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Lect19_SHO1 - Systems near equilibrium The harmonic...

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Lecture 19: The Quantum Simple- Harmonic Oscillator PHY851 Quantum Mechanics I Fall, 2008 M.G. Moore Systems near equilibrium The harmonic oscillator Hamiltonian is: Or alternatively, using Why is the SHO so important? Answer: any system near a stable equilibrium is equivalent to an SHO 2 2 2 2 1 2 X m m P H ! + = 2 2 2 1 2 kX m P H + = m k = ! A Random Potential Stable equilibrium points Definition of stable equilibrium point: 0 ) ( 0 = ! x V Expand around x 0 : 2 0 2 0 0 0 0 0 2 1 ) )( ( 2 1 ) )( ( ) ( ) ( y k V x x x V x x x V x V x V + = + ! " " + ! " + = K 0 x x y ! =
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Dimensionless Variables To solve the QM SHO it is very useful to introduce dimensionless units: Let Also let Make the choice: To find: X X ! " Constant with units of length New variable with no units H m H 2 2 ! h " 2 2 2 2 2 2 2 2 2 1 2 1 X m P m H m ! " ! ! + = h h P P ! h " 2 2 2 1 2 1 X P H + = ! " m h = Dimensionless Commutation Relations Lest compute the commutator for the new variables: X , P [ ] = X P " P X ! ! X X X X = " = P P P P h h ! ! = " = X , P [ ] = X " " P h # " P h X " ( ) PX XP ! = h 1 [ ] P X , 1 h = X , P [ ] = i
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Some New Variables Now for a change of variables: It’s more common to use: a, a
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