Introduction Constructing scar functions Unveiling homoclinic motions Homoclinic quantum numbers Homoclinic and Heteroclinic Motions in Quantum Dynamics F . Borondo Dep. de Química; Universidad Autónoma de Madrid, Instituto Mixto de Ciencias Matemáticas CSIC-UAM-UC3M-UCM Stability and Instability in Mechanical Systems: Applications and Numerical Tools Barcelona, 1 December 2008 F. Borondo Homo and Heteroclinic Motions in QM 1/ 67
Introduction Constructing scar functions Unveiling homoclinic motions Homoclinic quantum numbers Outline Introduction 1 Models Tools Periodic orbits in quantum mechanics: Scars Constructing scar functions 2 Unveiling homoclinic motions 3 Homoclinic quantum numbers 4 F. Borondo Homo and Heteroclinic Motions in QM 2/ 67
Introduction Models Constructing scar functions Tools Unveiling homoclinic motions Periodic orbits in quantum mechanics: Scars Homoclinic quantum numbers Outline Introduction 1 Models Tools Periodic orbits in quantum mechanics: Scars Constructing scar functions 2 Unveiling homoclinic motions 3 Homoclinic quantum numbers 4 F. Borondo Homo and Heteroclinic Motions in QM 3/ 67
Introduction Models Constructing scar functions Tools Unveiling homoclinic motions Periodic orbits in quantum mechanics: Scars Homoclinic quantum numbers In his pioneering work on chaos Poincaré showed the importance of Periodic orbits Homoclinic solutions Heteroclinic solutions F. Borondo Homo and Heteroclinic Motions in QM 4/ 67
Introduction Models Constructing scar functions Tools Unveiling homoclinic motions Periodic orbits in quantum mechanics: Scars Homoclinic quantum numbers In this talk, we will discuss the importance of: Periodic orbits Homoclinic motions Heteroclinic motions in Quantum Mechanics F. Borondo Homo and Heteroclinic Motions in QM 5/ 67
Introduction Models Constructing scar functions Tools Unveiling homoclinic motions Periodic orbits in quantum mechanics: Scars Homoclinic quantum numbers Outline Introduction 1 Models Tools Periodic orbits in quantum mechanics: Scars Constructing scar functions 2 Unveiling homoclinic motions 3 Homoclinic quantum numbers 4 F. Borondo Homo and Heteroclinic Motions in QM 6/ 67
Introduction Models Constructing scar functions Tools Unveiling homoclinic motions Periodic orbits in quantum mechanics: Scars Homoclinic quantum numbers Model: Quartic oscillator 2 x 2 y 2 + ε 4 ( x 4 + y 4 ) , H = 1 2 ( P 2 x + P 2 y ) + 1 ε = 0 . 01 Smooth, homogeneous potential Mechanical similarity � 1 / 4 � 1 / 2 � 3 / 4 � − 1 / 4 � � � � q E , P E , S E , T E q 0 = P 0 = S 0 = T 0 = E 0 E 0 E 0 E 0 Free from hassles due to phase space evolution (bif’s) SOS: y = 0 , P y > 0 Very chaotic dynamics Thought hyperbolic for ε → 0 Dahlqvist and Russberg (1990) found POs for ε = 0 Also Waterland el at. for ε = 1 / 240 F. Borondo Homo and Heteroclinic Motions in QM 7/ 67
Introduction Models Constructing scar functions Tools Unveiling homoclinic motions Periodic orbits in quantum mechanics: Scars Homoclinic quantum numbers Model: Billiards Bunimovitch stadium billiard Hyperbolic dynamics F. Borondo Homo and Heteroclinic Motions in QM 8/ 67
Introduction Models Constructing scar functions Tools Unveiling homoclinic motions Periodic orbits in quantum mechanics: Scars Homoclinic quantum numbers Billiards: Models in Nanotechnology Eigler F. Borondo Homo and Heteroclinic Motions in QM 9/ 67
Introduction Models Constructing scar functions Tools Unveiling homoclinic motions Periodic orbits in quantum mechanics: Scars Homoclinic quantum numbers Billiards: Models for Microcavity Lasers A. Douglas Stone, 1997 F. Borondo Homo and Heteroclinic Motions in QM 10/ 67
Introduction Models Constructing scar functions Tools Unveiling homoclinic motions Periodic orbits in quantum mechanics: Scars Homoclinic quantum numbers Microdisk laser, Douglas Stone, PNAS, 2004 Top and side view of a GaAs microdisk ( ∼ 5.2 µ m diameter) on top of an Al 0 . 7 Ga 0 . 3 pedestal. A thin InAs quantum well layer in the middle layer serves as active medium. Image obtained with a scanning electron microscope. F. Borondo Homo and Heteroclinic Motions in QM 11/ 67
Introduction Models Constructing scar functions Tools Unveiling homoclinic motions Periodic orbits in quantum mechanics: Scars Homoclinic quantum numbers Directional Laser emission Directional laser emission has direct applications in optical communications and optoelectronics F. Borondo Homo and Heteroclinic Motions in QM 12/ 67
Introduction Models Constructing scar functions Tools Unveiling homoclinic motions Periodic orbits in quantum mechanics: Scars Homoclinic quantum numbers More on microlasers ... r ( φ ) = a ( 1 + η 0 cos 2 φ + ǫη 0 cos 4 φ ) F. Borondo Homo and Heteroclinic Motions in QM 13/ 67
Introduction Models Constructing scar functions Tools Unveiling homoclinic motions Periodic orbits in quantum mechanics: Scars Homoclinic quantum numbers More on microlasers ... η = 0 . 09 η = 0 . 10 η = 0 . 12 η = 0 . 16 Exp. Th. A Th. B F. Borondo Homo and Heteroclinic Motions in QM 14/ 67
Introduction Models Constructing scar functions Tools Unveiling homoclinic motions Periodic orbits in quantum mechanics: Scars Homoclinic quantum numbers More on microlasers ... F. Borondo Homo and Heteroclinic Motions in QM 15/ 67
Introduction Models Constructing scar functions Tools Unveiling homoclinic motions Periodic orbits in quantum mechanics: Scars Homoclinic quantum numbers Outline Introduction 1 Models Tools Periodic orbits in quantum mechanics: Scars Constructing scar functions 2 Unveiling homoclinic motions 3 Homoclinic quantum numbers 4 F. Borondo Homo and Heteroclinic Motions in QM 16/ 67
Introduction Models Constructing scar functions Tools Unveiling homoclinic motions Periodic orbits in quantum mechanics: Scars Homoclinic quantum numbers Quantum Mechanics P = 2 π � De Broglie Hypothesis: λ = h P Wave function: ψ ( q , t ) , q =positions, t =time Stationary Schrödinger equation: with ˆ H φ n ( q ) = E n φ n ( q ) Heisenberg Uncertainty Principle: ∆ q ∆ p ≥ � / 2 and ∆ E τ ≥ � / 2 F. Borondo Homo and Heteroclinic Motions in QM 17/ 67
Introduction Models Constructing scar functions Tools Unveiling homoclinic motions Periodic orbits in quantum mechanics: Scars Homoclinic quantum numbers Example Helmholtz equation: ∇ 2 φ n = k 2 n φ n φ n ( boundary ) = 0 F. Borondo Homo and Heteroclinic Motions in QM 18/ 67
Introduction Models Constructing scar functions Tools Unveiling homoclinic motions Periodic orbits in quantum mechanics: Scars Homoclinic quantum numbers Simpler example (even trivial) ψ I = ψ III = 0 d 2 ψ II − � 2 dx 2 + V ψ II = E ψ II 2 m √ d 2 ψ II 2 mE dx 2 + k 2 ψ II , k = � But, don’t forget the dynamics: k = P � F. Borondo Homo and Heteroclinic Motions in QM 19/ 67
Introduction Models Constructing scar functions Tools Unveiling homoclinic motions Periodic orbits in quantum mechanics: Scars Homoclinic quantum numbers Solution ψ ( x ) = a sin kx + b cos kx First boundary condition: ψ ( 0 ) = 0 − → c = 0 ψ = b sin kx Normalization condition: � L � 0 | ψ | 2 dx = 1 − 2 → a = L Second boundary condition: → k n = n π ψ ( L ) = 0 − L Solutions: ψ n ( x ) = � 2 L sin n π x L , n = 1 , 2 , . . . F. Borondo Homo and Heteroclinic Motions in QM 20/ 67
Introduction Models Constructing scar functions Tools Unveiling homoclinic motions Periodic orbits in quantum mechanics: Scars Homoclinic quantum numbers But, don’t forget the dynamics . . . k = P � Classical action: � L � L 0 k � dx = 2 k � L = 2 n π � Pdx = 2 0 Pdx = 2 L � L = nh Action is quantized in QM! F. Borondo Homo and Heteroclinic Motions in QM 21/ 67
Introduction Models Constructing scar functions Tools Unveiling homoclinic motions Periodic orbits in quantum mechanics: Scars Homoclinic quantum numbers Quantization of the action. How? Einstein–Brillouin–Kramers (EBK) Method n j + α j � N � � � i P i dq i = h C j 4 Classical info = Quantum condition Associated WKB (Wentzel–Kramers–Brillouin) wave function j A j e iS j ( q ) / � ψ ( q ) = � F. Borondo Homo and Heteroclinic Motions in QM 22/ 67
Introduction Models Constructing scar functions Tools Unveiling homoclinic motions Periodic orbits in quantum mechanics: Scars Homoclinic quantum numbers Phase space representations of QM Wigner transform (1932) "On the quantum corrections to statistical thermodynamics" ds e isP ψ ∗ � � q − s � � q + s � W ( q , P ) = ψ 2 2 F. Borondo Homo and Heteroclinic Motions in QM 23/ 67
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