The biharmonic oscillator and the asymmetric linear well are two confining power-law-type potentials for which complete bound-state solutions are possible in both classical and quantum mechanics. We examine these problems in detail, beginning with studies of their trajectories in position and momentum space, evaluation of the classical probability…

The transmission through a stack of identical slabs that are separated by gaps with random widths is usually treated by calculating the average of the logarithm of the transmission probability. We show how to calculate the average of the transmission probability itself with the aid of a recurrence relation and derive analytical upper and lower…

We describe a method for the accurate calculation of bound-state and resonance energies for one-dimensional potentials. We calculate the shape resonances for symmetric two-barrier potentials and compare them with those coming from the Siegert approximation, the complex scaling method and the box-stabilization method. A comparison of the…

Here, we present complex resonance states (or Siegert states) that describe the tunnelling decay of a trapped quantum particle from an intuitive point of view that naturally leads to the easily applicable Siegert approximation method. This can be used for analytical and numerical calculations of complex resonances of both the linear and nonlinear…

The Landau-Zener formula provides the probability of non-adiabatic transitions occurring when two energy levels are swept through an avoided crossing. The formula is derived here in a simple calculation that emphasizes the physics responsible for non-adiabatic population transfer. (Contains 2 figures.)

Monte Carlo simulations are a classical tool to analyse physical systems. When unlikely events are to be simulated, the importance sampling technique is often used instead of Monte Carlo. Importance sampling has some drawbacks when the problem dimensionality is high or when the optimal importance sampling density is complex to obtain. In this…

We study different ways of determining the mean distance (r[subscript n]) between a reference point and its nth neighbour among random points distributed with uniform density in a D-dimensional Euclidean space. First, we present a heuristic method; though this method provides only a crude mathematical result, it shows a simple way of estimating…