A path-integral approach to composite impurities

Quantum mechanics - the first ingredient of this proposal - is the physical theory describing how matter behaves at very small scales. The quantum world is far from being intuitive: here particles can penetrate through walls, a cat can be dead and alive at the same time, while electron`s position and velocity cannot be simultaneously determined. These counterintuitive phenomena are usually understood through the wavefunction, a mathematical object defining how the probability of finding a particle varies in time. However, there is an alternative explanation, introducing the so-called "path integral". Within this formalism a particle does not follow the trajectory one would expect from classical physics. Instead, one has to consider an opportune weighted average including all trajectories, as absurd and unusual they may be. Surprisingly this procedure exactly reproduces all the extraordinary features of quantum mechanics, thus finding many applications in various fields of physics. Now the second ingredient: consider a single molecule, we shall call it an impurity, within a uniform environment, composed of smaller, identical particles. The impurity`s properties are modified by the interaction with the environment, for instance its motion is slowed as the tiny particles forming the environment hit the impurity. Being slower, the particle behaves as if it had a larger mass. If the molecule rotates, the rotational motion is affected as well, and the molecule behaves as if it had a higher moment of inertia. The understanding of impurity problems, beside this intuitive picture, is made difficult by the fact the impurity interacts with a very high number of particles in the environment -- a so-called many-body problem. Thus understanding impurities paves the way for the understanding of complex systems, in particular the so-called strongly-correlated systems. Additional motivation for studying molecules as impurities lies in the recent possibility of realizing such a scenario experimentally, trapping and cooling a molecule, or by embedding it in extremely small helium droplets. Combining the two ingredients one gets a path-integral formulation for a molecular impurity, which is the topic of present proposal. What are the innovations of the proposed research? First, it allows to map the problem of molecules interacting with an environment into a different problem: an impurity interacting only with itself in the past. The complex many-body problem is mapped to a single-particle one, a critical breakthrough in the understanding of molecular impurities. Second, the path-integral theory works for any interaction strength, whereas existing theories work only for certain values of the interaction. Finally, the proposed research will study for the first time molecular impurities at finite temperature, rather than at zero temperature, and their dynamical properties, i.e. how an impurity returns to an equilibrium position after being kicked away from it.

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