PACKET

Construction of Wave Packets

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

1 Mode with 1 value of momentum. The program automatically starts in this mode, but you can return to it anytime by clicking on 1. This mode illustrates the relationship between the 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 click on +p, 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. If you click on -p, the position of the vertical line moves to the left. Moving the momentum back and forth and watching how the wave changes should help you better understand the relationship between momentum and wavelength.
2 Mode with 2 values of momentum. Before you click on 2, think about what would happen if two waves with different wavelengths are superimposed. Now click on 2. 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. Clicking on +p moves the two lines further apart, and clicking on -p moves the two lines closer together.
N Mode with N values of momentum. Now we construct a wave packet. Each time you click on add one, 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 click on add all, all of the remaining new waves will be automatically added one at a time. 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.
case2 Second wave packet. When you click on case 2, a second x and p graph appears at the bottom of the screen. Each time you click on add one, an additional wave with a different momentum and wavelength will be added to the wave displayed on the bottom 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. Clicking on case 1 returns you to the top half of the screen.

1	Mode with 1 value of momentum. The program automatically starts in this mode, but you can return to it anytime by clicking on 1. This mode illustrates the relationship between the 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 click on +p, 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. If you click on -p, the position of the vertical line moves to the left. Moving the momentum back and forth and watching how the wave changes should help you better understand the relationship between momentum and wavelength.
2	Mode with 2 values of momentum. Before you click on 2, think about what would happen if two waves with different wavelengths are superimposed. Now click on 2. 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. Clicking on +p moves the two lines further apart, and clicking on -p moves the two lines closer together.
N	Mode with N values of momentum. Now we construct a wave packet. Each time you click on add one, 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 click on add all, all of the remaining new waves will be automatically added one at a time. 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.
case2	Second wave packet. When you click on case 2, a second x and p graph appears at the bottom of the screen. Each time you click on add one, an additional wave with a different momentum and wavelength will be added to the wave displayed on the bottom 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. Clicking on case 1 returns you to the top half of the screen.