# Chaos theory and quantum mechanics

liquidgrey01
Is there any correlation between these two fields? Has chaos theory been used as an explanation for quantum randomness? Did chaos theory develop out of quantum mechanics?

jeff
Originally posted by liquidgrey01
Is there any correlation between these two fields? Has chaos theory been used as an explanation for quantum randomness? Did chaos theory develop out of quantum mechanics?

Although quantized chaotic systems have been studied, chaos theory did not originate in and cannot expain quantum behaviour.

Chaotic behaviour originates in systems that interact with themselves in a way that results in a critical dependence of their evolution on initial conditions.

For example, a baseball thrown in slightly different ways will trace slightly different trajectories so this system is not chaotic.

On the other hand, since the evolutionary paths of weather systems from slightly different initial conditions very quickly diverge from each other, weather systems are chaotic. In fact, it's their chaotic nature that makes their behaviour so difficult to predict beyond a day or two ahead.

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Extensive and unsuccessful efforts

have been made to use chaos theory to explain quantum randomness, and there is a large literature on the subject, but the two are not directly related. Some quantum systems (wavefunctions) evolve in a chaotic way, most don't.

Chaos theory evolved out of a mathematician's observations of how non-linear iterated functions behaved on his pocket calculator.

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Check out xxx.lanl.org and search with the keywords quantum billiards or random matrix theory...

Stochastic Shrodinger equation

Dear frands!
Prompt please references to works in which it was considered the Schrodinger equation with stochastic (random) Gaussian delta-correlated potential which
time-dependent and spaces-dependent and with zero average (gaussian delta-correlated noise). I am interesting what average wave function is equal.

U - potential.
<> - simbol of average.

P(F) - density of probability of existence of size F.

Delta-correlated potential which
time-dependent and spaces-dependent:
<U(x,t)U(x,t)>=A*delta(x-x) *delta(t-t)
delta - delta-function of Dirack.
A - const.

Zero average:
<U(x,t)>=0

Gaussian potential (existence of probability is distributed on Gauss law):
P(U)=C*exp(U^2/delU^2)

C - normalizing constant.
delU - root-mean-square fluctuation of U.