The helical quantum two-body problem and its wave packet dynamics

Abstract

We explore the helical quantum two-body problem i.e. two repulsively Coulomb interacting particles confined to move along a helix. The effective potential possesses a tunable number of potential wells superimposed on the repulsive Coulomb interaction that can be varied by changing the ratio of the pitch and radius of the helix. The anharmonicity of these wells depends crucially on this ratio and on the order of the well which can be seen also by analyzing the individual wells energy eigenvalue spacing. Our main focus is the investigation of the quantum dynamics of differently prepared wave packets that scatter from the multi-well potential landscape. We show that there exists a rich pattern forming transient evolution which depends also on the number of bound states of the individual wells. We demonstrate how the multiple wells leave their fingerprints in the dynamics leading, among others, to oscillatory structures on different spatial scales, the formation of beats and pulsed emission from single well localized wave packets due to their intrawell dynamics.

Figures (15)

• Figure 1: (a) Sketch of a helix with the radius $R$ and pitch $h$. The black spheres on the helix indicate equilibria of two particle positions: if one particle is on the top of the winding the other is approximately at the bottom of the next or all following windings. Whether such a configuration represents a minimum and consequently forms a potential well depends on the detailed values of the radius and pitch, see text. (b) The helical potential $V$ as a function of the path length $s$ for three cases: $h=5.8,R=4$ (solid line, three potential wells), $h=10,R=10$ (dashed line, six potential wells) and $h=3,R=5$ (dotted line, 17 potential wells, not all shown).
• Figure 2: A zoom into a specific window of the helical potential ${\cal{V}}$ as a function of the path length $s$ for $h/R=0.25$ for $R=0.25,0.5,1.0,2.0,4.0$ corresponding to the solid, dashed, dot-dashed, dotted and solid curves from top to bottom, in order to illustrate the appearance/shape of the individual potential wells.
• Figure 3: (a) Energy eigenvalue spacing in the isolated innermost well of the helical potential ${\cal{V}}$ for $h=20,R=20,M=1$. The eigenvalue spacing increases linearly for the $18$ bound states except at the threshold to unboundedness. (b) the same but for the parameter values with $h=12.2,R=10,M=100$ where $32$ bound states exist. The linear slope is here much smaller, indicating the approximate equidistant spectrum.
• Figure 4: The Coulomb potential and snapshots of the density time evolution for times $t=0,400,1000$ (from bottom to top) for an initial Gaussian wave packet located at $s_0 = 220.0$ with zero momentum and width $\Delta s = 4.0$ for $M=1$. The underlying sine DVR grid has 2301 points in the interval $[-150,1000]$. Note that both the potential and the densities are scaled (and shifted) to fit the same coordinate scale.
• Figure 5: (a) The helical potential (h=5.8,R=4.0) possessing three distinct minima and snapshots of the density time evolution for times $t=0,500,700,1000$ (from bottom to top) for an initial Gaussian wave packet located at $s_0 = 220.0$ with zero momentum and width $\Delta s = 4.0$ for $M=1$. The underlying sine DVR grid has 2301 points in the interval $[-150,1000]$. Note that both the potential and the densities are scaled (and shifted) to fit the same coordinate scale. (b) The evolution of the density integrated over the first innermost (solid line), second (dash-dotted line) and third outermost (dotted line) well of the potential.