Quantum-mechanical Superposition Exercises

 

Infinite Square Well Revivals shows four initially localized Gaussian wave packets in an infinite square well.  The energy scale, which determines the time scale for the animation, is set 2/π which forces Trev = 1.  This is a convenient time scale for the study of revivals and fractional revivals which occur at Trev and fractions of Trev, respectively.  The time evolution of these states are shown in position and momentum space.  In addition, one can choose to look at the expectation value of x and p and also the position-space quantum carpet.

 

For simplicity, the energy scale of the animations to 2/π as this recasts the time in terms of the revival time.  All wave packets in the infinite square well of length L = 1 will revive at t = 1.

Consider the pre-set animations of increasing initial momentum of the wave packet with m = 0.5, L = 1, x0 = 0.5.  Choose a particular visualization depending on the question asked.

 

~~1. Calculate the classical period for a particle of mass m = 0.5 in an infinite square well of length L = 1 and p0 = 80π. 

 

~~2. What would the trajectory of the classical particle look like (x vs. t)?

 

~~3. What does the trajectory of the wave packet look like (<x> vs. t) as p0 increases?  Is there any noticeable correlation with the classical case?  If so, when?

 

~~4. What would p vs. t of the classical particle look?

 

~~5. What does <x> vs. t of the wave packet look like as p0 increases?  Is there any noticeable correlation with the classical case?  If so, when?

 

~~6. Can you see fractional revivals for p0 = 40π? for p0 = 80π?

 

~~7. 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)?