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Resonance and Revivals in Quantum Rotors --- Comparing Half-integer Spin and Integer Spin

Alvason Zhenhua LiWilliam Harter

Microelectronics-Photonics Program, Physics Department University of Arkansas, Fayetteville, AR 72701

68th Spectroscopy, June 20, 2013

68th Spectroscopy, June 20, 2013

Outline

SummarySummary

MotivationMotivation

Integer Spin(Symmetric Rotor)

Half-integer Spin(Symmetric Rotor)

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68th Spectroscopy, June 20, 2013

Quantum RotorMotivationMotivation

33/20W. G. Harter, Principles of Symmetry, Dynamics, and Spectroscopy.W iley-VCH,1ed.,1993.

68th Spectroscopy, June 20, 2013

Wigner-D Matrix ---- a Rotational Matrix for Any SpinList-plotList-plot

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W. G. Harter, Quantum Lecture Manual

A. Z. Li, Quantum Resonant Beats and Revivals in the Morse Oscillators and Rotors, Ph.D. thesis, University of Arkansas (2013).

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68th Spectroscopy, June 20, 2013

Wigner-D Matrix ---- a Rotational Matrix for Any SpinDensity-plotDensity-plot

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A. Z. Li, Quantum Resonant Beats and Revivals in the Morse Oscillators and Rotors, Ph.D. thesis, University of Arkansas (2013).

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68th Spectroscopy, June 20, 2013

Wigner-D Matrix ---- a Rotational Matrix for Any SpinDensity-plot MovieDensity-plot Movie

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68th Spectroscopy, June 20, 2013

Rotor Wave Functions of Integer Spin (Boson) System

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J=3J=3

68th Spectroscopy, June 20, 2013

Dynamics of Integer Spin (Boson) System

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J=10J=10

68th Spectroscopy, June 20, 2013

Rotor Wave Packet Propagation of Integer Spin (Boson) System Space-Time-plotSpace-Time-plot

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A. Z. Li, Quantum Resonant Beats and Revivals in the Morse Oscillators and Rotors, Ph.D. thesis, University of Arkansas (2013).

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68th Spectroscopy, June 20, 201310/20

68th Spectroscopy, June 20, 201311/20

Half-integer Spinning Rotors exhibit Farey-sum Revivals

Farey−sum−Rule:aA⊕bB=a+bA+B

0

1⊕

12=0 +11+2

=13

0

1⊕13=0 +11+ 3

=14

1

3⊕

12=1+13+2

=25

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25=1+23+5

=38

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14=0 +11+ 4

=15

68th Spectroscopy, June 20, 2013

Outline

SummarySummary

MotivationMotivation

Integer Spin(Symmetric Rotor)

Half-integer Spin(Symmetric Rotor)

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(a)

(d)

(c)(b)

(e) (f)

Rotor Wave Functions of Half-Integer Spin (Fermion) SystemUnique 4π RotationUnique 4π Rotation

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68th Spectroscopy, June 20, 2013

Rotor Wave Functions of Half-Integer Spin (Fermion) System

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J=5/2J=5/2

68th Spectroscopy, June 20, 2013

Dynamics of Integer Spin (Fermion) System

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J=9.5J=9.5

68th Spectroscopy, June 20, 2013

Rotor Wave Packet Propagation of Half-integer Spin (Fermion) System Space-Time-plotSpace-Time-plot

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A. Z. Li, Quantum Resonant Beats and Revivals in the Morse Oscillators and Rotors, Ph.D. thesis, University of Arkansas (2013).

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68th Spectroscopy, June 20, 201318/20

Half-integer Spinning Rotors exhibit Farey-sum Revivals

Farey−sum−Rule:aA⊕bB=a+bA+B

0

1⊕

12=0 +11+2

=13

0

1⊕13=0 +11+ 3

=14

1

3⊕

12=1+13+2

=25

68th Spectroscopy, June 20, 201319/20

The coming next talk will address the curious connection of Farey-sum and Ford-circles

A. Z. Li and W. G. Harter, Physical Review Letters (submitted 2013).

Summary

68th Spectroscopy, June 20, 2013

Pave the way for more rich dynamic behaviors in Asymmetric Rotors

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Both Integer and Half-integer Spinning Rotors exhibit Farey-sum Revivals

Thanks for your time!