Shrinking potentials in the Schrödinger equation

Chris Quigg, Jonathan L. Rosner

Koustav D. Chakladar, Santanu Mondal, Sukhamoy Bhattacharyya +1 more

The effect of a penetrable repulsive single-barrier potential on the structural properties of hydrogen atom in ground and different excited (n,l) states [n=1-3, l=0-2] is studied. The Lagrange mesh method is adopted to solve the corresponding Schrodinger equation numerically for energy eigenvalues and eigenfunctions. Different novel features and phenomena e.g. shrinking the size of the atom, atomic swelling, orbital fusion and fission etc. are noted when the strength of the barrier is changed by tuning its position and height. It is remarkable that all such alterations of the atomic orbital are well articulated from the Shannon entropy profile.

Alexander V. Zolotaryuk, Yaroslav Zolotaryuk

Abstract A heterostructure composed of N parallel homogeneous layers is studied in the limit as their widths l 1 , …, l N shrink to zero. The problem is investigated in one dimension and the piecewise constant potential in the Schrödinger equation is given by the strengths V 1 , …, V N as functions of l 1 , …, l N , respectively. The key point is the derivation of the conditions on the functions V 1 ( l 1 ), …, V N ( l N ) for realizing a family of one-point interactions as l 1 , …, l N tend to zero along available paths in the N -dimensional space. The existence of equations for a squeezed structure, the solution of which determines the system parameter values, under which the non-zero tunneling of quantum particles through a multi-layer structure occurs, is shown to exist and depend on the paths. This tunneling appears as a result of an appropriate cancellation of divergences.

Lucjan Piela

Mauro Antonio ANDREATA, V. V. Dodonov

We study the effect of initial ‘shrinking’ of radial ‘ring-shaped’ wave packets with zero initial velocity distribution in two and more dimensions. Considering time evolution of probability densities described by explicit exact solutions of the free Schrödinger equation in d dimensions, we introduce and compare two different families of quantitative measures of ‘expansion’. The measures of the first type are based on the mean values of arbitrary powers of radius, they characterize the ‘total extension’ of the packet. The measures of the second type quantify the ‘internal’ size of the packet (or an effective width of the ‘ring’). We find that the effect of initial ‘in-spreading’, which is very small in two dimensions and absent in higher dimensions with respect to the mean value of radius, is much more pronounced, if one uses mean values of ⟨ r α ⟩ with α < 1 and especially α < 0 as the measure of packet extension. In this sense, the case of two dimensions is not distinguished, and shrinking packets exist in more than two dimensions, as well. On the other hand, we show that the effect of initial ‘shrinking’ or ‘expansion’ strongly depends on the chosen measure of spatial extension (‘total’ or ‘internal’) of the packet. Moreover, the conclusions concerning the initial evolution of the ‘internal’ packet extensions based on the ‘volume’ probability density |ψ| 2 may be sometimes opposite to the conclusions based on the ‘radial probability density’ r d −1 |ψ| 2 .

M. Roncadelli