Homoclinic and Heteroclinic Motions in Quantum Dynamics F . - - PowerPoint PPT Presentation

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homoclinic and heteroclinic motions in quantum dynamics
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Homoclinic and Heteroclinic Motions in Quantum Dynamics F . - - PowerPoint PPT Presentation

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Introduction Constructing scar functions Unveiling homoclinic motions Homoclinic quantum numbers Homoclinic and Heteroclinic Motions in Quantum Dynamics F . Borondo Dep. de Qumica; Universidad Autnoma de Madrid, Instituto Mixto de

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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

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Introduction Constructing scar functions Unveiling homoclinic motions Homoclinic quantum numbers

Outline

1

Introduction Models Tools Periodic orbits in quantum mechanics: Scars

2

Constructing scar functions

3

Unveiling homoclinic motions

4

Homoclinic quantum numbers

  • F. Borondo

Homo and Heteroclinic Motions in QM 2/ 67

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Introduction Constructing scar functions Unveiling homoclinic motions Homoclinic quantum numbers Models Tools Periodic orbits in quantum mechanics: Scars

Outline

1

Introduction Models Tools Periodic orbits in quantum mechanics: Scars

2

Constructing scar functions

3

Unveiling homoclinic motions

4

Homoclinic quantum numbers

  • F. Borondo

Homo and Heteroclinic Motions in QM 3/ 67

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Introduction Constructing scar functions Unveiling homoclinic motions Homoclinic quantum numbers Models Tools Periodic orbits in quantum mechanics: Scars

In his pioneering work on chaos Poincaré showed the importance of Periodic

  • rbits

Homoclinic solutions Heteroclinic solutions

  • F. Borondo

Homo and Heteroclinic Motions in QM 4/ 67

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Introduction Constructing scar functions Unveiling homoclinic motions Homoclinic quantum numbers Models Tools Periodic orbits in quantum mechanics: Scars

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

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Introduction Constructing scar functions Unveiling homoclinic motions Homoclinic quantum numbers Models Tools Periodic orbits in quantum mechanics: Scars

Outline

1

Introduction Models Tools Periodic orbits in quantum mechanics: Scars

2

Constructing scar functions

3

Unveiling homoclinic motions

4

Homoclinic quantum numbers

  • F. Borondo

Homo and Heteroclinic Motions in QM 6/ 67

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Introduction Constructing scar functions Unveiling homoclinic motions Homoclinic quantum numbers Models Tools Periodic orbits in quantum mechanics: Scars

Model: Quartic oscillator H = 1

2(P2 x + P2 y) + 1 2x2y2 + ε 4(x4 + y4),

ε = 0.01 Smooth, homogeneous potential Mechanical similarity

q q0 =

  • E

E0

1/4 , P

P0 =

  • E

E0

1/2 , S

S0 =

  • E

E0

3/4 , T

T0 =

  • E

E0

−1/4 Free from hassles due to phase space evolution (bif’s) 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 SOS: y = 0, Py > 0

  • F. Borondo

Homo and Heteroclinic Motions in QM 7/ 67

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Introduction Constructing scar functions Unveiling homoclinic motions Homoclinic quantum numbers Models Tools Periodic orbits in quantum mechanics: Scars

Model: Billiards Bunimovitch stadium billiard Hyperbolic dynamics

  • F. Borondo

Homo and Heteroclinic Motions in QM 8/ 67

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Introduction Constructing scar functions Unveiling homoclinic motions Homoclinic quantum numbers Models Tools Periodic orbits in quantum mechanics: Scars

Billiards: Models in Nanotechnology Eigler

  • F. Borondo

Homo and Heteroclinic Motions in QM 9/ 67

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Introduction Constructing scar functions Unveiling homoclinic motions Homoclinic quantum numbers Models Tools Periodic orbits in quantum mechanics: Scars

Billiards: Models for Microcavity Lasers

  • A. Douglas Stone, 1997
  • F. Borondo

Homo and Heteroclinic Motions in QM 10/ 67

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Introduction Constructing scar functions Unveiling homoclinic motions Homoclinic quantum numbers Models Tools Periodic orbits in quantum mechanics: Scars

Microdisk laser, Douglas Stone, PNAS, 2004 Top and side view of a GaAs microdisk (∼5.2µm diameter) on top of an Al0.7Ga0.3 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

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Introduction Constructing scar functions Unveiling homoclinic motions Homoclinic quantum numbers Models Tools Periodic orbits in quantum mechanics: Scars

Directional Laser emission Directional laser emission has direct applications in optical communications and optoelectronics

  • F. Borondo

Homo and Heteroclinic Motions in QM 12/ 67

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Introduction Constructing scar functions Unveiling homoclinic motions Homoclinic quantum numbers Models Tools Periodic orbits in quantum mechanics: Scars

More on microlasers ... r(φ) = a(1 + η0 cos 2φ + ǫη0 cos 4φ)

  • F. Borondo

Homo and Heteroclinic Motions in QM 13/ 67

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Introduction Constructing scar functions Unveiling homoclinic motions Homoclinic quantum numbers Models Tools Periodic orbits in quantum mechanics: Scars

More on microlasers ... Exp.

  • Th. A
  • Th. B

η = 0.09 η = 0.10 η = 0.12 η = 0.16

  • F. Borondo

Homo and Heteroclinic Motions in QM 14/ 67

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Introduction Constructing scar functions Unveiling homoclinic motions Homoclinic quantum numbers Models Tools Periodic orbits in quantum mechanics: Scars

More on microlasers ...

  • F. Borondo

Homo and Heteroclinic Motions in QM 15/ 67

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Introduction Constructing scar functions Unveiling homoclinic motions Homoclinic quantum numbers Models Tools Periodic orbits in quantum mechanics: Scars

Outline

1

Introduction Models Tools Periodic orbits in quantum mechanics: Scars

2

Constructing scar functions

3

Unveiling homoclinic motions

4

Homoclinic quantum numbers

  • F. Borondo

Homo and Heteroclinic Motions in QM 16/ 67

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Introduction Constructing scar functions Unveiling homoclinic motions Homoclinic quantum numbers Models Tools Periodic orbits in quantum mechanics: Scars

Quantum Mechanics De Broglie Hypothesis: λ = h

P = 2π P

Wave function: ψ(q, t), q=positions, t=time Stationary Schrödinger equation: with ˆ Hφn(q) = Enφn(q) Heisenberg Uncertainty Principle: ∆q∆p ≥ /2 and ∆E τ ≥ /2

  • F. Borondo

Homo and Heteroclinic Motions in QM 17/ 67

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Introduction Constructing scar functions Unveiling homoclinic motions Homoclinic quantum numbers Models Tools Periodic orbits in quantum mechanics: Scars

Example Helmholtz equation: ∇2φn = k2

nφn

φn(boundary) = 0

  • F. Borondo

Homo and Heteroclinic Motions in QM 18/ 67

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Introduction Constructing scar functions Unveiling homoclinic motions Homoclinic quantum numbers Models Tools Periodic orbits in quantum mechanics: Scars

Simpler example (even trivial) ψI = ψIII = 0 − 2

2m d2ψII dx2 + VψII = EψII d2ψII dx2 + k2ψII,

k =

√ 2mE

  • But, don’t forget the dynamics:

k = P

  • F. Borondo

Homo and Heteroclinic Motions in QM 19/ 67

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Introduction Constructing scar functions Unveiling homoclinic motions Homoclinic quantum numbers Models Tools Periodic orbits in quantum mechanics: Scars

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 −

→ a =

  • 2

L

Second boundary condition: ψ(L) = 0 − → kn = nπ

L

Solutions: ψn(x) =

  • 2

L sin nπx L ,

n = 1, 2, . . .

  • F. Borondo

Homo and Heteroclinic Motions in QM 20/ 67

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Introduction Constructing scar functions Unveiling homoclinic motions Homoclinic quantum numbers Models Tools Periodic orbits in quantum mechanics: Scars

But, don’t forget the dynamics . . . k = P

  • Classical action:
  • Pdx = 2

L

0 Pdx = 2

L

0 kdx = 2kL = 2nπ L L = nh

Action is quantized in QM!

  • F. Borondo

Homo and Heteroclinic Motions in QM 21/ 67

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Introduction Constructing scar functions Unveiling homoclinic motions Homoclinic quantum numbers Models Tools Periodic orbits in quantum mechanics: Scars

Quantization of the action. How? Einstein–Brillouin–Kramers (EBK) Method

  • Cj

N

i Pi dqi = h

  • nj + αj

4

  • Classical info = Quantum condition

Associated WKB (Wentzel–Kramers–Brillouin) wave function ψ(q) =

j Aj eiSj(q)/

  • F. Borondo

Homo and Heteroclinic Motions in QM 22/ 67

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Introduction Constructing scar functions Unveiling homoclinic motions Homoclinic quantum numbers Models Tools Periodic orbits in quantum mechanics: Scars

Phase space representations of QM Wigner transform (1932)

"On the quantum corrections to statistical thermodynamics" W(q, P) =

  • ds eisP ψ∗

q − s

2

  • ψ
  • q + s

2

  • F. Borondo

Homo and Heteroclinic Motions in QM 23/ 67

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Introduction Constructing scar functions Unveiling homoclinic motions Homoclinic quantum numbers Models Tools Periodic orbits in quantum mechanics: Scars

But ... W(q, P) can be negative Why?: Heisenberg’s uncertainty principle Solution: Husimi function

Gaussian average in cells of area N H(q, P) =

N dq′ dP′Gq,P(q′, P′) W(q′, P′)

Coherent state representation H(q, P) =

1 (2π)N |φq,P|ψ|2

φ minimum uncertainty coherent state φ(x, y, Px, Py) = 2α

π

1/4 e−α(x−x0)2 e−α(y−y0)2 eiP0

xx eiP0 yy

  • F. Borondo

Homo and Heteroclinic Motions in QM 24/ 67

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Introduction Constructing scar functions Unveiling homoclinic motions Homoclinic quantum numbers Models Tools Periodic orbits in quantum mechanics: Scars

But ... W(q, P) can be negative Why?: Heisenberg’s uncertainty principle Solution: Husimi function

Gaussian average in cells of area N H(q, P) =

N dq′ dP′Gq,P(q′, P′) W(q′, P′)

Coherent state representation H(q, P) =

1 (2π)N |φq,P|ψ|2

φ minimum uncertainty coherent state φ(x, y, Px, Py) = 2α

π

1/4 e−α(x−x0)2 e−α(y−y0)2 eiP0

xx eiP0 yy

  • F. Borondo

Homo and Heteroclinic Motions in QM 24/ 67

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Introduction Constructing scar functions Unveiling homoclinic motions Homoclinic quantum numbers Models Tools Periodic orbits in quantum mechanics: Scars

But ... W(q, P) can be negative Why?: Heisenberg’s uncertainty principle Solution: Husimi function

Gaussian average in cells of area N H(q, P) =

N dq′ dP′Gq,P(q′, P′) W(q′, P′)

Coherent state representation H(q, P) =

1 (2π)N |φq,P|ψ|2

φ minimum uncertainty coherent state φ(x, y, Px, Py) = 2α

π

1/4 e−α(x−x0)2 e−α(y−y0)2 eiP0

xx eiP0 yy

  • F. Borondo

Homo and Heteroclinic Motions in QM 24/ 67

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Introduction Constructing scar functions Unveiling homoclinic motions Homoclinic quantum numbers Models Tools Periodic orbits in quantum mechanics: Scars

Outline

1

Introduction Models Tools Periodic orbits in quantum mechanics: Scars

2

Constructing scar functions

3

Unveiling homoclinic motions

4

Homoclinic quantum numbers

  • F. Borondo

Homo and Heteroclinic Motions in QM 25/ 67

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Introduction Constructing scar functions Unveiling homoclinic motions Homoclinic quantum numbers Models Tools Periodic orbits in quantum mechanics: Scars

Periodic orbits in quantum mechanics: Scars What are scars? Expected: Chaotic classical dynamics − → uniformly distributed quantum density

  • F. Borondo

Homo and Heteroclinic Motions in QM 26/ 67

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Introduction Constructing scar functions Unveiling homoclinic motions Homoclinic quantum numbers Models Tools Periodic orbits in quantum mechanics: Scars

Scarred functions But in numerical calculations (McDonald & Kaufman) ... Heller in 1984 coined the term scar to name an enhanced localization of quantum probability density of certain eigenstates on classical unstable periodic orbits

  • F. Borondo

Homo and Heteroclinic Motions in QM 27/ 67

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Introduction Constructing scar functions Unveiling homoclinic motions Homoclinic quantum numbers Models Tools Periodic orbits in quantum mechanics: Scars

Scars in Optical Fibers

  • F. Borondo

Homo and Heteroclinic Motions in QM 28/ 67

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Introduction Constructing scar functions Unveiling homoclinic motions Homoclinic quantum numbers Models Tools Periodic orbits in quantum mechanics: Scars

  • F. Borondo

Homo and Heteroclinic Motions in QM 29/ 67

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Introduction Constructing scar functions Unveiling homoclinic motions Homoclinic quantum numbers Models Tools Periodic orbits in quantum mechanics: Scars

Scars in Microcavity lasers

  • F. Borondo

Homo and Heteroclinic Motions in QM 30/ 67

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Introduction Constructing scar functions Unveiling homoclinic motions Homoclinic quantum numbers Models Tools Periodic orbits in quantum mechanics: Scars

Scars in Microcavity lasers

  • F. Borondo

Homo and Heteroclinic Motions in QM 31/ 67

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Introduction Constructing scar functions Unveiling homoclinic motions Homoclinic quantum numbers Models Tools Periodic orbits in quantum mechanics: Scars

Heller’s dynamical explanation for scars Recurrences Fourier transform between: correlation function C(t) = φ(0)|φ(t), and corresponding spectrum I(E) =

  • dt eiEt/ C(t)
  • F. Borondo

Homo and Heteroclinic Motions in QM 32/ 67

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Introduction Constructing scar functions Unveiling homoclinic motions Homoclinic quantum numbers Models Tools Periodic orbits in quantum mechanics: Scars

Peaks Where? Bohr–Sommerfeld quantization condition on the action: S =

  • P · dq = 2π
  • n + α

4

  • Why?

Constructive interference in the WKB wavefunction ψ(q) = A eiS(q)/

  • F. Borondo

Homo and Heteroclinic Motions in QM 33/ 67

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Introduction Constructing scar functions Unveiling homoclinic motions Homoclinic quantum numbers Models Tools Periodic orbits in quantum mechanics: Scars

BUT ... What happens to the density that does not come back in the recurrence along the scarring periodic orbit?

  • F. Borondo

Homo and Heteroclinic Motions in QM 34/ 67

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Introduction Constructing scar functions Unveiling homoclinic motions Homoclinic quantum numbers

Outline

1

Introduction Models Tools Periodic orbits in quantum mechanics: Scars

2

Constructing scar functions

3

Unveiling homoclinic motions

4

Homoclinic quantum numbers

  • F. Borondo

Homo and Heteroclinic Motions in QM 35/ 67

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Introduction Constructing scar functions Unveiling homoclinic motions Homoclinic quantum numbers

How to systematically construct scar function Borondo et al., PRL 73, 1613 (1994) version 2007 Wavepacket initially localized on the PO ψtube(x, y) = N T

0 dt e−αx(x−xt)2−αy(y−yt)2

× cos

  • St − µπt

2T + Pxt(x − xt) + Pyt(y − yt)

  • F. Borondo

Homo and Heteroclinic Motions in QM 36/ 67

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Introduction Constructing scar functions Unveiling homoclinic motions Homoclinic quantum numbers

In phase space Quantum SOS based on Husimi function: H(x, Px) =

  • ∞

−∞ dx′ e−(x−x′)2/(2α2

H)−iPxx′ψ(x′, y′ = 0)

  • 2
  • F. Borondo

Homo and Heteroclinic Motions in QM 37/ 67

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Introduction Constructing scar functions Unveiling homoclinic motions Homoclinic quantum numbers

This can be improved Propagate ψtube(x, y) in time and Fourier transform at EBS ψscar(x, y) = N TE

−TE dt cos

  • πt

2TE

  • e−i(ˆ

H−EBS)t ψtube(x, y)

  • F. Borondo

Homo and Heteroclinic Motions in QM 38/ 67

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Introduction Constructing scar functions Unveiling homoclinic motions Homoclinic quantum numbers

Same in phase space ψtube(x, y) ψscar(x, y)

  • F. Borondo

Homo and Heteroclinic Motions in QM 39/ 67

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Introduction Constructing scar functions Unveiling homoclinic motions Homoclinic quantum numbers

Outline

1

Introduction Models Tools Periodic orbits in quantum mechanics: Scars

2

Constructing scar functions

3

Unveiling homoclinic motions

4

Homoclinic quantum numbers

  • F. Borondo

Homo and Heteroclinic Motions in QM 40/ 67

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Introduction Constructing scar functions Unveiling homoclinic motions Homoclinic quantum numbers

  • F. Borondo

Homo and Heteroclinic Motions in QM 41/ 67

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Introduction Constructing scar functions Unveiling homoclinic motions Homoclinic quantum numbers

  • F. Borondo

Homo and Heteroclinic Motions in QM 42/ 67

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Introduction Constructing scar functions Unveiling homoclinic motions Homoclinic quantum numbers

Peaks at ω = 1.67, 2.76, 2.95

  • F. Borondo

Homo and Heteroclinic Motions in QM 43/ 67

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Introduction Constructing scar functions Unveiling homoclinic motions Homoclinic quantum numbers

Where do these frequencies come from? Additional quantization condition for the circuits in phase space: Si − π

2 νi = 2πn

When this condition is fulfilled the recurrence along the circuit reinforce the recurrence along the scarring periodic orbit!!

  • F. Borondo

Homo and Heteroclinic Motions in QM 44/ 67

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Introduction Constructing scar functions Unveiling homoclinic motions Homoclinic quantum numbers

  • F. Borondo

Homo and Heteroclinic Motions in QM 45/ 67

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Introduction Constructing scar functions Unveiling homoclinic motions Homoclinic quantum numbers

Circuit Type Si ω′

i

ωi 1 Homoclinic 1.464 3.519 2.764 2 Heteroclinic 1.212 2.913 2.913 3 Heteroclinic 4.014 3.365 2.918 4 Heteroclinic 3.299 1.646 1.646 5 Heteroclinic 1.927 4.632 1.651 That agree reasonably well with the previously found values ω = 1.67, 2.76, 2.95

  • F. Borondo

Homo and Heteroclinic Motions in QM 46/ 67

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Introduction Constructing scar functions Unveiling homoclinic motions Homoclinic quantum numbers

  • F. Borondo

Homo and Heteroclinic Motions in QM 47/ 67

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Introduction Constructing scar functions Unveiling homoclinic motions Homoclinic quantum numbers

  • F. Borondo

Homo and Heteroclinic Motions in QM 48/ 67

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Introduction Constructing scar functions Unveiling homoclinic motions Homoclinic quantum numbers

  • F. Borondo

Homo and Heteroclinic Motions in QM 49/ 67

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Introduction Constructing scar functions Unveiling homoclinic motions Homoclinic quantum numbers

Homoclinic motion and wave functions

  • Phys. Rev. Lett. 97, 094101 (2006)

|φscar = T

−Tdt cos

πt

2T

  • ei(EBS−ˆ

H)t/ |φtube

  • F. Borondo

Homo and Heteroclinic Motions in QM 50/ 67

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Introduction Constructing scar functions Unveiling homoclinic motions Homoclinic quantum numbers

Peaks coincide with the value of primary homoclinic areas

  • F. Borondo

Homo and Heteroclinic Motions in QM 51/ 67

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Introduction Constructing scar functions Unveiling homoclinic motions Homoclinic quantum numbers

Husimis for 4 scar function with quantization/antiquantization conditions on the homoclinic torus (all quantized on the PO)

  • F. Borondo

Homo and Heteroclinic Motions in QM 52/ 67

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Introduction Constructing scar functions Unveiling homoclinic motions Homoclinic quantum numbers

Scar function n = 224 Homoclinic quantization: nh1 = 189.01, nh2 = 168.07 Extra quantization on heteroclinic orbits: kShe = 2πnhe nhe1 = 19.00, nhe2 = 5.98 Husimis for T = 0.9tE, 1.2tE and 3.3tE

  • F. Borondo

Homo and Heteroclinic Motions in QM 53/ 67

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Introduction Constructing scar functions Unveiling homoclinic motions Homoclinic quantum numbers

Classical phase space

  • F. Borondo

Homo and Heteroclinic Motions in QM 54/ 67

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Introduction Constructing scar functions Unveiling homoclinic motions Homoclinic quantum numbers

Relative relevance of the different circuits Why some circuits are more important than others? We propose a quantity to measure this:

1

Let us consider pieces of homo or heteroclinic trajectories, as those shown before,

2

They have initial, (xi, Pxi), and ending points, (xf , Pxf ), close to the fixed point, (xF, PxF)

3

Substract

4

Apply symplectic transformation in order to write down these differences in terms of new coordinates, (u, s), living

  • n the unstable and stable directions

5

Define A ≡ uisjeλT, T being the time necessary for the trajectory to go from i → j

6

Remark: A is symplectic and canonical invariant

  • F. Borondo

Homo and Heteroclinic Motions in QM 55/ 67

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Circuit Type Si ω′

i

ωi Ai 1 Homoclinic 1.464 3.519 2.764 2.79 2 Heteroclinic 1.212 2.913 2.913 2.64 3 Heteroclinic 4.014 3.365 2.918 2.64 4 Heteroclinic 3.299 1.646 1.646 1.34 5 Heteroclinic 1.927 4.632 1.651 1.34 The agreement between Ai and the importance we found is astonishing !!!! In the sense that

1

A takes the same value for equivalent circuits

2

More important circuits have smaller values of A

  • F. Borondo

Homo and Heteroclinic Motions in QM 56/ 67

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Introduction Constructing scar functions Unveiling homoclinic motions Homoclinic quantum numbers

One question Is Ai related to the Lazutkin’s invariant? I don’t know, but ... ask Ernest Fontich

  • F. Borondo

Homo and Heteroclinic Motions in QM 57/ 67

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Introduction Constructing scar functions Unveiling homoclinic motions Homoclinic quantum numbers

Outline

1

Introduction Models Tools Periodic orbits in quantum mechanics: Scars

2

Constructing scar functions

3

Unveiling homoclinic motions

4

Homoclinic quantum numbers

  • F. Borondo

Homo and Heteroclinic Motions in QM 58/ 67

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Introduction Constructing scar functions Unveiling homoclinic motions Homoclinic quantum numbers

Scar functions Back to the quartic potential Scar functions along the vertical PO

  • F. Borondo

Homo and Heteroclinic Motions in QM 59/ 67

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Introduction Constructing scar functions Unveiling homoclinic motions Homoclinic quantum numbers

Quantum numbers How to compute/assign quantum numbers? Zeros in the Husimi function Leboeuf and Voros Zeros inside the homoclinic circuit

  • F. Borondo

Homo and Heteroclinic Motions in QM 60/ 67

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First zero in . . .

  • F. Borondo

Homo and Heteroclinic Motions in QM 61/ 67

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Second zero in . . .

  • F. Borondo

Homo and Heteroclinic Motions in QM 62/ 67

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Second zero in . . .

  • F. Borondo

Homo and Heteroclinic Motions in QM 63/ 67

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Counting the zeros in the Husimi function

  • F. Borondo

Homo and Heteroclinic Motions in QM 64/ 67

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Introduction Constructing scar functions Unveiling homoclinic motions Homoclinic quantum numbers

Quantizing . . . Let us make the argument quantitative . . . −Ai 2π(n) + µi 4 = mi, i = 1, 2      −A1 + A2 4π(n) + µ1 + µ2 8 = m A2 − A1 2π(n) − µ2 − µ1 4 = ∆m,

  • F. Borondo

Homo and Heteroclinic Motions in QM 65/ 67

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Quantizing . . .

Table: Homoclinic quantum numbers m

m n ∆m n 43 141 1 163 1 713 2 282 3 402 4 521 5 641 6 760

  • F. Borondo

Homo and Heteroclinic Motions in QM 66/ 67

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Thanks for your attention

  • F. Borondo

Homo and Heteroclinic Motions in QM 67/ 67