Muhammad Umair (Stockholm University, Department of Physics)
Three particles interacting via Coulomb forces represent a fundamental problem in quantum mechanics with no known
exact solution. We have investigated resonance states composed of three particles interacting via Coulombic and more
general potentials in non-relativistic quantum mechanics, using the complex scaling method. Our calculations have been
applied to three different systems.
(i) An investigation of resonances in the positron-alkali (Li, Na, K) systems has been conducted. Some calculations
have previously been reported on the resonances in positron-alkali systems; however, most of the work was limited to the
lower partial wave, such as S-wave resonances. In this thesis, we have extended the calculations to higher partial waves
and extracted the resonance positions and widths using the more accurate complex scaling method. A dipole series of
resonances has been found under positronium n = 2 threshold, for natural parity and n = 3 threshold for unnatural parity
states. Furthermore, these resonances were found to agree well with an analytically derived scaling law. This series in the
positron-alkali system are caused by the attractive potential formed by the dipole moment of positronium (the bound state
of an electron and a positron). This dipole moment is a hydrogen-like system, and hence its energy levels are degenerate
with respect to orbital angular momentum. We have also predicted several new resonances.
(ii) A calculation of resonances in positron-hydrogen scattering, which shows that we can represent this system with
the accuracy needed for future scattering calculations. Such cross sections are of interest since this is a way to form antihydrogen.
(iii) A search for possible resonances in the pμe system, which has been suggested as a possible reason for unexpected
results from a recent measurement of the proton radius in muonic hydrogen. We have ruled out the possibility of such
In all calculations we used the Couple Rearrangement Channel Method, where the wave function is represented by
Gaussians expressed in Jacobi coordinates. Thus effects due to mass polarization are automatically.