Demos.FreeParticle.Gaussian1D

For the special case of a Gaussian distribution of wavenumbers (momenta), one obtains a localized, Gaussian-shaped wavefunction in position space representation, too. This is due to the properties of the Fourier transformation connecting position and momentum space representation of wavefunctions.

Initial wavefunction

For a Gaussian wavepacket, the uncertainty of the position and momentum space wavefunctions are inversely proportional to each other. The product of the initial uncertainties equals the minimum required by Heisenberg's uncertainty relation δR · δP = ħ/2. Hence, such wavepackets are referred to as minimum uncertainty wavepackets.

Time evolution without initial momentum

Method	Visualization
Quant.-mech. wavefunction
Classical trajectories

As time evolves, a Gaussian wavepacket remains a Gaussian wavepacket both in position and momentum representation. In phase space representation the elliptic Wigner quasi-density starts to rotate which has a different effect on the two marginal distributions: While the momentum uncertainty remains unchanged, the position uncertainty is rapidly increasing (dispersion of the wavepacket). Hence, the uncertainty product monotonously increases in accord with Heisenberg's uncertainty relation: δR · δP ≥ ħ/2. Note that real and imaginary part of the corresponding wavefunction undergo rapid oscillations both in position/momentum and in time. The effect of dispersion can be easily understood considering that the initial wavepacket contains contributions of different (non-zero) momenta.

Matlab version	C++ version
Animation of evolving wavepacket	Animation
Animation of trajectory swarm
Input data file	Input file and equivalent Python script
Logfile output	Logfile output

Time evolution with non-vanishing initial momentum

Method	Visualization
Quant.-mech. wavefunction
Classical trajectories

This example beautifully demonstrates Ehrenfests principle stating that the expectation values of quantum mechanical observables obey the rules of classical mechanics. For the case of a free particle the expectation value of the momentum does not change in the absence of forces while the expectation value of position moves with constant velocity. Hence, the momentum density remains centered around $P\_0=0$ while the corresponding position density travels with constant velocity and becomes wider at the same time (wavepacket dispersion). Note that upon reaching the outer boundary of the spatial grid, the wavepacket is smoothly absorbed by the absorbing boundary condition (indicated by dashed vertical line).

Matlab version	C++ version
Animation of evolving wavepacket
Animation of trajectory swarm
Input data file	Input file and equivalent Python script
Logfile output	Logfile output