M.Sc Student | Harari Gal |
---|---|

Subject | Quantum Mechanical Analysis of Harmonic Oscillator with Time-Dependent Parameters with Application to Ion Trap |

Department | Department of Physics |

Supervisors | Professor Jacob Ben-Aryeh |

Professor Ady Mann | |

Full Thesis text |

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.