A model featuring a one-dimensional particle in a tilted potential which may be trapped in a delta type potential well is considered. The time dependence of a state where the particle is trapped and where it decays via the quantum mechanical tunnel effect is calculated explicitly. Since the background potential decreases without limit (in one direction), the decay is purely exponential for long times and the decay rate is equal to the imaginary part of the eigenenergy in a Schrodinger equation that is solved for a quasi-stationary state.
Employing a time-dependent mean-field-hydrodynamic model we study the generation of black solitons in a degenerate fermion-fermion mixture in a cigar-shaped geometry using variational and numerical solutions. The black soliton is found to be the first stationary vibrational excitation of the system and is considered to be a nonlinear continuation of the vibrational excitation of the harmonic oscillator state. We illustrate the stationary nature of the black soliton, by studying different perturbations on it after its formation.
A quasi-static process is realized in a purely quantum-mechanical model which is described by oscillator (or particle) systems having relative-phase interactions. Time development of a mixture of two oscillator (or particle) systems which have different temperatures is investigated in the quantum mechanical framework. We study how they go to a thermal equilibrium. Time dependence of the temperature of the object is numerically studied and the difference of the time dependence between classical objects and quantum objects is clearly pointed out.