Course - Wavefunctions in chaotic quantum systems

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VI Autocorrelation fct and rate of qerg – Summary Origin of fluctuations around J 0 (r): deviations from quantum ergodicity at finite energies Thus: Autocorrelation function allows to study the rate of quantum ergodicity! Remarks on σ 2 (E, r): – Efficient quantity to measure the dependence of the rate of quantum ergodicity on different length scales. – For larger r ≡ |δx|, one needs to incorporate higher terms in l which corresponds to expectation values of faster oscillating observables. Arnd Bäcker ⇓ ⇐ ⇒ Σ ⊕ 86
VII More recent results – Poincaré Husimi representation Arnd Bäcker ⇓ ⇐ ⇒ Σ ⊕ 87 Question: Poincaré representation of eigenstates? AB, S. Fürstberger, R. Schubert: Poincaré Husimi representation of eigenstates in quantum billiards (2003) Natural starting point: normal derivative of the eigenfunction u n (s) := 〈ˆn(s), ∇ψ n (x(s))〉 , (94) Coherent states on the billiard boundary ∂Ω ( k ) 1/4 c(q,p),k b (s) := ∑ e ik[p(s−q+mL)+ i 2σ (s−q+mL)2] , (95) πσ where (q, p) ∈ ∂Ω × R. m∈Z Husimi function: h n (q, p) = ∣ ∣〈c(q,p),k b , u n〉 ∣ 2 (96)
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Wavefunctions in chaotic quantum sy
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II Classical billiards Arnd Bäcker
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II Classical billiards - Mathematic
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II Classical billiards - Mathematic
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II Classical billiards - ergodicity
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II Classical billiards - mixing Def
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II Classical billiards — periodic
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III Quantum billiards — some eige
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III Quantum billiards - integrable
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III Quantum billiards — spectral
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III Quantum billiards — mean beha
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IV SoE - Statistics of eigenfunctio
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IV SoE - Random wave model - chaoti
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IV SoE - Random wave model - conseq
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IV SoE - Maximum norms - (some) kno
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Arnd Bäcker ⇓ ⇐ ⇒ Σ ⊕ 31
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V Quantum ergodicity Arnd Bäcker
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Page 37 and 38: V QE — observables and operators
Page 39 and 40: V Quantum ergodicity theorem Arnd B
Page 41 and 42: V QET - Example - observable in pos
Page 43 and 44: V Quantum ergodicity theorem —
Page 45 and 46: V Quantum ergodicity theorem —
Page 47 and 48: V QET — quantum limits Arnd Bäck
Page 49 and 50: V QET - quantum limits (sketch of i
Page 51 and 52: V QET - eigenfunctions stadium Arnd
Page 53 and 54: V QET - BBMs - Counting function Qu
Page 55 and 56: V BBMs - Counting function Counting
Page 57 and 58: V BBMs - Counting function Arnd Bä
Page 59 and 60: V BBMs - Counting function Arnd Bä
Page 61 and 62: V QET - quasimodes (Arnold ’72) A
Page 63 and 64: V QET — Scars Arnd Bäcker ⇓
Page 65 and 66: V QET — Scars Arnd Bäcker ⇓
Page 67 and 68: V QET — Scars Some results on thi
Page 69 and 70: V QET and random wave model Amplitu
Page 71 and 72: V Sketch: proof of the QET Theorem
Page 73 and 74: V Sketch: proof of the QET Arnd Bä
Page 75 and 76: V QET - Summary Arnd Bäcker ⇓
Page 77 and 78: VI Autocorrelation function and rat
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Page 89 and 90: VII Poincaré Husimi functions - ex
Page 91 and 92: VII Poincaré Husimi representation
Page 93 and 94: VII Poincaré Husimi representation
Page 95 and 96: VII Relation between Husimi functio
Page 97 and 98: VII Poincaré Husimi functions —
Page 99 and 100: VIII Time evolution Arnd Bäcker
Page 101 and 102: VIII Time evolution Arnd Bäcker
Page 103 and 104: VIII Time evolution Plot of (x(t),
Page 105 and 106: Arnd Bäcker ⇓ ⇐ ⇒ Σ ⊕ 10
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Course - Wavefunctions in chaotic quantum systems

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