Quantum mechanics is a very old theory, having arrived at its modern version almost one hundred years ago. Interestingly, it is significantly older than the classical concept of chaos, which was only established in the 1950s.

The discovery of chaotic behavior posed a major barrier to earlier efforts attempting to rigorously bridge classical and quantum mechanics. For consistency reasons, the latter should always allow for the emerge of the former as its classical limit is taken. Such limit turned out to be rather nontrivial, requiring the use of sophisticated mathematical concepts, the most important being stationary phase approximations. These approximations give birth to a formalism valid in an intermediate regime in which quantum numbers are not large enough for the system to be considered near classical, but not small enough for it to be considered purely quantum – the semiclassical regime.

Semiclassical physics uses stationary phase approximations in order to take classical information as input and to output quantum interference. The formalism has two very important characteristics: Firstly, it allows us to visualize quantum effects as emerging from interfering classical trajectories, providing quantum mechanics with geometry; Secondly, since it models quantum mechanics based on classical trajectories, it does not scale exponentially with the number of degrees of freedom. The first aspect is a philosophical one that has helped

physicists devise and interpret a plethora of experiments, and the second is a computational one that basically states that semiclassical methods can bypass the dimensionality curse found in quantum mechanics. For the latter reason such methods have been consistently employed in many fields of science (especially quantum chemistry) for the past 50 years, being able to model systems with dozens of degrees of freedom (which are intrinsically semiclassical due to their sizes).

Semiclassical methods, however, suffer from two major problems: They typically diverge when the underlying classical dynamics is strongly chaotic; And, sometimes, the stationary phase approximations fail catastrophically. The computation of mean values, for example, is one such occasion in which an absurd result is obtained if stationary phase approximations are blindly employed. In order to remediate this problem, a previous work by researchers at Universite Paris-Saclay focused on establishing a semiclassical method that could deal with mean values, although the method itself had not yet been put to the test [1].

An IBS researcher at the Center of Theoretical Physics of Complex Systems and collaborators have employed the method to compute mean values in a prototype system lying simultaneously in the deep semiclassical and strongly chaotic regimes, both known to be remarkably problematic [2]. For comparisons, one of the most successful and well-known methods in quantum chemistry was also implemented for the same system. Not only was their semiclassical method extremely accurate, but it was also able to deal with regimes in which the standard method produces nothing but numerical noise. Moreover, a rare glimpse into the inner workings of mean values in quantum mechanics is provided, together with how they connect to the geometry of classical physics.

These results show that previous problems seen in semiclassical methods, which are very popular in chemistry, are intrinsic to the methods being used instead of a consequence of a general barrier to semiclassics. As a result, more accurate semiclassical methods can be devised that should be able to reach regimes previously thought impossible. These methods, such as our own, can then be expanded to systems with more degrees of freedom or to deal with the propagation of coherent states, which are situations more closely connected to experiments in both physics and chemistry.

Figure 1. Novel semiclassical method (labeled SMV, blue) works as well as a standard one (labeled HK, red) in the near-integrable regime, as can be seen in panels (a) and (c). The y axis here is an operator’s mean value, while the x axis controls how deep one is in the semiclassical regime: The larger the n, the deeper, which is why both methods improve as n grows. When chaos is increased, the novel semiclassical method produces extremely accurate results as seen in panel (b), while the standard semiclassical method returns nothing but numerical noise, as shown in panel (d).