Classical wavefunctions and hybrid classical-quantum systems

Since the early days in the development of quantum mechanics, the interaction of classical and quantum systems emerged as a fundamental open question mainly motivated by Bohr's description of quantum measurement. Over the decades, the problem of classical-quantum coupling has made its appearance in a variety of contexts, ranging from measurement theory to quantum gravity and from chemical physics to quantum biology.

Tackling the classical-quantum coupling problem

In quantum chemistry, the problem first appeared in the 1920s, when Born and Oppenheimer formulated their adiabatic theory of molecular dynamics, in which nuclei were treated as classical particles while letting the electronic wavefunction retain all the quantum features in the system. Over the years, this approach has led to several efforts addressed to take appropriate partial classical limits on factorized molecular wavefunctions. Another approach to classical-quantum coupling was to write down an equation for the evolution of a hybrid classical-quantum density matrix satisfying what is now known in chemistry as the `classical-quantum Liouville equation’. Unfortunately, this equation may violate quantum uncertainty as the density matrix of the quantum subsystem is allowed to become negative in time.

An alternative approach has also been developed by several authors: instead of obtaining a hybrid theory by taking the partial classical limit after starting from a fully quantum treatment, one considers the problem of treating one of the two systems as classical from the start. This is an exciting direction that is currently being developed along with other related topics in quantum hydrodynamics and geometric phases, which are being investigated by members of the Centre in the Department of Mathematics.