Semiclassical chaos, the uncertainty principle, and quantum dissipation Page: 8,492
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BONCI, RONCAGLIA, WEST, AND GRIGOLINI
atp y(q,p; t) Lp (q,p; t)
1 h n 2n +I1 a2n +1
- H,pw+ (2n +1)! 2i aq2nl V(q) p2n +1 pw(q, ;t) ,
H w+ (2n + 1)! I a21 a
where the curly brackets denote the Poisson brackets, H
is the c-number Hamiltonian, V(q) is the c-number po-
tential, and L is the Liouville operator implicitly defined
by (2.5). The main properties of the Wigner distribution
are reviewed by Hillery et al. .
The most important features of the above Wigner for-
malism are summarized as follows.
(i) The concept of a quantum phase space is introduced
in a natural way as the corresponding classical phase
space. On the other hand, the state of the quantum sys-
tem is not completely defined by its position in this phase
space. In fact, due to the intrinsically statistical nature of
quantum mechanics, even for a single quantum state, i.e.,
a pure state, the Wigner distribution cannot be a 6 func-
tion. Thus a pure quantum state is not a point in this
phase space, but is rather an ensemble of classical systems
distributed according to the Wigner density. However, if
this initial uncertainty undergoes a fast increase as an
effect of semiclassical chaos, the resulting enhanced
spreading must be interpreted as a quantum uncertainty
of a single quantum system (enhanced by truly quantum-
(ii) Equation (2.5) introduces another kind of dynami-
cal evolution for quantum systems. Following the stan-
dard treatment of statistical mechanics we define the
dynamical operator for the function A w(q,p; t) as
= Aw(q,p;t) . (2.6)
The operator P must satisfy the equation
( A(t)) = fdqfdp Aw[exp(Lt)pw]
= fdqfdp[exp(Pt)Aw,]pw, (2.7)
where we have used (2.2). Note that in the limit 1--+0 the
operator L recovers the exact classical result, i.e., it be-
comes the Poisson bracket with the Hamiltonian [cf.
(2.5)] which is just the classical Liouvillian.
Following the above procedure we have that (2.1) and
(2.6) give the phase-space equivalent of the Heisenberg
representation of quantum mechanics. The formalism is
our starting point in the analysis of the behavior of the
solutions to the Schr6dinger (or Heisenberg) equation and
we find it to be singularly useful for study of the quantum
system whose classical limit is nonintegrable and chaotic.
Note that the average over the initial conditions must
satisfy the inequality
U Aq Ap > i/2 , (2.8)
where Aq and Ap are the root-mean-square quantities
Aq(t)2 -- (q(t2)-(q( t )2 , (2.9)
Equation (2.8) is one expression of the quantum uncer-
tainty principle (QUP), which within the Wigner formal-
ism results in a broadened classical distribution at all
times, including the initial one. Note that in the present
illustrative case, the classical approximation cannot re-
sult in chaos, due to the one-dimensional character of the
system under study, i.e., we have a single q and a single p.
However, one can easily imagine the generalization of
this approach to the n-dimensional case, where classical
chaos is admitted. In the chaotic case the initial broaden-
ing of the Wigner density forced by the QUP is expected
to increase at a rate determined by the Liapunov
coefficients and independently of the QGD mechanism
. However, as we show with numerical methods in
Sec. V, the growth of U is limited by the finite size of the
phase space available to a deterministic system, and the
quantum fluctuations becoming macroscopic according
to the prediction of Fox  are possible only if this
phase space region is macroscopic. On the other hand,
an expectation value is the average over many trajec-
tories, which quickly lose correlations with the initial
conditions and among themselves when the trajectories
are chaotic. Thus the time evolution of an expectation
value is expected to manifest the character of irreversibil-
Actually, these remarks do not take into account the
fact that, due to the presence of the QGD mechanism
quantum mechanics is not simply obtained by averaging
over classical trajectories. Due to the presence of a
derivative of order greater than unity in (2.6), unlike the
classical case, the operator f is not deterministic, i.e.,
[exp( t)( AB)]#=[exp(ft)A][exp(it )B].
In the phase-space representation the evolution of
quantum systems is determined by the interplay of the
above two contributions, the classical Liouvillian and the
QGD mechanism: the first leads to an evolution of the
quantum system in a way completely equivalent to the
corresponding classical one, with initially broadened dis-
tributions mimicking the uncertainty principle. When
the classical or semiclassical approximation is affected by
chaos, it is expected to produce irreversibility and a rapid
increase in the strength of the quantum fluctuations; the
second operator modifies this evolution to take the con-
straints posed by quantization rules into account.
III. SPIN-BOSON HAMILTONIAN
Let us examine the application of the Wigner formal-
ism to a quantum-mechanical system of sufficient com-
plexity that its semiclassical equations of motion have
chaotic solutions. We investigate one of the simplest of
such systems, this being the spin-boson Hamiltonian:
Ap (t)2- (p(t2) -- p(t))2 .
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Bonci, Luca; Roncaglia, Roberto; West, Bruce J. & Grigolini, Paolo. Semiclassical chaos, the uncertainty principle, and quantum dissipation, article, June 15, 1992; [College Park, Maryland]. (digital.library.unt.edu/ark:/67531/metadc268887/m1/3/: accessed December 18, 2018), University of North Texas Libraries, Digital Library, digital.library.unt.edu; crediting UNT College of Arts and Sciences.