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Approximation of the Propagator for the Harmonic Oscillator

This project investigates the accuracy of certain approximate propagators for the harmonic oscillator potential. The numerical investigation compares the approximate propagators to the exact propagator, known for this case. The goal is to test the accuracy of the first-order propagator approximation compared to the actual propagator.

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Approximation of the Propagator for the Harmonic Oscillator

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  1. Approximation of the Propagator for the Harmonic Oscillator By Logan Thomas Advisor: Dana Fine

  2. Abstract •   The propagator, which is a function of an initial position, a final position and a time, completely determines the quantum mechanics of a point particle in a given potential. The approximate propagators are explicitly calculated for the case of the harmonic oscillator potential. This project presents numerical investigation of the accuracy of certain approximate propagators for various values of N and t. This is feasible for this case, as the exact propagator is known.

  3. The Problem • The propagator gives a probability amplitude of a point particle moving to a new place at a given time. • The actual Propagator for a 1 dimensional particle is known • The experimental first order Propagator was derived by N. Makri and W. Miller

  4. Problem cont. • My Job: • Test the accuracy of the first order propagator approximation compared to the actual propagator.

  5. Task 1: Simplification • Actually proved to be very hard, figuring out how to put these two equations into MATLAB.

  6. Simplification (cont.) • As specified in the paper the proposed approximation is valid when: • My final equations after the simplifications were:

  7. Task 2: Matlab • Running an error analyses: • ((experimental – actual)/actual)*100 • The paper showed their error plot as being:

  8. Current State • At this current moment, My MATLAB program, is producing this graph for the error: • Obviously there is an error in my MATLAB program somewhere.

  9. Future Plans • Iron out whatever problems/figure out the source of the problem in MATLAB • Find the actual % error of their approximation as N gets larger. • Attempt to make the approximations better. • Also try for different values of x.

  10. References • Nancy Makri, William H Miller, Correct short term propagator for feynman Path integration by power series expansion in $/delta t$. Department of Chemistry, University of California. (1988)

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