Superoscillations in the Quantum Harmonic Oscillator

A superoscillatory function is one that oscillates faster than its fastest Fourier component - A phenomenon created by subtle interference between Fourier components. In this thesis, the work of Professor Sir Michael Berry [1, 2] and Professor Sandu Popescu on superoscillations in free space [2] has been applied to the case of the quantum harmonic oscillator. Superoscillations in free space are seen to persist in time longer than expected and are also seen to reform periodically after times much greater than their disappearance - for the quantum harmonic oscillator superoscillations reform periodically faster. The time of disappearance, td, depends upon a larger set of variables, this is due to the added complexity of the quantum harmonic oscillator. The evolution of the wavepacket is investigated using an expansion in terms of eigenfunctions and in terms of the propagator using both an exact integration and a saddle point approximation. The creation and disappearance of superoscillations is shown to depend on the behaviour of the saddle points. The frequency of the quantum harmonic oscillator, ! is found to be an extra control parameter which dictates the strength and duration of the superoscillations.