טכניון מכון טכנולוגי לישראל
הטכניון מכון טכנולוגי לישראל - בית הספר ללימודי מוסמכים  
M.Sc Thesis
M.Sc StudentHarari Gal
SubjectQuantum Mechanical Analysis of Harmonic Oscillator with
Time-Dependent Parameters with Application to Ion
Trap
DepartmentDepartment of Physics
Supervisors Professor Jacob Ben-Aryeh
Professor Ady Mann
Full Thesis textFull thesis text - English Version


Abstract

This research is yet another treatment of the time-dependent harmonic oscillator (HO). The HO is a fundamental model in physics. It occurs whenever one approximates a potential near an equilibrium point. The solution to the simplest case (constant mass and frequency) goes back to the beginning of quantum mechanics and plays a crucial role in numerous applications since. One notable application of the time-dependent HO is the ion-trap which is based on a sinusoidal spring constant.

Our method is based on Glauber's treatment of the ion-trap potential. The devised solution is claimed to be the simplest possible closed-form solution. A complete set of wave-functions and propagator of the time-dependent HO is presented, including the linear terms at no further complexity. The solution is elegant and simple and depends only on the classical equation of motion without the linear terms, i.e. a linear, homogeneous second order equation.

The devised solution enables calculation of the propagator in many cases inaccessible to date due to the complexity of contemporary methods (unitary transformations, Feynman path integrals, Lewis and Riesenfeld's Invariant method). In particular the method has been applied in the current research to the periodical spring-constant (without the need for perturbation theory) including the ion-trap potential, forced HO and forced free particle. Transition amplitudes for the general case have also been computed and in particular for the case of periodical spring constant.