Lect22_CohStates - Lecture 22 Coherent States Phy851 Fall...

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Lecture 22: Coherent States Phy851 Fall 2009
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Summary Properties of the QM SHO: 1 = n n n A 1 1 + + = n n n A n = A ( ) n n ! 0 ψ 0 ( x ) = π λ [ ] 1/ 2 e x 2 2 λ 2 ψ 1 ( x ) = 2 π λ [ ] 1/ 2 2 x λ e x 2 2 λ 2 ψ n ( x ) = π 2 n n ! λ [ ] 1/ 2 H n x / λ ( ) e x 2 2 λ 2 ψ n ( x ) = 2 n x λ ψ n 1 ( x ) n 1 n ψ n 2 ( x ) n n n H ) 2 / 1 ( + = ω h 2 2 2 2 1 2 X m m P H ω + = + = P i X A h λ λ 2 1 ( ) 2 A A X + = λ ( ) 2 A A i P = λ h A = 1 2 X λ i λ h P + = 2 1 A A H ω h 2 / 1 + = Δ n X λ 2 / 1 + = Δ n P λ h ω λ m h =
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What are the `most classical states of the SHO? In HW6.4, we saw that for a minimum uncertainty wavepacket with: The uncertainties in position and momentum would remain constant. The interesting thing was that this was true independent of x 0 and p 0, the initial expectation values of X and P . We know that other than the case x 0 =0 and p 0 =0 , the mean position and momentum oscillate like a classical particle This means that for just the right initial width, the wave-packet moves around like a classical particle, but DOESN’T SPREAD at all. 2 osc x λ = Δ osc osc M ω λ h =
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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 α can be any complex number i.e. there is a different coherent state for every possible choice of α (Roy Glauber, Nobel Prize for Quantum Optics Theory 2005) These states are not really any more ‘coherent’ then other pure states, they do maintain their coherence in the presence of dissipation somewhat more efficiently In QM the term ‘coherence’ is over-used and
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