Quantum Chaos. Shagesh Sridharan. Department of Physics Rutgers University. Quantum Mechanics II, 2018
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1 Quantum Chaos Shagesh Sridharan Department of Physics Rutgers University Quantum Mechanics II, 2018 Shagesh Sridharan (Rutgers University) Quantum Chaos Quantum Mechanics II, / 26
2 Outline 1 Classical Chaos Definition 2 Quantized Chaos Motivation Types 3 Kicked rotor Classical rotor Quantum rotor Poincaré surfaces Husimi function plots 4 Spectral statistics Measures Application Shagesh Sridharan (Rutgers University) Quantum Chaos Quantum Mechanics II, / 26
3 Outline 1 Classical Chaos Definition 2 Quantized Chaos Motivation Types 3 Kicked rotor Classical rotor Quantum rotor Poincaré surfaces Husimi function plots 4 Spectral statistics Measures Application Shagesh Sridharan (Rutgers University) Quantum Chaos Quantum Mechanics II, / 26
4 Classical Chaos Definition Boundedness Shagesh Sridharan (Rutgers University) Quantum Chaos Quantum Mechanics II, / 26
5 Classical Chaos Definition Boundedness Infinite Recurrence Shagesh Sridharan (Rutgers University) Quantum Chaos Quantum Mechanics II, / 26
6 Classical Chaos Definition Boundedness Infinite Recurrence Exponential Sensitivity Shagesh Sridharan (Rutgers University) Quantum Chaos Quantum Mechanics II, / 26
7 Outline 1 Classical Chaos Definition 2 Quantized Chaos Motivation Types 3 Kicked rotor Classical rotor Quantum rotor Poincaré surfaces Husimi function plots 4 Spectral statistics Measures Application Shagesh Sridharan (Rutgers University) Quantum Chaos Quantum Mechanics II, / 26
8 Bunimovich stadium billiard Figure: Exponential sensitivity to initial conditions [2] Shagesh Sridharan (Rutgers University) Quantum Chaos Quantum Mechanics II, / 26
9 Outline 1 Classical Chaos Definition 2 Quantized Chaos Motivation Types 3 Kicked rotor Classical rotor Quantum rotor Poincaré surfaces Husimi function plots 4 Spectral statistics Measures Application Shagesh Sridharan (Rutgers University) Quantum Chaos Quantum Mechanics II, / 26
10 Quantum Chaos Motivation Old Quantum Theory J := pdr = 2nπ Shagesh Sridharan (Rutgers University) Quantum Chaos Quantum Mechanics II, / 26
11 Quantum Chaos Motivation Old Quantum Theory J := pdr = 2nπ Correspondence Principle 0 Shagesh Sridharan (Rutgers University) Quantum Chaos Quantum Mechanics II, / 26
12 Outline 1 Classical Chaos Definition 2 Quantized Chaos Motivation Types 3 Kicked rotor Classical rotor Quantum rotor Poincaré surfaces Husimi function plots 4 Spectral statistics Measures Application Shagesh Sridharan (Rutgers University) Quantum Chaos Quantum Mechanics II, / 26
13 Quantum Chaos Types Quantized Chaos Shagesh Sridharan (Rutgers University) Quantum Chaos Quantum Mechanics II, / 26
14 Quantum Chaos Types Quantized Chaos Semiquantum Chaos Shagesh Sridharan (Rutgers University) Quantum Chaos Quantum Mechanics II, / 26
15 Quantum Chaos Types Quantized Chaos Semiquantum Chaos True Quantum Chaos Shagesh Sridharan (Rutgers University) Quantum Chaos Quantum Mechanics II, / 26
16 Outline 1 Classical Chaos Definition 2 Quantized Chaos Motivation Types 3 Kicked rotor Classical rotor Quantum rotor Poincaré surfaces Husimi function plots 4 Spectral statistics Measures Application Shagesh Sridharan (Rutgers University) Quantum Chaos Quantum Mechanics II, / 26
17 Kicked rotor Classical rotor Hamiltonian H(x, p) = p2 2 + V (x) n= δ(t n) Shagesh Sridharan (Rutgers University) Quantum Chaos Quantum Mechanics II, / 26
18 Kicked rotor Classical rotor Hamiltonian H(x, p) = p2 2 + V (x) n= δ(t n) Mapping Equations p n+1 = p n V (x n ) x n+1 = x n + p n+1 Shagesh Sridharan (Rutgers University) Quantum Chaos Quantum Mechanics II, / 26
19 Kicked rotor Classical rotor Hamiltonian H(x, p) = p2 2 + V (x) n= δ(t n) Mapping Equations p n+1 = p n V (x n ) x n+1 = x n + p n+1 Potential V (x) = K 4π 2 cos(2πx) Shagesh Sridharan (Rutgers University) Quantum Chaos Quantum Mechanics II, / 26
20 Outline 1 Classical Chaos Definition 2 Quantized Chaos Motivation Types 3 Kicked rotor Classical rotor Quantum rotor Poincaré surfaces Husimi function plots 4 Spectral statistics Measures Application Shagesh Sridharan (Rutgers University) Quantum Chaos Quantum Mechanics II, / 26
21 Kicked rotor Quantum rotor System evolution φ(x; t = (n + 1)) = Ûφ(x; t = n) Shagesh Sridharan (Rutgers University) Quantum Chaos Quantum Mechanics II, / 26
22 Kicked rotor Quantum rotor System evolution φ(x; t = (n + 1)) = Ûφ(x; t = n) Propagator m Û m = 1 exp(iπ(m m ) 2 /M)exp(i KM im 2π cos(2π m + 1/2 M ) Shagesh Sridharan (Rutgers University) Quantum Chaos Quantum Mechanics II, / 26
23 Outline 1 Classical Chaos Definition 2 Quantized Chaos Motivation Types 3 Kicked rotor Classical rotor Quantum rotor Poincaré surfaces Husimi function plots 4 Spectral statistics Measures Application Shagesh Sridharan (Rutgers University) Quantum Chaos Quantum Mechanics II, / 26
24 Poincaré surfaces Figure: The left section shows integrable dynamics, the middle shows outset of chaos, while the right is almost completely chaotic [2] Shagesh Sridharan (Rutgers University) Quantum Chaos Quantum Mechanics II, / 26
25 Outline 1 Classical Chaos Definition 2 Quantized Chaos Motivation Types 3 Kicked rotor Classical rotor Quantum rotor Poincaré surfaces Husimi function plots 4 Spectral statistics Measures Application Shagesh Sridharan (Rutgers University) Quantum Chaos Quantum Mechanics II, / 26
26 Husimi function Figure: Plots of Husimi functions for two different eigenstates for varying coupling constants [2] Shagesh Sridharan (Rutgers University) Quantum Chaos Quantum Mechanics II, / 26
27 Outline 1 Classical Chaos Definition 2 Quantized Chaos Motivation Types 3 Kicked rotor Classical rotor Quantum rotor Poincaré surfaces Husimi function plots 4 Spectral statistics Measures Application Shagesh Sridharan (Rutgers University) Quantum Chaos Quantum Mechanics II, / 26
28 Spectral Statistics Measures Mean density of states Shagesh Sridharan (Rutgers University) Quantum Chaos Quantum Mechanics II, / 26
29 Spectral Statistics Measures Mean density of states Variance of density of states Shagesh Sridharan (Rutgers University) Quantum Chaos Quantum Mechanics II, / 26
30 Spectral Statistics Measures Mean density of states Variance of density of states Nearest neighbour density Shagesh Sridharan (Rutgers University) Quantum Chaos Quantum Mechanics II, / 26
31 Outline 1 Classical Chaos Definition 2 Quantized Chaos Motivation Types 3 Kicked rotor Classical rotor Quantum rotor Poincaré surfaces Husimi function plots 4 Spectral statistics Measures Application Shagesh Sridharan (Rutgers University) Quantum Chaos Quantum Mechanics II, / 26
32 Spectral Statistics Figure: Nearest neighbour density for the Bunimovich stadium billard (chaotic) and for the semicircular billiard (integrable) [2] Shagesh Sridharan (Rutgers University) Quantum Chaos Quantum Mechanics II, / 26
33 Outline 1 Classical Chaos Definition 2 Quantized Chaos Motivation Types 3 Kicked rotor Classical rotor Quantum rotor Poincaré surfaces Husimi function plots 4 Spectral statistics Measures Application Shagesh Sridharan (Rutgers University) Quantum Chaos Quantum Mechanics II, / 26
34 Spectral Statistics Application Figure: Nearest neighbour distribution for Hydrogen atom in weak (top) and strong (bottom) magnetic field [2] Shagesh Sridharan (Rutgers University) Quantum Chaos Quantum Mechanics II, / 26
35 Summary Chaos, in the rigorous sense, hasn t been found yet in the quantum regime. Wavefunctions and spectral statistics point to residues of chaotic behaviour. Shagesh Sridharan (Rutgers University) Quantum Chaos Quantum Mechanics II, / 26
36 References An Introduction to Quantum Chaos Mason A. Porter arxiv:nlin/ v2, Introduction to Quantum Chaos Denis Ullmo and Steven Tomsovic Fundamentals of Physics, Encyclopedia of Life Support Systems (EOLSS), Shagesh Sridharan (Rutgers University) Quantum Chaos Quantum Mechanics II, / 26
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