An exploration of techniques for determining propagators In quantum mechanics

Abstract

In this report, we will build the foundation for the understanding of the propagator in an attempt to search for a correspondence between the classical and quantum mechanics (QM). We will derive the propagator ($K$) from both spectral representation and path integral, and observe the role of the classical Lagrangian in the solution. At the same time, the Quantum Harmonic Oscillator (QHO) was used as a illustrative example for which the propagators from both methods were shown to be equal. The idea of a Fourier transformed propagator ($\tilde{K}$) will also be explored, where it is shown that the transform complex function's residues are eigenfunctions and its simple poles are its bound state energies. Given that understanding, we will verify a given $\tilde{K}$ for the non-harminic P\"{o}schl Teller (PT) potential against the numerical solution of the spectral representation using MAPLE. Furthermore, motivated by the similar shape of the potential between the QHO and the PT potential, we will verify that the PT potential system will tend to a QHO in the appropriate limit. Lastly, we will extend our understanding of propagators into Supersymmetric Quantum Mechanics (SUSY QM), where we will seek for a possible relationship between the pair propagators for partner potentials, and checked its validity for the QHO.