Demos.HarmOscillator.Gaussian1D
If a Gaussian wavepacket is chosen as an initial wavepacket, the wavefunction remains Gaussian (bell-) shaped at all later times. This is illustrated for the famous examples of coherent (Glauber) states and "squeezed" states using the numerical propagation schemes of WavePacket.
Equations of motion
A Gaussian wavepacket is an exact solution of the time-dependent Schrödinger equation for the harmonic oscillator potential. Hence, if you choose such an initial state, the wavefunction remains Gaussian shaped for all later times: both in position and in momentum space representation its center follows the corresponding classical trajectory. Analytical expressions exist also for the width of the Gaussian, see the seminal work by E. J. Heller.
Squeezed states
| Method | Visualization |
|---|---|
| Quant.-mech. wavefunction |
 |
| Classical trajectories |
 |
In general, the center of a Gaussian wavepacket oscillates according to the classical (e.g. Newton's or Hamilton's) equations of motion. This is a manifestation of Ehrenfest's theorem stating that the expectation values of a quantum mechanical wavefunction obey the laws of classical mechanics. However, both the width and the phase of the wavefunction vary in a complicated manner with time. Due to the interchange of minima and maxima of the width in position and momentum space representation, these states are referred to as squeezed states in the quantum optics community. The dynamics of width and phases give rise to two classes of quantum phenomena: (1) The finite width of the wavepacket implies that the edge of a wavepacket penetrates to a certain amount into the classically forbidden region (tunneling). (2) The (complex) phase of the wavepacket governs the interference of wavepackets and various kinds of spectroscopic properties.
Case 1
| Matlab version | C++ version |
|---|---|
| Animated wavepacket | Animated wavepacket |
| Animated trajectories | |
| Input data file | Input file and equivalent Python script |
| Logfile output | Logfile output |
Case 2
| Matlab version | C++ version |
|---|---|
| * Animation of evolving wavepacket | Animated wavepacket |
| * Animation of trajectory swarm | |
| * Input data file | Input file and equivalent Python script |
| * Logfile output | Logfile output |
Coherent (Glauber, quasi-classical) states
| Method | Visualization |
|---|---|
| Quant.-mech. wavefunction |
 |
| Classical trajectories |
 |
If the initial width of a quantum mechanical wavepacket exactly equals the width of the (stationary) ground state wavefunction of the harmonic oscillator, the following type of quantum dynamics is observed: Again, the wavepacket oscillates with its center following the classical equations of motion. In addition, the width of the WavePacket remains constant. This is because the curvature of the potential exactly compensates the tendency for wavepacket spreading (dispersion) observed, e.g., for a free particle dynamics. This type of quantum mechanics is the closest correspondence with classical mechanics. In the literature, this phenomenon is termed Glauber state, coherent state, or quasi-classical state.
| Matlab version | C++ version |
|---|---|
| * Animation of evolving wavepacket | Animated wavepacket |
| * Animation of trajectory swarm | |
| * Input data file | Input file and equivalent Python script |
| * Logfile output | Logfile output |