Lect30_SymmetryI - Symmetries and Consequences in QM...

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Lecture 30: Symmetry I PHY851 Quantum Mechanics I Fall, 2008 M.G. Moore Symmetries and Consequences in QM Symmetry: A transformation which leaves the Hamiltonian unchanged: Example: Let: Let the transformation T be defined via: If H is symmetric under T then: Why wouldn’t this be true in general? P P X X ! " ! " H H ! = " H = H ( " X , " P ) ) , ( P X H H =
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Common symmetries encountered in 1D problems T – translational invariance: Symmetry under shift of the origin Example of continuous symmetry ! – Parity invariance Symmetry under reflection about the origin Example of discrete symmetry Consider a classical particle: The kinetic energy term is symmetric under T and ! : Let: Let: So symmetry of H depends on symmetry of V ( x ) ) ( 2 1 2 x V x m H + = & x x & & = ! " d x x ! = " x x ! = " x x & & ! = " # ( ) 2 2 x x & & = ! " Symmetries of V (x) Translation invariance: We must have (for any d ): The only solution is: Consequently, we have: Which means that momentum is a constant of motion. Another possible symmetry is time-translation invariance: For all h : So Energy is a constant of motion iff V ( x,t ) invariant under time translation ) ( ) ( x V d x V = ! 0 ) ( V x V = 0 = ! ! " = x H p & V ( x , t + h ) = " " t V ( x , t ) # V = 0 ) , ( 2 ) ( 2 t x V m p t E + = V x V p m p E & & & & + ! + = ( ) 0 = = + ! + ! " = V V m p V V m p E & & &
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Symmetries lead to constants of motion Translational Invariance ! Conservation of Linear Momentum Time Translational Invariance !
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