Science: 'Scars' connect classical and quantum theory

6 January 1990

CHAOTIC systems are unstable and extremely sensitive to initial conditions.
Examples of them abound in the world of classical physics. But physicists
have been unable to demonstrate that the same kind of chaotic behaviour
occurs atall in quantum, or microscopic, systems.

The search for quantum chaos has not been fruitless, however. Physicists
are discovering interesting and subtle new connections between classical
theory and quantum theory. One of the most interesting is the phenomenon
of so-called ‘scars’, which cut through the wave function of a particle.
The wave function contains information on the particle’s dynamics and evolves
in time according to the Schrodinger equation.

Dieter Wintgen and A. Honig of the Max PlanckInstitute for NuclearPhysics
in Heidelberg haveinvestigated scarring of the wave function of a hydrogen
atom in magnetic field (Physical Review Letters, preprint). Their work provides
the most effective demonstration so far of the scarring for a real system
– that is, one which exists in nature.

The physicist Richard Feynman showed that the wave function for a particle
moving between any two points receives contributions from all the possible
trajectories it might take between those points. In effect, the particle
‘samples’ an infinite number of paths. Feynman also showed that the classical
trajectories of particles – the paths that classical physics predicts they
should follow – are special. This is because it is along these paths that
there are the fewest cancellation effects due to interference from neighbouring


The hydrogen atom in a magnetic field is one of the classic problems
of chaotic dynamics (New Scientist, 19 November 1987, p 44). The atom’s
single electron is influenced by both the positive charge of the nucleus
and the magnetic field.

At low energies, the electron stays close to the nucleus, and the magnetic
field only weakly perturbs its motion. When physicists look at the problem,
using classical physics and neglecting the field, they find that the electron
follows regular periodic orbits, tracing and retracing its path repeatedly.

At high energies, on the other hand, something very different happens.
The electron moves so far from the nucleus that the influence of the magnetic
field competes with the attractive force of the nucleus. In general, the
electron follows an orbit that does not close on itself. It is unstable
and extremely sensitive to initial conditions. In other words, the motion
is chaotic.

However, periodic orbits still survive. They are unstable and isolated
among infinitely more chaotic trajectories. But it turns out that such periodic
orbits have an importance out of all proportion to their numbers. These
trajectories are singled out in the particle’s wave function.

Wintgen and Honig have plotted the probability density of a hydrogen
atom in a highly excited state (see Figure). In quantum theory, the probability
of finding a particle at any point in space is given simply by the square
of the value of the wave function at that point. Wintgen and Honig’s results
show that the probability density is concentrated along the track of the
periodic orbits. In other words, the particleis more likely to be foundalong
these orbits. It is this concentration of probability (see Figure) that
had prompted physicists to refer to the ‘scarring’ of the wave function.

The strength of a scar alters for different quantum states of the hydrogen
atom. In the same wave function, contributions from different scars are
often superimposed. Sometimes, the enhancement is along the path of the
orbit, at other times symmetrically distributed alongside. Only the stablest
periodic orbits, with the shortest periods, produce obvious scars in the
wave function.

Michael Berry of the University of Bristol and others have now gone
a long way towards developing a theory for the interference effects which
produce the scarring. When they began their work, they did not think that
the periodic orbits would resultin such pronounced features in the wavefunction.
This is because non-periodic orbits occur infinitely more frequently than
periodic orbits in the chaotic regime.