This java applet is a quantum mechanics simulation that shows the behavior of a particle in a two dimensional harmonic oscillator.

At the top of the screen, you will see a cross section of the potential, with the energy levels indicated as gray lines. The red line is the expectation value for energy.

In the center of the applet, you will see the probability distribution of the particle's position. The color indicates the phase. At the bottom of the screen is a set of phasors showing the magnitude and phase of some of the possible states.

You may select a single state by picking one of the phasors at the bottom and double-clicking on it. Or, you may click on the phasor and drag its value to modify the magnitude and phase. In this way, you can create a combination of states. When you move the mouse over a state, it is highlighted in yellow, and other states with the same energy are also highlighted.

You may create a gaussian wave packet by clicking and dragging on the position graph.

Between each graph is a horizontal line which may be dragged up and down to adjust the size of each graph.

The Mouse Popup determines what happens when the mouse is clicked. The choices are:

• Create Gaussian: this allows you to create a gaussian distribution on either the position or momentum graph. Click on the graph to locate the center of the gaussian. After clicking, drag the mouse to select the size of the distribution.
• Gaussian w/ Momentum: this allows you to create a gaussian distribution on the position graph, with some initial momentum. Click on the graph to locate the center of the gaussian. The size of the gaussian will be the same size as the last time you used Create Gaussian. After clicking, drag the mouse to select the initial momentum. Note that if you drag too far, the gaussian may not be formed properly.
• Rotate Function: this allows you to rotate the wave function with the mouse. This changes the state of the particle to a new state which is at a different orientation. States with nx+ny > 13 cannot be rotated; this may cause a high-energy particle's position distribution to abruptly change shape when you attempt to rotate it. Also, if the aspect ratio is not set to 1, then rotation won't work.
• Translate Function: this allows you to move the wave function around with the mouse. If you take a stationary state and translate it slightly, then it will oscillate about its original position. If you move a state too close to the edge of the screen, then it may not translate properly.
• Scale Function: this allows you to resize the wave function around with the mouse. If you take a stationary state and scale it slightly, then it will oscillate about its original size. If you try to scale a state too large or too small, then it may not work properly.

The Clear button clears out all states.

The Normalize button normalizes the set of particle states. (By default, the states are not shown normalized because the interface is easier to use if they are not. They are normalized internally when calculating the wave functions, however.)

The Maximize button changes the magnitude of the particle states so that they are all as large as possible. This makes them easier to see. (It won't change the wave function at all because the states are normalized internally.)

The Ground State button selects the ground state wave function.

The Stopped checkbox stops the evolution of the wave function.

The Alternate Rendering checkbox is used to speed up rendering, but it actually slows things down on some machines. (Internally, it uses the MemoryImageSource class instead of drawing a bunch of rectangles.) It should be set to the proper setting by default, but try selecting or unselecting it to see if it speeds things up.

The Simulation Speed slider changes the speed of the wave function evolution.

The Brightness slider controls the brightness, just like on a TV set.

The Resolution slider changes the resolution of the applet. There are only two possible resolutions.

The Aspect Ratio slider will adjust the strength of the "spring constant" in the x direction as compared to the y direction.

The View Menu has the following items:

• Energy: show the energy/potential graph (on by default)
• Position: show the position graph (can't be turned off)
• Linear Momentum: show the linear momentum graph
• Angular Momentum: show the angular momentum graph. This is only available if the aspect ratio is 1. States with nx+ny > 13 do not appear on the angular momentum graph.
• Rectangular State: show the set of rectangular state phasors (on by default)
• Angular State: show the set of angular state phasors, allowing you to pick states by angular momentum. This is only available if the aspect ratio is 1.
• Expectation Values: show expectation values as red lines
• Uncertainties: show uncertainties as blue lines to the left and right of the expectation value. The distance from the blue lines to the red line is the uncertainty.
• Wave Function: display the wave function in one of three ways: as a probability (magnitude squared), as a probability with the phase shown using colors, or as a magnitude with the phase shown using colors.

The Measure Menu has the following items:

• Measure Energy: take a measurement of the energy by picking a random set of states with the same energy (using the state coefficients to determine probability) and putting the particle in those states.
• Measure Angular Momentum: take a measurement of the angular momentum, picking a random set of states with the same m (using the state coefficients to determine probability) and putting the particle in those states. Only works if the aspect ratio is 1.

The Options Menu has the following items:

• Always Normalize: always normalize the state coefficients.
• Always Maximize: always maximize the state coefficients.

  If neither one of these items is selected, then the applet will maximize the state coefficients whenever major changes are made, and leave them alone otherwise.

On the right side of the phasor panel, you will see a phasor all by itself. This phasor's rotation has the same period as the oscillation of the wave function. Create a gaussian to see this. It is only present if the aspect ratio is square.

If you like this applet you may be interested in the book Visual Quantum Mechanics.