rj04 resonance and revivals i. quantum rotor and infinite-well dynamics william g. harter and...

RJ04

RESONANCE AND REVIVALS I. QUANTUM ROTOR AND INFINITE-WELL DYNAMICS

William G. Harter and Alvason Zhenhua LiUniversity of Arkansas - Fayetteville

Physics Department and Microelectronics-Photonics Program

m=0±1±2

±3

±4

±5

±6

n=1

2

3

4

5

6

RJ04




m=0±1±2

±3

±4

±5

±6

n=1

2

3

4

5

6

Rotor revival structure includes anything ∞-well can do…. …and is easier to explain.

Won’t talk about ∞-well

RJ04




m=0±1±2

±3

±4

±5

±6

n=1

2

3

4

5

6

Rotor revival structure includes anything ∞-well can do…. …and is easier to explain.

Some Early History of Quantum RevivalsJ.H. Eberly et. al. Phys. Rev. A 23,236 (1981) Laser QuantumCavityDynamic revivalsR.S. McDowell, WGH, C.W. Patterson LosAlmos Sci. 3, 38(1982) Symmetric-top revivalsS.I. Vetchinkin, et. al. Chem. Phys. Lett. 215,11 (1993) 1D ∞-Square well revivalsAronstein, Stroud, Berry, …, Schleich,.. (1995-1998) “ “ “ “WGH, J. Mol. Spectrosc. 210, 166 (2001) Bohr-rotor revivals

So we thought we’d put this revival business to bed! Then...

Won’t talk about ∞-well

More recent story of Quantum RevivalsAnne B. McCoy Chem. Phys. Lett. 501, 603(2011)...reminds me that Morse potential is integer-analytic.

Some Early History of Quantum RevivalsJ.H. Eberly et. al. Phys. Rev. A 23,236 (1981) Laser QuantumCavityDynamic revivalsR.S. McDowell, WGH, C.W. Patterson LosAlmos Sci. 3, 38(1982) Symmetric-top revivalsS.I. Vetchinkin, et. al. Chem. Phys. Lett. 215,11 (1993) 1D ∞-Square well revivalsAronstein, Stroud, Berry, …, Schleich,.. (1995-1998) “ “ “ “WGH, J. Mol. Spectrosc. 210, 166 (2001) Bohr-rotor revivals

So we thought we’d put this revival business to bed! Then this...

Leads to cool Morse revivals in: Following Talk RJ05 by Li:Resonance&Revivals II. MORSE OSCILLATOR AND DOUBLE MORSE WELL DYNAMICS.

So now we’re having a revival-revival!

...and, in words by Joannie Mitchell, I find: “I didn’t really know... revivals … at all.”

What do revivals look like?(...in space-time…)

What do revivals look like?(...in space-time…)

OK,let’s try that again...

withquantum revivals...

Time t (units of fundamental period ) τ1

Coordinate φ (units of 2π )

0/1

1/10

1/5

3/10

2/5

1/2

3/5

7/10

4/5

9/10

1/1

3/4

1/4

1/21/40-1/4-1/2

1/2

1/4

0-1/4

-1/2

(Imagine "wrap-around" φ-coordinate) +π /2

−π /2

+π−π

3/4

1/4

0/1

1/2

1/1

Time t

Observable dynamics of N-level-system state Depends on Fourier spectrum of probability distribution

Ψ = e−iωnt

n=0

N

∑ ψ n

ω 0 ω1 ω 2 ω 3 ω 4

...But individual eigenfrequencies are not directly observable...

ω n

ΨΨ Ψ

Ψ = e−iωnt

n=0

N

∑ ψ n

Ψ = e+iωmt

m=0

N

∑ ψ *m

ω 0 ω1 ω 2 ω 3 ω 4

ω 4

ω 3

ω 2

ω1

ω 0



ΨΨ Ψ

Ψ = e−iωnt

n=0

N

∑ ψ n

Ψ = e+iωmt

m=0

N

∑ ψ *m

Ψ Ψ = ei(ωm −ωn )t

n=0

N

∑ ψ *mψ n

= eiΔm nt

m,n=0

N

∑ ρ mnω 0 ω1 ω 2 ω 3 ω 4

ω 4

ω 3

ω 2

ω1

ω 0



ΨΨ Ψ

ω3

Ψ = e−iωnt

n=0

N

∑ ψ n

Ψ = e+iωmt

m=0

N

∑ ψ *m

ω0 ω1 ω2 ω3 ωN

ω0

ω1

ω2

ωN


n=0

N

∑ ψ *mψ n

= eiΔm nt

m,n=0

N

∑ ρ mn



ΨΨ Ψ

Ψ = e−iωnt

n=0

N

∑ ψ n

Ψ = e+iωmt

m=0

N

∑ ψ *m


n=0

N

∑ ψ *mψ n

= eiΔm nt

m,n=0

N

∑ ρ mnω 0 ω1 ω 2 ω 3 ω 4

ω 4

ω 3

ω 2

ω1

ω 0

...But individual eigenfrequencies are not directly observable… ...only differences Δmn = (ωm −ωn )

Δ04

Δ14

Δ03

Δ23

Δ13

Δ4 0 Δ41


ΨΨ Ψ

Ψ = e−iωnt

n=0

N

∑ ψ n

Ψ = e+iωmt

m=0

N

∑ ψ *m


n=0

N

∑ ψ *mψ n

= eiΔm nt

m,n=0

N

∑ ρ mnω 0 ω1 ω 2 ω 3 ω 4

ω 4

ω 3

ω 2

ω1

ω 0


Δ04

Δ14

Δ03

Δ23

Δ13

Δ4 0 Δ41

N=5 eigenfrequenciesgives: N(N-1)/2=10 positive and : N(N-1)/2=10negative

Δm>n = ωm −ωn

Δm<n = ωm −ωn


ΨΨ Ψ

Ψ = e−iωnt

n=0

N

∑ ψ n

Ψ = e+iωmt

m=0

N

∑ ψ *m


n=0

N

∑ ψ *mψ n

= eiΔm nt

m,n=0

N

∑ ρ mnω 0 ω1 ω 2 ω 3 ω 4

ω 4

ω 3

ω 2

ω1

ω 0


Δ04

Δ14

Δ03

Δ23

Δ13

Δ4 0 Δ41


Δm>n = ωm −ωn

Δm<n = ωm −ωn

...that is, N(N-1)/2=10 observable beats Δm<n = ωm −ωn


ΨΨ Ψ

Observable dynamics of 2-level-system state Fourier spectrum of has ONE beat frequency

Ψ = e−iωnt

n=0

N

∑ ψ n

Ψ = e+iωmt

m=0

N

∑ ψ *m


n=0

N

∑ ψ *mψ n

= eiΔm nt

m,n=0

N

∑ ρ mnω1 ω 2

ω 2

ω1

Δ12

Δ21


Δ2>1 = ω2 −ω1

Δ1<2 = ω1 −ω2

...that is, N(N-1)/2=1 observable beat Δm<n = ωm −ωn

ΨΨ Ψ Δ21 = −Δ12

2-level-system and C2 symmetry beat dynamics

symmetric A1

vs.

antisymmetric A2

C2 Character Table describes eigenstates

1=r0 r =r1

0mod21 1

±1mod2 1 −1


symmetric A1

vs.

antisymmetric A2


1=r0 r =r1

0mod21 1

±1mod2 1 −1

Phasor C2 Characters describe local state beats

Initial sum

C2 Phasor-Character Table


symmetric A1

vs.

antisymmetric A2


1=r0 r =r1

0mod21 1

±1mod2 1 −1


Initial sum

1/4-beat



symmetric A1

vs.

antisymmetric A2


1=r0 r =r1

0mod21 1

±1mod2 1 −1


Initial sum

1/4-beat

1/2-beat



symmetric A1

vs.

antisymmetric A2


1=r0 r =r1

0mod21 1

±1mod2 1 −1


Initial sum

1/4-beat

1/2-beat

3/4-beat



m=+1, 0, -1


What do revivals look like?...in per-space-time…

(… that is:frequency radian/sec.

vs

k-vector km radian/cm)

ωm

N-level-system and revival-beat wave dynamics Bm2=B42

Bm2=B32

Bm2=B22

Bm2=B

N-level-system and revival-beat wave dynamics

0123456789

1011121314151617

Harmonic Oscillator level spectrum contains the Rotor Levels as a subset

N-level-system and revival-beat wave dynamics

N-level-system and revival-beat wave dynamics (Just 2-levels (0, ±1) (and some ±2) excited)

N-level-system and revival-beat wave dynamics (Just 2-levels (0, ±1) (and some ±2) excited) (4-levels (0, ±1,±2,±3) (and some ±4) excited)

N-level-system and revival-beat wave dynamics (9 or10-levels (0, ±1, ±2, ±3, ±4,..., ±9, ±10, ±11...) excited)

Zeros (clearly) and “particle-packets” (faintly) have paths labeled by fraction sequences like: 0

7,1

7,2

7,3

7,4

7,5

7,6

7,1

1

1

7

2

7

3

7

4

7

5

7

6

7

4

7

5

7

6

7

1

1

0

1

Farey Sum algebra of revival-beat wave dynamicsLabel by numerators N and denominators D of rational fractions N/D

A Lesson in Rational Fractions N/D (...that you can take home for your kids!)

Farey Sum related to vector sum

andFord Circles1/1-circle has

diameter 1


andFord Circles

1/2-circle hasdiameter 1/22=1/4

1/1-circle hasdiameter 1


andFord Circles


1/3-circles havediameter 1/32=1/9


andFord Circles


1/3-circles havediameter 1/32=1/9

n/d-circles havediameter 1/d2

Cm algebra of revival-phase dynamics

Summary

Quantum rotor revivals obey wonderfully simplegeometry, number, and group theoretical analysis

andas the next talk will show...

Summary

Quantum rotor revivals obey wonderfully simplegeometry, number, and group theoretical analysis

andas the next talk will show…

“I still don’t really know... revivals … at all.”

QuickTime™ and aH.264 decompressor

are needed to see this picture.

Simulation of revival-intensity dynamics

QuickTime™ and aH.264 decompressor

are needed to see this picture.

rj04 resonance and revivals i. quantum rotor and infinite-well dynamics william g. harter and...

Documents