# Chapter 6: Correlation Functions and Spectra

For a bound-state problem, we can expand a given wavepacket in the eigenfunctions of the problem:

The "spectrum", is then given by

where i.e. the spectrum is simply the sum of the absolute squares of the energy components of the wavepacket, positioned at the eigenvalue energies. We can also compute the spectrum from the Fourier transform of the wavepacket autocorrelation function:

The "spectrum", is then given by

where i.e. the spectrum is simply the sum of the absolute squares of the energy components of the wavepacket, positioned at the eigenvalue energies. We can also compute the spectrum from the Fourier transform of the wavepacket autocorrelation function:

- Autocorrelation function and spectrum of a displaced Gaussian in a harmonic potential (analytic)
- Autocorrelation function and spectrum of a displaced Gaussian in a harmonic potential (numeric)
- Spectral method for calculating eigenstates, by constructing the eigenstate as a Fourier transform of the wavepacket at energy E.
- Spectral method for calculating eigenstates in a 2-d anharmonic potential
- Relaxation method for calculating eigenstates
- Adiabatic switching method for calculating eigenstates
- Fourier Grid Hamiltonian method for calculating eigenstates
- Pseudospectral method for calculating eigenvalues