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Quantum analog of Ulam's conjecture can guide molecules, reactions
Kloeppel, Physical Sciences Editor
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by L. Brian Stauffer
|Martin Gruebele, a William H. and Janet Lycan Professor of Chemistry, and colleagues have created a quantum mechanical analog of Ulam's conjecture that expands the flexibility and controllability of quantum mechanical systems.
CHAMPAIGN, Ill. —
Like navigating spacecraft through the solar system by means of gravity
and small propulsive bursts, researchers can guide atoms, molecules
and chemical reactions by utilizing the forces that bind nuclei and
electrons into molecules (analogous to gravity) and by using light for
propulsion. But, knowing the minimal amount of light required, and how
that amount changes with the complexity of the molecule, has been a
Now, by creating a quantum mechanical analog of Ulam’s conjecture,
researchers at the University of Illinois and the University of California
have expanded the flexibility and controllability of quantum mechanical
“Using photons, we can harness chaotic motion to control chemical
reactions and to move quantum objects, such as nanoclusters, molecules
and buckyballs,” said Martin Gruebele, a William H. and Janet
Lycan Professor of Chemistry, and the director of the Center
for Biophysics and Computational Biology at Illinois.
Gruebele and co-author Peter Wolynes, a professor of chemistry and biochemistry
at the University of California, describe their work in a paper accepted
for publication in Physical Review Letters and posted on the journal’s
Given sufficient time, classical chaotic motion will spontaneously connect
two points in phase space with arbitrary precision. In 1956, American
mathematician Stanislaw Ulam conjectured that owing to this phase space-filling
aspect of chaotic trajectories, a minimal series of energy expenditures
would suffice to transfer a body from one point to another much more
rapidly than by spontaneous motion.
Ulam’s conjecture is now routinely used to steer spacecraft around
the solar system with minimal energy expenditure.
“The idea is that a complex system like our solar system has lots
of planets, moons, and asteroids that can fling spacecraft gravitationally
anywhere you want,” said Gruebele, who is also a professor of physics and biophysics, and
a researcher at the Beckman Institute.
“Rather than powering a rocket on a brute force, direct route,
you can shoot your spacecraft near some moon, and let the moon do most
of the work.”
Using photons as an energy source, electrons within molecules can move
in much the same way as spacecraft in the solar system. But, there is
a hitch: Quantum mechanics, not Newtonian dynamics, must be used to
describe the motions. In quantum mechanics, the system is described
by a wave function, or quantum state.
In their quantum mechanical analog of Ulam’s conjecture, Gruebele
and Wolynes show there are limits on how efficiently an external force
can nudge a system from a given initial state to a target state. They
use the concept of a “state space” to describe all the possible
quantum states of the system.
“We can calculate where this initial state will most likely go,
and we can calculate where the target state will most likely come from,”
Gruebele said. “We can then identify places in state space where
the two are closest to one another.”
Those locations are where energy is most efficiently applied to perform
the desired quantum transformation from initial state to target state.
The researchers’ equations also tell them how many photons are
needed, and set fundamental limits on the time required.
“We can wait for the best possible moment to use the least amount
of energy,” Gruebele said. “What we have is a fast and accurate
method for computing the most efficient way of steering a quantum system
between two specified states.”
The work was funded by the National Science Foundation
Editor’s note: To reach Martin Gruebele, call 217-333-1624; e-mail: firstname.lastname@example.org.
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