Exercises for Quantum Revivals:

Position Expectation Value

For simplicity, set the energy scale of the animations to 2/π as this recasts the time in terms of the revival time.  Now all wave packets in the infinite square well of length L = 1 will revive at t = 1.
Consider the pre-set animations of wave packets with p0 = 40π. 
Calculate the classical period for a particle of mass m = 0.5 in an infinite square well of length L = 1 and p0 = 40π. 
What would the trajectory of the classical particle look like (x vs. t)?
The classical time scale for the quantum-mechanical case is the same as for the classical case.  Give the scaling we did above (set the energy scale of the animations to 2/π), what is the new classical time scale?
Do these packets remain "classical" packets for the classical time scale?
Can you see fractional revivals for p0 = 40π?
If so, how does the number of mini-packets you can see depend on the initial packet width?  At what time do these mini-packets occur? Is there a relationship between the number of mini-packets and the time they occur (as a fraction of the revival time)?

 

Questions? Contact us at mabelloni@davidson.edu or wochristian@davidson.edu.
© Mario Belloni and Wolfgang Christian (2004).