We present here some representative animations showing both the short-term, semi-classical, and longer-term revival structure of Gaussian-like wave packets in a two-dimensional infinite square well or 2-D square billiard. The theoretical background, and details of the wave packet construction are contained in the paper "Wave packet construction in two-dimensional billiards: blueprints for the square, equilateral triangle, and circular cases" by M. A. Doncheski, S. Heppelmann, R. W. Robinett, and D. C. Tussey (submitted to The American Journal of Physics.)

For most of the animations, we include plots of the time-dependent

• the position-space probability density, |psi(x,y,t|² versus x,y, (upper left)
• the momentum--space probability density, |phi(p_x,p_y,t|² versus p_x, p_y, (upper right)
• the autocorrelation function (|A(t)|² versus t) (lower left) and
• the quantum mechanical expectation values versus

• A Gaussian wave packet corresponding to p_y/p_x = 1/2, over the first three classical periods. Note how...
  • the autocorrelation function returns to something close to unity, at each classical period, but not quite due to quantum mechanical spreading.
  • the quantum mechanical expectation value follows the classical trajectory (denoted by the straight lines), except near the 'collisoins' with the walls, and
  • the impulsive change in momentum values also corresponding to 'collisions'.
• The same quantities, but now over n entire revival period.
  • Note the exact revival and the appearance of obvious fractional at T_rev/4 and 3T_rev/4

• Plots corresponding to a Gaussian wave packet characterized by p_y/p_x = 2/3, over various time intervals, looking at the fine detail of the time-dependence:
• Periods 0-4, showing the short-term, semi-classical dynamics.
• Periods 24-28, showing the one of the 'quarter' revivals.
• Periods 36-40, showing the one of the 'third' revivals.
• Periods 52-56, showing the reformation of the wave packet, going in the 'wrong' direction in momentum-space, out of phase with the initial packet.

Copyright D. C. Tussey and R. W. Robinett (2002)