Wave Packet In Harmonic Oscillator Example

This example illustrates the use of the qdyn library to simulate simple wave packet dynamics. A Gaussian wave packet is placed in a harmonic potential, where it oscillates.

Running the Example

You must compile the example by running make. Then, run the resulting harmonic program by executing

./harmonic r001

This will run the simulation and write the wave function to r001/psi.dat at regular time intervals. You may visualize the resulting wave packet dynamics by executing

cd r001
gnuplot psi.plt

This will create a PDF file r001/psi.pdf which contains a plot for all wave functions (absolute square). Flipping through the pages of the PDF provides an animated visualization of the dynamics.

Moreover, the expectation values for position and momentum are written to r001/exp_values.dat, and the energy expectation values to r001/exp_energies.dat.

Note that there is also an Jupyter Notebook in the example folder that illustrate how to run the example in a Python environment. An online version of the notebook can be found here:

Background

The behavior of a harmonic wave packet in a harmonic oscillator is well-known analytically (cf. [1, Chapter 3.3]).

When looking at \(|\Psi|^2\) of the wave packet with an initial Gaussian shape

\[\Psi(x) = \frac{1}{(\sqrt{\pi} \sigma)^\frac{1}{2}} \exp\left[ -\frac{(x-x_0)^2}{2 \sigma^2} \right] \exp\left[ i k_0 (x - x_0) \right]\]

in a harmonic potential

\[V(x) = \frac{1}{2} m \omega^2 x^2\]

the shape will always stay Gaussian, but the expectation value of \(x\) will oscillate according to

\[x_t = x_0 \cos \omega t + \frac{\hbar k_0}{m \omega} \sin \omega t\,.\]

along with the width of the Gaussian shape.

Details

In this example, The harmonic potential has the parameters \(\omega = 10\), \(m = 1\), and is centered around zero.

The initial wave packet is centered at \(x = 4\) and with a width of \(\sigma = 1\) and has zero momentum.

The simulation shows the oscillation of the center and width of the wave packet’s Gaussian shape over the time period from 0 to 1.

Bibliography

  • [1] D. J. Tannor, Introduction to Quantum Mechanics: A Time-Dependent Perspective (University Science Books, Sausalito, California, 2007).