Introduction Warming up Stable case General propagation of c.s. Unstable case Questions of symbols Conclusion Long time semiclassical evolution T. PAUL C.N.R.S. and D.M.A., ´ Ecole Normale Sup´ erieure, Paris pour Sandro, 27 augusto 2008
Introduction Warming up Stable case General propagation of c.s. Unstable case Questions of symbols Conclusion Quantum evolution � i � ∂ t ψ t H ψ t = ψ t =0 = ψ
Introduction Warming up Stable case General propagation of c.s. Unstable case Questions of symbols Conclusion Quantum evolution � i � ∂ t ψ t ˙ H ψ t � i � [ O t , H ] 1 = O t = OR ψ t =0 O t =0 = ψ = O
Introduction Warming up Stable case General propagation of c.s. Unstable case Questions of symbols Conclusion Quantum evolution � i � ∂ t ψ t ˙ H ψ t � i � [ O t , H ] 1 = O t = OR ψ t =0 O t =0 = ψ = O t ≤ T � → + ∞ as � → 0
Introduction Warming up Stable case General propagation of c.s. Unstable case Questions of symbols Conclusion Quantum evolution � i � ∂ t ψ t ˙ H ψ t � i � [ O t , H ] 1 = O t = OR ψ t =0 O t =0 = ψ = O t ≤ T � → + ∞ as � → 0 T � ∼ log ( � − 1 ) : unstable case T � ∼ 1 � : stable case.
Introduction Warming up Stable case General propagation of c.s. Unstable case Questions of symbols Conclusion Quantum evolution � i � ∂ t ψ t ˙ H ψ t � i � [ O t , H ] 1 = O t = OR ψ t =0 O t =0 = ψ = O t ≤ T � → + ∞ as � → 0 T � ∼ log ( � − 1 ) : unstable case T � ∼ 1 � : stable case. Do we recover Classical Mechanics ?
Introduction Warming up Stable case General propagation of c.s. Unstable case Questions of symbols Conclusion Quantum evolution � i � ∂ t ψ t ˙ H ψ t � i � [ O t , H ] 1 = O t = OR ψ t =0 O t =0 = ψ = O t ≤ T � → + ∞ as � → 0 T � ∼ log ( � − 1 ) : unstable case T � ∼ 1 � : stable case. Do we recover Classical Mechanics ? Answer : not always.
Introduction Warming up Stable case General propagation of c.s. Unstable case Questions of symbols Conclusion Quantum evolution � i � ∂ t ψ t ˙ H ψ t � i � [ O t , H ] 1 = O t = OR ψ t =0 O t =0 = ψ = O t ≤ T � → + ∞ as � → 0 T � ∼ log ( � − 1 ) : unstable case T � ∼ 1 � : stable case. Do we recover Classical Mechanics ? Answer : not always. New phenomena : delocalization, reconstruction, ubiquity ...... contained in the (classical) infinite time.
Introduction Warming up Stable case General propagation of c.s. Unstable case Questions of symbols Conclusion Bambusi-Graffi-P 1998, Bouzuoina-Robert 2002 for Egorov Haguedorn, Combescure-Robert, de Bi` evre-Robert .....1995-2002 for coherent states a lot of papers in physics, including experimental
Introduction Warming up Stable case General propagation of c.s. Unstable case Questions of symbols Conclusion Why long time ? Quantum Mechanics : stability, stationnary states, eigenvectors Schr¨ odinger (linear) equation i � ∂ t ψ = H ψ Very different form Classical Mechanics : � ˙ x = ∂ ξ h ( x , ξ ) ˙ ξ = − ∂ x h ( x , ξ )
Introduction Warming up Stable case General propagation of c.s. Unstable case Questions of symbols Conclusion Why long time ? Quantum Mechanics : stability, stationnary states, eigenvectors Schr¨ odinger (linear) equation i � ∂ t ψ = H ψ Very different form Classical Mechanics : � ˙ x = ∂ ξ h ( x , ξ ) ˙ ξ = − ∂ x h ( x , ξ ) How to “construct” eigenvectors ? Link with models in atomic physics (cold atoms) How do we understand the transition Quantum/Classical ?
Introduction Warming up Stable case General propagation of c.s. Unstable case Questions of symbols Conclusion Why semiclassical approximation ? Asymptotic method (very efficient) Semiclassical limit ⊂ Quantum Mechanics ex. atomic systems (scalings) systems of spins ( N spins- 1 2 (symmetrized) ∼ 1 spin-2 N ) Corresponds to experimental situations
Introduction Warming up Stable case General propagation of c.s. Unstable case Questions of symbols Conclusion Why coherent states ? Natural way of taking semiclassical limit More precise than, e.g., Egorov theorem Generalize to more geometrical situations (ex. spins)
Introduction Warming up Stable case General propagation of c.s. Unstable case Questions of symbols Conclusion Why coherent states ? Natural way of taking semiclassical limit More precise than, e.g., Egorov theorem Generalize to more geometrical situations (ex. spins) Coherent state at ( q , p ) and symbol-vacuum a : a ( x ) = � − n / 4 a ( x − q ) e i px ψ qp √ � �
Introduction Warming up Stable case General propagation of c.s. Unstable case Questions of symbols Conclusion Main ideas • Coherent state follows the classical flow, and a follows the linearized flow, up to a certain time T 0 ( � )
Introduction Warming up Stable case General propagation of c.s. Unstable case Questions of symbols Conclusion Main ideas • Coherent state follows the classical flow, and a follows the linearized flow, up to a certain time T 0 ( � ) • After, quantum effects are persistent, and classical paradigm is lost
Introduction Warming up Stable case General propagation of c.s. Unstable case Questions of symbols Conclusion Main ideas • Coherent state follows the classical flow, and a follows the linearized flow, up to a certain time T 0 ( � ) • After, quantum effects are persistent, and classical paradigm is lost • New “dynamics” enter the game, that we can sometimes compute
Introduction Warming up Stable case General propagation of c.s. Unstable case Questions of symbols Conclusion Main ideas • Coherent state follows the classical flow, and a follows the linearized flow, up to a certain time T 0 ( � ) • After, quantum effects are persistent, and classical paradigm is lost • New “dynamics” enter the game, that we can sometimes compute • The wave packet can reconstruct, but with (always) a singular vacuum
Introduction Warming up Stable case General propagation of c.s. Unstable case Questions of symbols Conclusion Main ideas • Coherent state follows the classical flow, and a follows the linearized flow, up to a certain time T 0 ( � ) • After, quantum effects are persistent, and classical paradigm is lost • New “dynamics” enter the game, that we can sometimes compute • The wave packet can reconstruct, but with (always) a singular vacuum • Overlapping between quantum undeterminism and classical unpredictability
Introduction Warming up Stable case General propagation of c.s. Unstable case Questions of symbols Conclusion Outline Introduction 1 Warming up 2 Stable case 3 General propagation of c.s. 4 Unstable case 5 Questions of symbols 6 Conclusion 7
Introduction Warming up Stable case General propagation of c.s. Unstable case Questions of symbols Conclusion Free evolution on the circle i � ∂ t ψ = − � 2 ψ ∈ L 2 ( S 1 ) 2 ∆ ψ σ ( − � 2 � � 2 m 2 � , phases : e it � m 2 2 ∆) = , m ∈ Z 2 2
Introduction Warming up Stable case General propagation of c.s. Unstable case Questions of symbols Conclusion Free evolution on the circle i � ∂ t ψ = − � 2 ψ ∈ L 2 ( S 1 ) 2 ∆ ψ σ ( − � 2 � � 2 m 2 � , phases : e it � m 2 2 ∆) = , m ∈ Z ⇒ 2 2 Quantum Flow is 4 π � -periodic.
Introduction Warming up Stable case General propagation of c.s. Unstable case Questions of symbols Conclusion Free evolution on the circle i � ∂ t ψ = − � 2 ψ ∈ L 2 ( S 1 ) 2 ∆ ψ σ ( − � 2 � � 2 m 2 � , phases : e it � m 2 2 ∆) = , m ∈ Z ⇒ 2 2 Quantum Flow is 4 π � -periodic. Classical flow is NOT.
Introduction Warming up Stable case General propagation of c.s. Unstable case Questions of symbols Conclusion Free evolution on the circle i � ∂ t ψ = − � 2 ψ ∈ L 2 ( S 1 ) 2 ∆ ψ σ ( − � 2 � � 2 m 2 � , phases : e it � m 2 2 ∆) = , m ∈ Z ⇒ 2 2 Quantum Flow is 4 π � -periodic. Classical flow is NOT. (except with quantized momenta ( m � ) but
Introduction Warming up Stable case General propagation of c.s. Unstable case Questions of symbols Conclusion Free evolution on the circle i � ∂ t ψ = − � 2 ψ ∈ L 2 ( S 1 ) 2 ∆ ψ σ ( − � 2 � � 2 m 2 � , phases : e it � m 2 2 ∆) = , m ∈ Z ⇒ 2 2 Quantum Flow is 4 π � -periodic. Classical flow is NOT. (except with quantized momenta ( m � ) but quantum period = 2 × classical one (like harm. osc.))
Introduction Warming up Stable case General propagation of c.s. Unstable case Questions of symbols Conclusion Free evolution on the circle i � ∂ t ψ = − � 2 ψ ∈ L 2 ( S 1 ) 2 ∆ ψ σ ( − � 2 � � 2 m 2 � , phases : e it � m 2 2 ∆) = , m ∈ Z ⇒ 2 2 Quantum Flow is 4 π � -periodic. Classical flow is NOT. (except with quantized momenta ( m � ) but quantum period = 2 × classical one (like harm. osc.)) odinger cats : consider fractional times : t = p 4 π Schr¨ � ⇒ q
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