The Causal Interpretation of the Triangular Quantum Billiard

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The quantum mechanical billiard is given by the Schrödinger equation with Dirichlet boundary condition. It describes a wave function of shape fixed at the boundary . The simplest quantum billiard, which has an analytic but nonseparable wave function, is the equilateral triangle of side . The causal interpretation of quantum mechanics is a nonlocal theory that would solve many of the paradoxes of quantum mechanics, such as the measurement problem, Schrödinger's cat problem, and the collapse of the wave function. In the causal interpretation, every particle has a definite position and momentum at all times, but the trajectories are not measurable directly. The quantum motion of the triangular quantum billiard shows a large variety of dynamical behavior ranging from regular motion over to mixed dynamics and then to chaotic behavior. In general, the appearance of chaos is associated with the existence of "nodal points", where becomes null and/or where the phase is singular. In our case, the chaotic character of the quantum mechanical orbits occurs when the frequencies are commensurable, but the amplitudes of the superposed eigenfunctions have a complex ratio. As an example, the superposition of two eigenstates and a constant phase factor is chosen:

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,

with and , where , , , and are arbitrary integers. If and ( an integer), an exponential divergence of neighboring trajectories appears, because the "moving nodal points" and the motion of the quantum particles depend sensitively on their initial conditions, which applies directly to the classical definition of chaos. If , ( an integer), , and , the first-order equations for the velocities become an autonomous system and the squared wave function becomes time-independent. Due to the periodicity of the wave function in space and the influence of the constant phase factor, the trajectories could leave the billiard regime. The initial starting positions (in the graphic the bigger points) for the quantum particles are near to a local maximum. The graphic shows the squared wave function, the trajectories, and the initial and actual position of quantum particles. The parameters of the system have to be chosen carefully, because of the singularities of the velocities and the large oscillations of the trajectories that can lead to very unstable trajectories.

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Contributed by: Klaus von Bloh (August 2009)
Open content licensed under CC BY-NC-SA


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Details

The energy eigenvalues and wave functions for the 2D equilateral triangular billiard can be derived from those of the infinite square well solutions of the 2D Schrödinger equation. In the simplest closed form (adopting , where is the mass), the eigenfunctions are

,

,

with the energy spectrum and with . By differentiation it is easy to show that any linear combination of satisfies Schrödinger's equation: .

References

W.-K. Li and S. M. Blinder, "Solution of the Schrödinger Equation for a Particle in an Equilateral Triangle," Journal of Mathematical Physics, 26(11), 1985 pp. 2784–2786.

M. A. Doncheski, S. Heppelmann, R. W. Robinett, and D. C. Tussey, "Wave Packet Construction in Two-Dimensional Quantum Billiards: Blueprints for the Square, Equilateral Triangle, and Circular Cases," American Journal of Physics, 71, (6), 2003 pp. 541–557.

C. Efthymiopoulos, C. Kalapotharakos, and G. Contopoulos, "Nodal Points and the Transition from Ordered to Chaotic Bohmian Trajectories," Journal of Physics A: Mathematical and Theoretical, 40, 2007 pp. 12945–12972.



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