Lect22_CoherentStates

# Lect22_CoherentStates - Summary P2 1 m 2 X 2 2m 2 memorize...

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Lecture 22: Coherent States PHY851 Quantum Mechanics I Fall, 2008 M.G. Moore Summary What we have learned about the QM SHO: 1 ! = n n n A 1 1 + + = n n n A n = A ( ) n n ! 0 " 0 ( x ) = # \$ [ ] % 1/ 2 e % x 2 2 \$ 2 " 1 ( x ) = 2 # \$ [ ] % 1/ 2 2 x \$ e % x 2 2 \$ 2 " n ( x ) = # 2 n n ! \$ [ ] % 1/ 2 H n x / \$ ( ) e % x 2 2 \$ 2 " n ( x ) = 2 n x # " n \$ 1 ( x ) \$ n \$ 1 n " n \$ 2 ( x ) n n n H ) 2 / 1 ( + = ! h 2 2 2 2 1 2 X m m P H ! + = ! " # \$ % & + = P i X A h 2 1 ( ) 2 A A X + = ! ( ) 2 A A i P ! ! = " h A = 1 2 X " # i " h P \$ % & ( ) ! " # \$ % & + = 2 1 A A H h 2 / 1 + = ! n X " 2 / 1 + = ! n P " h ! " m h = memorize

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‘Coherent States’ Coherent states, or as they are sometimes called ‘Glauber Coherent States’ are the eigenstates of the annihilation operator Here ! can be any complex number i.e. there is a different coherent state for every possible choice of ! (Roy Glauber, Nobel Prize for Quantum Optics Theory 2005) These states are not really any more ‘coherent’ then other pure states, they just maintain their coherence in the presence of dissipation In QM the term ‘coherence’ is over-used and often abused, so do not think that it always has a precise meaning Glauber Coherent States are very important: They are the ‘most classical’ states of the harmonic oscillator They describe the quantum state of a laser Replace the number of ‘quanta’ with the number of
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