Here ‘ G ’ is the observable and ! is the degree of transformation Example: position shift operator A system is invariant under the transformation when: In which case the generator , G is a constant of motion U = " UOU O = ! U " ( ) = e # iG T d ( ) = e " i h Pd H , G [ ] = 0 T d ( ) x = x + d p ( g , t ) = p ( g ,0)

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Active Transformations Question: when we want to make a symmetry transformation, should we transform both the state and all observables? A symmetry transformation is an active transformation, in that the physical properties of the transformed system are intended to be changed by the transformation Example: suppose we have a wavepacket that is centered at . x = x 0 . Suppose we want to move the position of our origin to x = d . Let x _ be the position measured in the new coordinate system. It is related to x via: With respect to our new coordinate system, the wavepacket of our state should now be centered at x _ = x 0 - d d x x ! = " Classical versus Quantum symmetry transformations Notice that in classical mechanics, the state and of the observables are the same thing. So there is no question as to which should be transformed. In QM state and observables are separate
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Lect31_SymmetryII - Review A symmetry is a transformation...

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