Chaos in de Broglie - Bohm quantum mechanics
Christos Efthymiopoulos
Research Center for Astronomy and Applied Mathematics
Academy of Athens
Collaboration with: G. Contopoulos, N. Delis, and C Kalapotharakos
J.Phys.A39, 1819 (2006)J.Phys.A40, 12945 (2007)Cel. Mech. Dyn. Astron.102, 219 (2008)Nonlin. Ph. Compl. Sys.11, 107 (2008)Phys. Rev. E79, 036203 (2009)Annals of Physics 327, 438 (2012)Int. J. Bifurcations and Chaos (2012) J. Phys. A.: Math. Theor. (2012).
Bohm’s (de Broglie) theory
Demonstration that a non-local realistic theory is possible
Non-relativistic QM
Introduce polar co-ordinates
Quantum Hamilton-Jacobi equationContinuity equation
Quantum potential
Equations of motion(“pilot-wave”)
=| |2
Manifestedly non-local theory
Fully-consistentwith Bell-type correlationsBell (1987): “Speakable and un-speakable about QM”
Relativistic, spin and field theories
No “measurement problem”
Quantum relaxation - Quantum equilibrium
Serves as a “picture” of quantum mechanics
(solve Schrödinger’s equation by the quantum tajectories)
Quantum relaxation - quantum equilibrium
Quantum equilibrium hypothesis: if dBB(x,t=0) | (x,t=0)|2
then, under certain conditions, dBB(x,t) | (x,t)|2 as t
First result in this direction: Bohm - Vigier theory (1954)
Theorem (Bohm and Vigier 1954): one has dBB(x,t) | (x,t)|2 as tprovided that the following `mixing condition’is satisfied:
“a fluid element starting in an elementary element of volume dx’ , in a region where the fluid density is appreciable, has a nonzero probability of reaching any other element of volume dx in this region”
Typicality theorem (Dürr et al. 1992): in a large system, “most” Bohmian initial conditions imply extremely small fluctuationswith respect to the quantum equilibrium state.
Vallentini (1991): the “coarse-grained” particle distribution approaches | |2 on a finer and finer scale (works also for few degrees of freedom).
Valentini (1991)Valentini and Westman
(2005)
Quantum version of Boltzmann’s H-theorem
Timescale to relaxation for an -coarse graining:
Bohm’s method as a “propagator” of quantum dynamics
Consider particle-like “tracers” of the quantum flow
positions of particles at the time t:
=R2
Integrate the Newtonian equations for one time step
positions of particles at the time t+dt:
R(t+dt)= (t+dt)1/2
momenta of particles at the time t+dt:
S(x1,x2,...,xN, t+dt)
A two-fold approach to the importance of chaos in dBB theory:
1) Foundational aspects:dynamical relaxation to quantum equilibrium(Valentini)
2) Practical aspects:accuracy of Lagrangian (hydrodynamical) schemes for solving Schrödinger’s equation(Wyatt, Oriols, Towler)
New predictions?
I. Times of arrival
II. Non-equilibrium quantum physics (early cosmology)
Co-existence of ordered and chaotic trajectories
2D harmonic oscillator (Parmenter and Valentine 1995)
Superposition of three eigenstates
One moving nodal point
Domains of analyticity (devoid of nodal points)
a=b=1
c= 7 / 10 2/2=c
Theoretical limits on nodal point domains
Series expansions for regular orbits
(valid within the domain of analyticity)
where xn, yn are of order n in the superposition amplitudes a,b
Proposition: construction is consistent (no secular terms appear)
Convergence: guaranteed if c=rational (conjectured if c=irrational)
NumericalSeries
representation
Origin of chaos near moving nodal points
Early works: Frisk 1997, Konkel and Makowski 1998, Wu and Sprung 1999, Makowski et al. 2000, Falsaperla and Fonte 2003,
Wisniaski and Pujals 2005, Wisniaski et al. 2006
Goals
1) Study the quantum flow structure in a frame of reference moving with the local vortex speed
2) Unravel the mechanism of generation of chaos
3) Quantitative estimates on Lyapunov exponentsin terms of:
a) the local parametersof a vortex, and b) the number of vortices in a given state and time
Vortex speed:Vx=dx0/dt, Vy=dy0/dt
Introduce local coordinates: u=x(t)-x0(t), v=y(t)-y0(t)
Expand (x,y,t) around (x0,y0) and calculate dBB equations of motion
Determine quantum flow structure in an adiabatic approximation
Nodal point
X point
Nodal point: limit of a spiral (point attractor or point repellor)
where is the averaged distance from the nodal point as a function of the polar angle
Nodal point
X point
)(φR
R( )
<f3> is in general, non-zero, only if the vortex velocity is non-zero(moving vortices generate chaos)
<f3> changes sign quasi-periodically in time (Hopf bifurcation: nodal point turns from attractor to repellor)
Nodal point
X point
R( )
Frequencies: 1=1, 2= 2/2
Example of a Hopf bifurcation
Channelof inward flow(typically verynarrow )
Nodal point:attractor
Nodal point:repellor
Limit cycle
Limit cycle reachesouter separatrix
No inward flow
Nodal point:repellor
Nodal point:repellor
`Avoidance’ rule
As a rule, quantum trajectories in systems with moving vortices avoid approaching very close to nodal points
Most of the time, the nodal point:
i) is a repellor, or
ii) is an attractor protected by a limit cycle, or
iii) is accompanied by a very narrow channel of inward flow
Flow is regular very close to a nodal point
However,
quantum trajectories are chaotically scattered by X-points!
X-point: existence is generic
Nodal point
X point
X-point: local eigenvalue analysis
1 2<0
p=2 (analytic estimate), p=1.5 (numerically)
Unstable manifolds U, UU
Stable manifolds S, SS
Nodal point
X point
Rx
Size of the vortex is inversely proportional to its speed: RX~1/V
Compare systems with one, two, or three vortices
= 00 + a 10 + b 11one nodal point, exists for all times
= 00 + a 20 + b 11two nodal points, exist in particulartime intervals
= 00 + a 30 + b 11one or three nodal points
Correlation of Lyapunov exponents with the size of vortices
Most chaotic scattering events satisfy dmin<O(RX)
Quantum relaxation
First probe case dBB(0)=| (0)|2
for numerical accuracy
900 (30X30) initial conditions
Define `smoothed density’:
noise level
Probe, now, case dBB(0) | (0)|2
Relaxation timescale correlateswith the `time of stabilization’ of the trajectories (positive) Lyapunovcharacteristic number
Quantum Nekhoroshev time
T15( )
Tcross( )
6.0
*0 exp~
ρρ
TTrelax
Exponentially long in 1/
result analogous to the classical Nekhoroshevtheorem (1977)
Relaxation in non-completely chaotic regime?
III. charged particle diffraction
and arrival time measurements
Wavepacket description of electron scattering
ingoing electronwavepacket
D: transverse quantum coherence length
llll: transverse quantum coherence length
outgoing (radial) electron wavepacketin direction 2
at t=2l0/v0
outgoing (radial) electron wavepacketin direction 1
at t=2l0/v0
crystal: source of a radial wavepacket propagating from the center outwards
=crystal number densityd=crystal thickness
Experimental requirements
LASER-inducedfield emission
“Start” event determined with a psec accuracy
Highly coherent electron
nanotip
laser pulse with high time resolution
photon detector
E
Experimental requirements
(axisymmetric diffraction pattern)
crystal: thin metal foil or polycrystalline film
Experimental requirements
Single electron detectors
with psecresponse time
Wavefunction modelling
Potential Eigenfunctions (in Born approximation)
sum over atomic nodes in the crystal
Final model
radial (Gaussian) wavepacket
Gaussian wavepacket
sum of phasors (produces diffraction pattern)
Outgoing wavefunction: fitting model when l>>D
Fraunhofer factor(depending on the reciprocal lattice vector g)
at distancesr<k 0D2a/done has:
Seff~( D)1/2d
at large distances
)2/(sin4
),;(122
0 θ
θ
k
rgS
reff≈
Gaussianwavepacketpropagatingoutwards
Outgoing wavefunction: fitting model when l<<D
Radial extent of the outgoing wave
If l>>D lIf l<<D D
Quantum current structure, separator and quantum vortices
central beam axis
D
separator: locus where | in|=| out|
Quantum trajectories:horizontal up to the point where separator is encounteredRadial outward afterwards
O(D)1
Ingoing term:exponential fall
Outgoing term:power-law fall
22 /~ DRin e−ψ
rout /1~ψ
`hard’ deflectionsdue to the approach to nodal point -X-point complexes
2
Quantum vortices: structure of quantum currents
Size of quantum vortices
Expand the wavefunction around a nodal point
Second order perturbation theory
close toBragg angles
domain of diffusescattering
Size of vortices
Bohmian trajectories(test numerically preservation of the continuity of the quantum flow)
Separator evolution and quantum vortices
Analytic approach to thelocus of initial conditions leading toscattering in a particular angle
The problem of time in quantum mechanics
(reviewed in Muga and Leavens 2000, and Muga et al. 2002,2009)
1) Times are experimentally observed (e.g. TOF spectroscopy for heavy ions)
2) Time, however, is not an operator-valued observable(Pauli’s theorem).Furthermore, measurements in QM are considered to occur oninstants of time
3) Several approaches, still ambiguous (and controversial)
- Kijowski (1972) arrival times distribution (based on Aharonov-Bohm operators). For asymptotically free wave packets, it turns to be analogous to the classical flux approach
- Sum-over-historiesapproach (Yamada and Takagi 1992, based on works by Hartle, Gel-Mann, Griffiths and Omnes)
- POVMs deduced from histories (Anastopoulos and Savvidou 2006)
- Bohmian approach (Leavens): consistent and simple
Arrival time distributions
t
l>>D
P(t)
t
t~max(l,D)/v0 (of order ~1psec for cold field electrons)
t
D>>l