Physics – Quantum Physics
Scientific paper
2003-07-09
Am. J. Phys. 71, 541 (2003)
Physics
Quantum Physics
46 pages, 6 embedded .eps figures
Scientific paper
10.1119/1.1538574
We present quasi-analytical and numerical calculations of Gaussian wave packet solutions of the Schr\"odinger equation for two-dimensional infinite well and quantum billiard problems with equilateral triangle, square, and circular footprints. These cases correspond to N=3, N=4, and $N \to \infty$ regular polygonal billiards and infinite wells, respectively. In each case the energy eigenvalues and wavefunctions are given in terms of familiar special functions. For the first two systems, we obtain closed form expressions for the expansion coefficients for localized Gaussian wavepackets in terms of the eigenstates of the particular geometry. For the circular case, we discuss numerical approaches. We use these results to discuss the short-time, quasi-classical evolution in these geometries and the structure of wave packet revivals. We also show how related half-well problems can be easily solved in each of the three cases.
Doncheski Michael A.
Heppelmann Steven
Robinett R. W.
Tussey D. C.
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