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
in a harmonic potential
the shape will always stay Gaussian, but the expectation value of \(x\) will oscillate according to
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).