Expansion and Time Development using Eigenstates

by Cemal Yalabik, Bilkent University, Physics Department. January 2001

This applet demonstates how an arbitrary function may be expanded in terms of the eigenfunctions of any one-dimensional quantum mechanical system. Since one knows the time development of the individual eigenfunctions, time-development of the arbitrary function (now an "initial contition") can be determined from the superposition. Just follow the steps below:

1. The red curve at the at the top of the applet window is the one dimensional potential experienced by the quantum particle. A box with finite walls is provided as a default, but you can move the cursor to that area and by pushing the left mouse button, modify this curve.
2. The u(x) curves are the 9 lowest energy eigenfunctions of the potential. For a one dimensional problem they can be constructed so that they are all real. Note that some higher energy eigenfunctions spill outside the box.

If you cannot see anything in the applet area, just sccribble something (with the mouse, right button pressed) at the very top of the applet area.
3. The energy eigenstates are indicated on the upper right hand side of the window, on the same scale as the potential.
4. The circles and the bars on the right hand side of the box indicate the phase and magnitude of the complex expansion coefficients for the arbitrary function plotted at the bottom of the window. Initially all phases are zero because all functions are real.
5. The lowest green graph is an arbitrary wavefunction, which will serve as an initial condition for time-development. A Gaussian function is given as a default, but you can modify it by placing the cursor in that region, and plotting while pressing the left mouse button. A white curve is generated, which is a result of the expansion in terms of the eigenstates. Notice how the expansion coefficients change as you modify the curve, and also note that the fit is not perfect (and is worse for non-smooth functions) because only a finite nuber of eigen-states are used. 6. Push the "Go" button to see the time-development of the arbitrary initila condition. Notice that the magnitudes of the expansion coefficients do not change, but their phases change as - E_n t / hbar. The phases of the higher energy eigenfunctions will therefore advance more rapidly.

You can stop the simulation by pressing the "Stop" button. Modifying the potential or the arbitrary intial wavefunction will also stop the time-development, but this will also reset the simulation time variable to zero, unlike pushing the "Stop" button.

1. The red curve at the at the top of the applet window is the one dimensional potential experienced by the quantum particle. A box with finite walls is provided as a default, but you can move the cursor to that area and by pushing the left mouse button, modify this curve. 2. The u(x) curves are the 9 lowest energy eigenfunctions of the potential. For a one dimensional problem they can be constructed so that they are all real. Note that some higher energy eigenfunctions spill outside the box.	If you cannot see anything in the applet area, just sccribble something (with the mouse, right button pressed) at the very top of the applet area.	3. The energy eigenstates are indicated on the upper right hand side of the window, on the same scale as the potential. 4. The circles and the bars on the right hand side of the box indicate the phase and magnitude of the complex expansion coefficients for the arbitrary function plotted at the bottom of the window. Initially all phases are zero because all functions are real.
	5. The lowest green graph is an arbitrary wavefunction, which will serve as an initial condition for time-development. A Gaussian function is given as a default, but you can modify it by placing the cursor in that region, and plotting while pressing the left mouse button. A white curve is generated, which is a result of the expansion in terms of the eigenstates. Notice how the expansion coefficients change as you modify the curve, and also note that the fit is not perfect (and is worse for non-smooth functions) because only a finite nuber of eigen-states are used.	6. Push the "Go" button to see the time-development of the arbitrary initila condition. Notice that the magnitudes of the expansion coefficients do not change, but their phases change as - E_n t / hbar. The phases of the higher energy eigenfunctions will therefore advance more rapidly.