PACKET

Construction of Wave Packets

This animation shows how wave packets are constructed. It also demonstrates the Heisenberg uncertainty principle

The program recognizes the following commands:

P Momentum mode. The program automatically starts in this mode, but you can return to it anytime by pressing P. This mode illustrates the relationship between momentum and wavelength of a particle. At the top is shown a wave. The horizontal axis is position x. Below the wave is shown the momentum of the particle. The horizontal axis is momentum p. The horizontal position of the vertical line is the momentum of the particle represented by the wave. If you press the space bar, the position of the vertical line moves to the right, indicating that the momentum of the particle is increasing. Since the wavelength of the particle is inversely proportional to its momentum, the wavelength of the wave decreases. Press the space bar again to hold the momentum constant. Press the space bar yet again to cause the momentum to decrease. Repeatedly pressing the space bar causes the momentum to stop changing or to start changing. If the momentum is stopped to the right of the center of the screen, pressing the space bar will cause it to decrease. If the momentum is stopped to the left of the center of the screen, pressing the space bar will cause it to increase. Moving the momentum back and forth and watching how the wave changes should help you better understand the relationship between momentum and wavelength.
B Beats mode. Before you press B, think about what would happen if two wave with different wavelengths are superimposed. Now press B. Two vertical lines appear on the momentum axis, indicating two waves with two different wavelengths. The superposition of these two waves produce "beats". In regions where the two waves are in phase, they constructively interfere and the amplitude is large. In regions where the two waves are out of phase, they destructively interfere and the amplitude is small.
1 First wave packet. Now we construct a wave packet. Each time you press the space bar, an additional wave with a different momentum and wavelength will be added to the wave displayed on the screen. All of these waves constructively interfere in the region near the center of the screen. If we add together enough of them, they destructively interfere outside of this center region. The length of the vertical lines indicate the relative amplitudes of the waves added together. Each time a new wave is added, the amplitude of the wave displayed on the screen is scaled so that its amplitude at the center of the screen remains constant. If you hold down the space bar, new waves will be added more rapidly. The resulting wave packet represents the particle. The position of the particle is somewhere within the bounds of the packet and is therefore uncertain. The momentum of the particle is also uncertain since the packet is made up of many waves with different wavelengths.
2 Second wave packet. When you press 2, the bottom half of the screen becomes red, and a second x and p graph appears. The top half of the screen remains as it was. Each time you press the space bar, an additional wave with a different momentum and wavelength will be added to the wave displayed on the red half of the screen, just like the first packet. This packet, however, uses a broader distribution in momentum. The result is a narrower packet. This illustrates the Heisenberg uncertainty principle. Decreasing the uncertainty in the position of a particle requires that the uncertainty in the momentum be increased.
Q Quit program.
? Help. These commands are listed.

P	Momentum mode. The program automatically starts in this mode, but you can return to it anytime by pressing P. This mode illustrates the relationship between momentum and wavelength of a particle. At the top is shown a wave. The horizontal axis is position x. Below the wave is shown the momentum of the particle. The horizontal axis is momentum p. The horizontal position of the vertical line is the momentum of the particle represented by the wave. If you press the space bar, the position of the vertical line moves to the right, indicating that the momentum of the particle is increasing. Since the wavelength of the particle is inversely proportional to its momentum, the wavelength of the wave decreases. Press the space bar again to hold the momentum constant. Press the space bar yet again to cause the momentum to decrease. Repeatedly pressing the space bar causes the momentum to stop changing or to start changing. If the momentum is stopped to the right of the center of the screen, pressing the space bar will cause it to decrease. If the momentum is stopped to the left of the center of the screen, pressing the space bar will cause it to increase. Moving the momentum back and forth and watching how the wave changes should help you better understand the relationship between momentum and wavelength.
B	Beats mode. Before you press B, think about what would happen if two wave with different wavelengths are superimposed. Now press B. Two vertical lines appear on the momentum axis, indicating two waves with two different wavelengths. The superposition of these two waves produce "beats". In regions where the two waves are in phase, they constructively interfere and the amplitude is large. In regions where the two waves are out of phase, they destructively interfere and the amplitude is small.
1	First wave packet. Now we construct a wave packet. Each time you press the space bar, an additional wave with a different momentum and wavelength will be added to the wave displayed on the screen. All of these waves constructively interfere in the region near the center of the screen. If we add together enough of them, they destructively interfere outside of this center region. The length of the vertical lines indicate the relative amplitudes of the waves added together. Each time a new wave is added, the amplitude of the wave displayed on the screen is scaled so that its amplitude at the center of the screen remains constant. If you hold down the space bar, new waves will be added more rapidly. The resulting wave packet represents the particle. The position of the particle is somewhere within the bounds of the packet and is therefore uncertain. The momentum of the particle is also uncertain since the packet is made up of many waves with different wavelengths.
2	Second wave packet. When you press 2, the bottom half of the screen becomes red, and a second x and p graph appears. The top half of the screen remains as it was. Each time you press the space bar, an additional wave with a different momentum and wavelength will be added to the wave displayed on the red half of the screen, just like the first packet. This packet, however, uses a broader distribution in momentum. The result is a narrower packet. This illustrates the Heisenberg uncertainty principle. Decreasing the uncertainty in the position of a particle requires that the uncertainty in the momentum be increased.
Q	Quit program.
?	Help. These commands are listed.