Petra Zdanska, IOCB June 2004 – Feb 2006 Resonances and background scattering in gedanken...
-
Upload
shanna-lynch -
Category
Documents
-
view
214 -
download
0
Transcript of Petra Zdanska, IOCB June 2004 – Feb 2006 Resonances and background scattering in gedanken...
Petra Zdanska, IOCBPetra Zdanska, IOCB
June 2004 – Feb 2006June 2004 – Feb 2006
Resonances and Resonances and background scattering in background scattering in
gedanken experiment with gedanken experiment with varying projectile fluxvarying projectile flux
IOCB IOCB February 20, 200February 20, 20066
22
Personal acknowledgement
Personal acknowledgement
• Milan Sindelka and Nimrod Moiseyev
• Vlada Sychrovsky and people attending my unfinished Summer course of resonances 2004
• Nimrod’s group and conferences
IOCB IOCB February 20, 200February 20, 20066
33
Resonance and direct scattering as two mechanisms
Resonance and direct scattering as two mechanisms
• Direct– density of states
changes evenly smooth spectrum
• Resonance– metastable states– density of states
includes peaks
IOCB IOCB February 20, 200February 20, 20066
44
Simultaneous occurrence of direct and resonance
scattering mechanisms?
Simultaneous occurrence of direct and resonance
scattering mechanisms?
IOCB IOCB February 20, 200February 20, 20066
55
Question:Question:
• Are direct and resonance scattering mechanisms separable at near resonance energy ?
• Mathematical answer: yes by complex scaling transformation.
• Physical answer: ?
IOCB IOCB February 20, 200February 20, 20066
66
Complex scaling method (CS)
Complex scaling method (CS)
• useful non-hermitian states – “resonance poles”– purely outgoing condition is a cause to
exponential divergence and complex energy eigenvalue
• complex scaling transformation of Hamiltonian– non-unitary similarity transformation for
taming diverging states
IOCB IOCB February 20, 200February 20, 20066
77
exp exp cos Re sin Im
exp sin Re cos Im
Imarctan arctan
Re Re
i
c
ipxe ix p p
x p p
p
p p
Ougoing condition for resonances and CS
Ougoing condition for resonances and CS
• Problem:• Solution:
exp exp Re Imipx ix p x p
IOCB IOCB February 20, 200February 20, 20066
88
Outgoing condition for resonances and CS
Outgoing condition for resonances and CS
IOCB IOCB February 20, 200February 20, 20066
99
Separation of direct and resonance scattering by CS
transformation
Separation of direct and resonance scattering by CS
transformation
Im E
Re Eboundstates resonance
rotated continuum
IOCB IOCB February 20, 200February 20, 20066
1010
States obtained by CS as scattering states for varying
projectile flux
States obtained by CS as scattering states for varying
projectile flux
IOCB IOCB February 20, 200February 20, 20066
1111
• Connection between gamma and theta:
IOCB IOCB February 20, 200February 20, 20066
1212
Proofs by semiclassical and quantum simulations
Proofs by semiclassical and quantum simulations
• Why semiclassical and not just quantum mechanics – only way to prove a correspondence between the
classical notion of flux of particles and quantum wavefunctions
• Cases I and II:– I. analytical proof for free-particle scattering– II. numerical evidence for direct scattering problem
• Case III:– a quantum simulation of resonance scattering for
varying projectile flux displaying the new effects
IOCB IOCB February 20, 200February 20, 20066
1313
Case I: Free-particle Hamiltonian
Case I: Free-particle Hamiltonian
• non-hermitian solutions of CS Hamiltonian:
2
Im E
Re E
2 22
2
ˆ ˆˆ ˆ2 2
ˆ
exp exp
i
i
i
p pH H e
H
E e
ipx ipxe
IOCB IOCB February 20, 200February 20, 20066
1414
Wavefunctions of rotated continuum
Wavefunctions of rotated continuum
• exponentially modulated plane waves:
grows in x
decays in time
IOCB IOCB February 20, 200February 20, 20066
1515
• time-dependence:
2
2
exp
ˆexp exp
exp
i
i
i i
ip xe
i it Ht E e t
ip xe E te
IOCB IOCB February 20, 200February 20, 20066
1616
Semiclassical solution to the expected physical process
behind these non-hermitian states:
Semiclassical solution to the expected physical process
behind these non-hermitian states:
• step I: construction of a corresponding density probability in classical phase space– 1st order emission in an asymptotic
distance xe with the rate :
IOCB IOCB February 20, 200February 20, 20066
1717
– density of particles in a close neighborhood of the emitter:
– analytical integration of the classical Liouville equation with the above boundary condition:
IOCB IOCB February 20, 200February 20, 20066
1818
Classical density for free particles:
Classical density for free particles:
IOCB IOCB February 20, 200February 20, 20066
1919
Step II: transformation of classical phase space density to a quantum wavefunction
Step II: transformation of classical phase space density to a quantum wavefunction– non-approximate, in the case of free-
Hamiltonian
IOCB IOCB February 20, 200February 20, 20066
2020
IOCB IOCB February 20, 200February 20, 20066
2121
Exact comparison with non-hermitian wavefunction as a
proof
Exact comparison with non-hermitian wavefunction as a
proof• the non-hermitian and scattering
wavefunctions have the same form and are equivalent supposed that,
– which was to be proven.
IOCB IOCB February 20, 200February 20, 20066
2222
Case II: Rotated complex continuum of Morse oscillator
Case II: Rotated complex continuum of Morse oscillator
• potential:
• semiclassical simulation of scattering experiment with parameters:– particles arrive with classical energy:– decay rate of the emitter:
11 . ., 1 . . , 10 . .D a u a u a u
IOCB IOCB February 20, 200February 20, 20066
2323
Construction of classical phase space density
Construction of classical phase space density
• classical orbit [x(t),p(t)] is evaluated
• phase space density:
IOCB IOCB February 20, 200February 20, 20066
2424
Construction of semiclassical wavefunction
Construction of semiclassical wavefunction
• dividing to incoming and outgoing parts:
• transformation of density to wf:
IOCB IOCB February 20, 200February 20, 20066
2525
IOCB IOCB February 20, 200February 20, 20066
2626
IOCB IOCB February 20, 200February 20, 20066
2727
The expected quantum counterpart
The expected quantum counterpart
• Non-hermitian solution of CS Hamiltonian with the energy:
IOCB IOCB February 20, 200February 20, 20066
2828
Solution of CS Hamiltonian in finite box:
Solution of CS Hamiltonian in finite box:
• box: • N=200 basis functions• solution of CS Hamiltonian:
• back scaled solution:
IOCB IOCB February 20, 200February 20, 20066
2929
Comparison of scattering wavefunction and rotated
continuum state:
Comparison of scattering wavefunction and rotated
continuum state:
IOCB IOCB February 20, 200February 20, 20066
3030
Case III: near resonance scattering
Case III: near resonance scattering
• Potential:
• Examined scattering energies:– resonance hit– very slightly off-resonance
IOCB IOCB February 20, 200February 20, 20066
3131
in complex energy plane:in complex energy plane:
Im E
Re E
-0.0034
-0.002
V(x)
x
0.7126 0.716
-0.004
IOCB IOCB February 20, 200February 20, 20066
3232
Quantum dynamical simulations of scattering
experiments
Quantum dynamical simulations of scattering
experiments• “particles” added as Gaussian
wavepackets in an asymptotic distance, 40 a.u.
• beginning of simulation: scattering experiment does not start abruptly but the intensity I(t) is modulated as follows:
IOCB IOCB February 20, 200February 20, 20066
3333
slow change of gammaslow change of gamma
Im E
Re E
-0.0034
-0.002
0.7126 0.716
-0.004
IOCB IOCB February 20, 200February 20, 20066
3434
Resonance hit:Resonance hit:
IOCB IOCB February 20, 200February 20, 20066
3535
Off-resonance:Off-resonance:
IOCB IOCB February 20, 200February 20, 20066
3636
Off-resonanceOff-resonance
IOCB IOCB February 20, 200February 20, 20066
3737
What is going on:What is going on:
• We reach stationary-like scattering states, which are characterized by a constant scattering matrix and by a constant (and complex) expectation energy value.
• Are these states the non-hermitian solutions to Hamiltonian obtained by CS method?
IOCB IOCB February 20, 200February 20, 20066
3838
Calculations of scattering matrix:
Calculations of scattering matrix:
• comparison of dynamical simulations with stationary solutions of complex scaled Hamiltonian
• gamma<Gamma_res :– rotated continuum
• gamma>Gamma_res :– resonance hit resonance pole– slightly off-resonance rotated
continuum
IOCB IOCB February 20, 200February 20, 20066
3939
Scattering matrix from simulations:
Scattering matrix from simulations:
IOCB IOCB February 20, 200February 20, 20066
4040
Inverted control over dynamics for gamma>Gamma_res
Inverted control over dynamics for gamma>Gamma_res
• incoming flux decays faster than the wavefunction trapped in resonance
• natural control: incoming flux disappears faster than outgoing flux – this occurs for discrete resonance energies
• inverted control: outgoing flux decays according to gamma and not Gamma_res. Reason: destructive quantum interference removes the trapped particle.
IOCB IOCB February 20, 200February 20, 20066
4141
• empirical rule in CS: rotated continuum for θ> θc (γ>Γres) is not responsible for resonance cross-sections.
IOCB IOCB February 20, 200February 20, 20066
4242
Conclusions:Conclusions:
• resonance phenomenon studied in a new context of scattering dynamics
• new light shed into complex scaling method, interference effect behind the long accepted empirical rule
• first physical realization of complex scaling eventually interesting for experiment