Chapter 1: The Time Dependent Schrödinger Equation

(Animation) A. — A. The oscillating wavepacket. Notice that at periodic times τ the wavepacket returns to the left side of the box, while at half periods it is on the right side.

The Actual Formul — B. For a bound-state problem, we can expand a given wavepacket in the eigenfunctions of the problem:
We show here the “spectrum”, defined by

(IMG) C. — C. We can also compute the spectrum from the Fourier transform of the autocorrelation function: The figure shows the autocorrelation function.

The movie shows the evolution of a particle initially in the ground state of half a box. At t=0, the box is extended to twice its size, and the particle, represented by a superposition of eigenstates of the full box, begins to oscillate. We define the “spectrum”, The Actual Formul

as

where

The spectrum is simply the sum of the absolute squares of the energy components of the wavepacket, positioned at the eigenvalue energies. The spectrum can also be computed from the Fourier transform of the wavepacket autocorrelation function: The Actual Formul