ISSN:
1089-7658
Source:
AIP Digital Archive
Topics:
Mathematics
,
Physics
Notes:
We compute the Fourier transform (ρ(L)) of the quantum mechanical energy level density for the problem of a particle in a two-dimensional circular infinite well (or circular billiard) as well as for several special generalizations of that geometry, namely the half-well, quarter-well, and the circular well with a thin, infinite wall along the positive x-axis (hereafter called a circular well plus baffle). The resulting peaks in plots of |ρ(L)|2 versus L are compared to the lengths of the classical closed trajectories in these geometries as a simple example of the application of periodic orbit (PO) theory to a billiard or infinite well system. We then solve the Schrödinger equation for the general case of a circular well with infinite walls both along the positive x-axis and at an arbitrary angle aitch-theta (a circular "slice") for which the half-well (aitch-theta=π), quarter-well (aitch-theta=π/2), and circular well plus baffle (aitch-theta=2π) are then all special cases. We perform a PO theory analysis of this general system and calculate |ρ(L)|2 for many intermediate values of aitch-theta to examine how the peaks in ρ(L) attributed to periodic orbits change as the quasi-circular wells are continuously transformed into each other. We explicitly examine the transitions from the half-circular well to the circle plus baffle case (half-well to quarter-circle case) as aitch-theta changes continuously from π to 2π (from π to π/2) in detail. We then discuss the general aitch-theta→0 limit, paying special attention to the cases where aitch-theta=π/2n, as well as deriving the formulae for the lengths of closed orbits for the general case. We find that such a periodic orbit theory analysis is of great benefit in understanding and visualizing the increasingly complex pattern of closed orbits as aitch-theta→0.© 1998 American Institute of Physics.
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1063/1.532314
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