From generalized Gaussian wave packet dynamics to homoclinic orbits

Steve TomsovicWashington State University, Pullman, WA USA

The general Gaussian integral

!!N

Det A

" 12

exp!

pT · A!1 · p4

"=

# "

!"dx exp

$!xT · A · x + pT · x

%

where (x,p) are N -dimensional vectors and A is an N " N symmetric,generally complex, matrix.

General forms of Gaussian wave packets [3]

#x|"$ = exp!

i

!

&(x ! x0)

T · A0 · (x ! x0) + pT0 · (x ! x0) + s0

'"

#x|"(t)$ = exp!

i

!

&(x ! xt)

T · At · (x ! xt) + pTt · (x ! xt) + st

'"

Im s0 =!4

ln!

Det2!! Im A0

"

The semiclassical time-dependent Green function [10, 2]

#x|"(t)$ =#

dx# #x|e!iHt/!x#$#x#|"$

#x|"(t)$sc =#

dx# Ksc(x,x#; t)#x#|"$

Ksc(x,x#; t) =!

12!i!

"N2 (

j

))))#2Sj(x,x#; t)

#x#x#

))))

12

exp*

i

!Sj(x,x#; t) ! i!

2$j

+

Sj(x,x#; t) =# t

0dt#L(x,x#; t#)

• 1-D example

S(x, x#; t) % S(xt, x0; t) + (x ! xt)!

#S#x

"

xtx0

+ (x# ! x0)!

#S#x#

"

xtx0

+(x ! xt)

2

2

!#2S#x2

"

xx

+(x# ! x0)

2

2

!#2S#x#2

"

xtx0

+ (x ! xt) (x# ! x0)!

#2S#x#x#

"

xtx0

The Wigner transform

P!(x,p) =!

12!!

"N # "

!"dq#"|x + q/2$#x ! q/2|"$eip·q/!

The (x,p) behavior in the resulting exponential argument for P!(x,p) is

Arg xp

[P!] =i

!

,(x ! x0)

T · (A0 ! A$0) · (x ! x0) + [p! p0 ! (A0 + A$

0) · (x ! x0)]T

·(A0 ! A$0)

!1 · [p! p0 ! (A0 + A$0) · (x ! x0)]

-

Returning to the quadratic expansion derivatives

pt = &xS(x,x#; t) , !p0 = &x!S(x,x#; t)&xpt = m11m!1

21 , !&x!p0 = m!111 m22

&2x,x!S(x,x#; t) = m21 ! m11m!1

21 m22

The stability matrix!

%pt

%xt

"= Mt

!%p0

%x0

"=

!m11 m12

m21 m22

"

t

!%p0

%x0

"

Its evolution

dMt

dt=

.

/! "2H(x,p;t)

"x"p

)))xt,pt

! "2H(x,p;t)"x2

)))xt,pt

"2H(x,p;t)"p2

)))xt,pt

"2H(x,p;t)"x"p

)))xt,pt

0

1Mt

Its initial condition

M0 = 1

Linearized Gaussian wave packet dynamics [3]

#x|"(t)$lwpd = exp!

i

!

&(x ! xt)

T · At · (x ! xt) + pTt · (x ! xt) + st

'"

x = &pH(x,p; t) p = !&xH(x,p; t)

At =12Bt · C!1

t where!

Bt

Ct

"= Mt

!2A0

1

"

st = s0 + S(xt,x0; t) +i!2

Tr [lnCt]

Quadratic expansion of the Schrodinger equation [4]

i! #

#t#x|"(t)$lwpd =

*! !2

2m&2

x + V (x)+#x|"(t)$lwpd

V (x) = V (xt) + (x ! xt)T ·&xV (x)|xt +12(x ! xt)T ·&x (&xV (x)) |xt · (x! xt)

Time scales of validity or breakdown

Integrable systems:

&b =

2!xT ·!x!vT ·!v

=

34

5

cI

!'

Tr(#$!·#$!)optimal

cI'!Tr(#$!·#$!)

not paying attention

Chaotic systems:

&b =16n 'n

lncc

!

Propagating wave packets - Circle

Propagating wave packets - Stadium

Generalized Gaussian wave packet dynamics [5]

Recall the basic Gaussian form

#x|"$ = exp!

i

!S0(x;x0,p0)"

= exp!

i

!

&(x ! x0)

T · A0 · (x ! x0) + pT0 · (x ! x0) + s0

'"

Note that it is invariant if you choose any new (x#0,p#

0) and s#0 such that

p#0 ! p0 = 2A0 · (x#

0 ! x0) and let

s#0 ! s0 = xT0 · A0 · x0 ! x#

0T · A0 · x#

0 + p#0T · x#

0 ! pT0 · x0

Complex Lagrangian manifold:

p(x) =#S0(x;x0,p0)

#x= p0 + 2A0 · (x ! x0) [x,p(x)]

Example quantized tori of 2-D coupled quartic oscillators [7]

Linearized Gaussian wave packet dynamics [3]

#x|"(t)$lwpd = exp!

i

!

&(x ! xt)

T · At · (x ! xt) + pTt · (x ! xt) + st

'"

x = &pH(x,p; t) p = !&xH(x,p; t)

At =12Bt · C!1

t where!

Bt

Ct

"= Mt

!2A0

1

"

st = s0 + S(xt,x0; t) +i!2

Tr [lnCt]

Propagating wave packet - 1-D Morse oscillator [5]

Propagating classical hydrogen orbits [1]

Propagating hydrogen wave packets [1]

Propagating classically chaotic orbits [8]

Propagating classically chaotic orbits in the stadium [8]

Heteroclinic (homoclinic) orbits in the stadium [8]

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Propagating wave packets in the stadium [8]

Propagating wave packets in the stadium [8]

Propagating wave packets in the stadium [8]

Validity time scales [8]

Validity time scales [6]

Validity time scales [6]

Constructing chaotic eigenstates in the stadium [9]

Constructing chaotic eigenstates in the stadium [9]

References

[1] I. Marianna Suarez Barnes, Michael Nauenberg, Mark Nockleby, andSteven Tomsovic. Classical orbits and semiclassical wave packet propa-gation in the coulomb potential. J. Phys. A: Math. Gen., 27:3299–3321,1994.

[2] M. C. Gutzwiller. Periodic orbits and classical quantization conditions.J. Math. Phys., 12:343, 1971. and references therein.

[3] E. J. Heller. Wavepacket dynamics and quantum chaology. In M. J. Gian-noni, A. Voros, and J. Zinn-Justin, editors, Chaos and Quantum Physics,pages 547–663. North-Holland, Amsterdam, 1991.

[4] Eric J. Heller. Time-dependent approach to semiclassical dynamics.J. Chem. Phys., 62:1544–1555, 1975.

[5] Daniel Huber, Eric J. Heller, and Robert G. Littlejohn. Generalized gaus-sian wave packet dynamics, schrodinger equation, and stationary phaseapproximation. J. Chem. Phys., 89:2003–2014, 1988.

[6] Miguel-Angel Sepulveda, StevenTomsovic, and Eric J. Heller. Semiclassicaldynamics: how long can it last. Phys. Rev. Lett., 69:402–406, 1992.

[7] S. Tomsovic. Chaos-assisted tunneling in the absence of symmetry.J Phys A: Math Gen 31:9469–9481 1998

[8] S. Tomsovic and E. J. Heller. The long-time semiclassical dynamics ofchaos: the stadium billiard. Phys. Rev. E, 47:282–299, 1993.

[9] S. Tomsovic and E. J. Heller. The semiclassical construction of chaoticeigenstates. Phys. Rev. Lett., 70:1405–1409, 1993.

[10] John H. Van Vleck. The correspondence principle in the statistical interpre-tation of quantum mechanics. Proc. Natl. Acad. Sci. U.S.A., 14:178–188,1928.