C. Harmonic Eigenfunctions being created by the spectral method:
The eigenfunctions are obtained by calculating the Fourier transform of the
moving wavepacket at the appropriate energy.
The colors represent the phase of the state that is being extracted.
Normally one thinks of wavepackets as coherent superpositions of eigenstates. There is a fascinating converse relationship in which an eigenstate is viewed as a superposition of wavepackets. This can he seen from the following formula:
Note that the LHS is the eigenstate of H with eigenvalue The RHS
is the Fourier transform of the moving wavepacket,
from time to energy at the eigenvalue energy If we recognize that the integral is simply the continuous limit of a sum, then the integral over time on the RHS can be viewed as the sum of wavepackets,
at different times t, added with the phase factors,
The choice of phase factor, through the choice of determines which eigenstate
is created via the integral in eq. 1. If does not match any of the eigenvalues of H, the integral will give 0.