Time dependent QM: Introduction

We do not have direct sensory experience about fundamental particles, the properties have to be inferred from experiments involving the measurement of counting rates in scattering experiments, measurement of optical spectra, etc. The outcomes of collision processes are usually analysed in terms of time-independent techniques, which do not tell anything about what happens during the scattering event. To know, how the state of the system evolves in time, one has to solve the quantum-mechanical equation of motion, the time-dependent Schrodinger equation, but there are only few and oversimplified examples for which this can be done analytically.

Recent developments in experimental technique as the availability of laser systems producing reproducible ultra-short pulses of 10 fs makes possible to follow real-time evolution on an atomic scale. Femtosecond lasers are making the investigation of time-dependent quantum phenomena in molecules a reality. Molecular vibrations, for example, occur on a 100 fs time scale. The femtosecond time scale is also suitable for exciting electronic states in multiple quantum well semiconductor devices and the time evolution of these states may provide valuable information about transport properties of mesoscopic devices.

The theoretical investigation of such problems requires the numerical solution of the time-dependent Schrodinger equation. In the past computing of the time evolution of more or less realistic quantum systems has posed formidable numerical obstacles, the computer solution of the time-dependent Schrodinger equation was a difficult task due to hardware and software limitations. These limitations are by now largely removed by the efficient numerical techniques developed in the recent years and by the high performance of microcomputers and workstations available. These computational advances made it possible to introduce the time dependent quantum-mechanical study of realistic physical systems into the educational process.