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Section 10.8: Exploring Wave Packet Revivals with Classical Analogies
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In Section 10.7 we saw that quantum wave packets in the infinite square well revive (reform at their original position and momentum with the exact same shape they had at t = 0). In Eq.(10.15) we stated the equation for the revival time, but what does this equation mean? In this Exploration we will give two two ways in which we can visualize this behavior.9
In the first set of animations, the ones with the racers, a number of objects (cars, runners, etc.) race around a track. For simplicity, we allow the racers to pass each other by going through another racer. The angular frequency of each racer is different in a special way: the angular frequency is an integer squared times the angular speed of the slowest racer. In the second set of animations, the ones with the arrows, a number of arrows in the complex plane (phasors) indicate the phase (from each e−iEnt/ħ contribution) of a particular state in the infinite square well are shown. Their lengths are fixed and do not represent amplitude. The angular frequency of each phasor is different in a special way: the angular frequency is an integer squared times the angular speed of the slowest racer. Such a depiction is often called a phase clock.
Answer the following questions for both the racer and phasor animations.
- How long does it take the slowest racer and the slowest phasor to return to its t = 0 position? What do these times signify?
- As you increase the number of racers and phasors, describe how the racers and phasors move relative to each other.
- Explain in your own words why a wave packet in the infinite square well revives.
9This Exploration is based in part on R. W. Robinett's talk, "Quantum Wave Packet Revivals" given at the 128th AAPT National Meeting, Miami Beach, FL, Jan. 24-28, 2004.