# Semiclassical chaos, the uncertainty principle, and quantum dissipation Page: 8,492

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BONCI, RONCAGLIA, WEST, AND GRIGOLINI

at

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(2.5)

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. [17].

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-

mechanical fluctuations).

(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

aAw(q,p;t)

= Aw(q,p;t) . (2.6)

at

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

[21]. 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 [21] 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-

ity.

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:

2 21(2.10)

8492

45

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.