Quantum mechanics on time-varying space domains

Abstract

The extension of quantum theory to time-varying space domains is often challenging, since domain evolution frequently results in non-autonomous and non-adiabatic evolution of corresponding wave function for a given quantum mechanical system. For generic domain evolution, we show that the evolution of the wave function is determined by the mixing of spatial modes or bound states, and this is the instigator for non-adiabatic wave function evolution. For some applications, it is desirable to retain adiabaticity of the wave function and with knowledge of how domain evolution causes loss of adiabaticity, we construct a control potential—comprising a harmonic term and a volume expansion/contraction term—which may be used to counteract this feature of domain evolution, thereby preserving wave function adiabaticity throughout the time evolution of the space domain. Examples of quantum mechanical systems on time-varying space domains are used to illustrate the theory, including analogues of classical examples such as the hydrogen atom and quantum harmonic oscillator on unbounded stretched space and a particle in a stretched and translated box. We also discuss how to combine our approach with numerical simulations, using the compressed hydrogen atom confined within an evolving sphere to demonstrate the method.