Complex dynamics and coherent states

Abstract

In the early years of the quantum theory Schrödinger already suggested that wave packets evolving along the classical trajectory represent the quantum description of the particle’s evolution. This idea is also close to de Broglie’s concept of the pilot wave [90] and to Bohm’s mechanics [56] [55], The classical limit would correspond to the zero width of the packet. Schrödinger gave an example of the wave packet of an oscillator. This wave packet is now called the coherent state. However, it was soon realized that in quantum mechanics only the time evolution generated by quadratic Hamiltonians can be expressed exactly by a classical evolution. There were numerous attempts to generalize the concept of the coherent state. We distinguish the studies which are concerned with coherent states localized in space and to some extent localized around classical trajectory. The coherent states which have the property of minimal uncertainty have been discussed by Nieto and Simmons [302] [304][303] (mainly in one dimension; higher dimensions are discussed in [305]). The authors show [173] that these states lose their localization at large time. Wave packets of an electron in the hydrogen atom attracted some attention recently because of their possible realization in experiments [292] [296] [81] [60]. The dynamics of such states is beyond the range of our methods. We are mainly interested in wave packets of anharmonic oscillators which can form localized states describing classical behaviour of quantum molecules [27].