טכניון מכון טכנולוגי לישראל
הטכניון מכון טכנולוגי לישראל - בית הספר ללימודי מוסמכים  
Ph.D Thesis
Ph.D StudentNarevicius Edvardas
SubjectNon Hermitian Quantum Mechanics: Fingerprints of Overlapping
Resonances in Observable Quantities
DepartmentDepartment of Chemistry
Supervisor Professor Emeritus Nimrod Moiseyev


Abstract

A question whether fingerprints of broad overlapping resonances maybe observed in measurable quantities has been raised a long time ago.

There are claims that broad overlapping resonances serve only as a basis set to expand the time evolving initial state of a quantum system. Using the non Hermitian quantum mecganucs we show in this thesis that there are particular physical systems where the contribution of broad overlapping resonances may be seen in the

experimental measurements. For example, under certain conditions  of initial state preparation the absorption cross section for the photodissociation of ArHCl van der Waals complex exhibits peak structure associated with broad resonance states.


Another example which illustatres the fact that broad overlapping resonances are observable is the electron scattering from the hydrogen molecule. The  experimental H­2­ vibrational excitation cross sections show a sharp

structure which was associated with H-2 resonance states. However the theoretical calculations have shown that the lifetime of these metastable states is small. The puzzle of how the metastable states with such a short lifetime could be observed in the experiment remained unsolved untill now. In this thesis we show that the sharp structure results from the interference effect between the broad overlapping resonances.


In principle the sharp structures in the hydrogen vibrational excitation  can be obtained using the conventional Hermitian quantum mechanics. However the usual Born-Oppenheimer approximation brakes down in this system. Consequently one should take into account both the nuclear and the electronic degrees of freedom in the numerical calculations.

This is impossible task due to the current limitations on the computing power. Within the non Hermitian

quantum mechanics the computational effort is enormously reduced. Two decay channels (vibrational dissociation and electron autoionization) can be included in the calculations by the use of a single complex

 electronic potential surface with its complex part proportional to the autoionization rate. We have developed a complex analogue to the Born-Oppenheimer approximation and used it successfully in order to explain

the sharp structures in the excitation cross section.


The problem of a sudden delocalization of the bacteria population in convective media can only be described by the means of non Hermitian diffusion equation. In the full contrast to the previous case the non Hermiticity here is an  inherent property. It has been shown by Nelson from Harvard University that the sudden delocalization of the bacteria population happens for the critical value of the convection velocity. We have observed that

in the periodical systems the delocalization is accompanied by the formation of the branch point in the complex eigenvalue plain of the non Hermitian diffusion operator. At this point the corresponding eigenfunctions

become self-orthogonal and the standard deviation of any observable that does not commute with the Hamiltonian diverge.