{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

# Lect19_SHO1 - Systems near equilibrium The harmonic...

This preview shows pages 1–4. Sign up to view the full content.

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 ! =

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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
Some New Variables Now for a change of variables: It’s more common to use: a, a

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}