# The Caged Anharmonic Oscillator in the Causal Interpretation of Quantum Mechanics

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This Demonstration studies the caged anharmonic oscillator—fundamental for quantum field theory as well as many-body theory—in the causal interpretation of quantum mechanics developed by David Bohm. Among the large number of possible potentials for the Schrödinger equation, only a very few are analytically solvable. The potential is that of the harmonic oscillator with rational frequency ratio, but additionally with a barrier term from inclusion of a repulsive force [1].

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Contributed by: Klaus von Bloh (July 2012)

Open content licensed under CC BY-NC-SA

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## Details

The simplified Schrödinger equation is:

, with , and so on, where is a complex-valued function and are real-valued constants. Here the common standard notation is used, where is the time and is the spatial coordinate. The time-dependent unnormalized solution gives the wavefunction:

,

where are the associated Laguerre polynomials and the quantized energy is given by where is an integer. For simplicity, the parameters and are set equal to 1 in the quantum potential. If , then the wave density is time independent and the particles are at rest. The guiding equation for the particle velocity is , which is calculated from the gradient of the phase from the total wavefunction in the eikonal form . The quantum potential is given by . Due to limitations of CPU power, AccuracyGoal, PrecisionGoal, and MaxSteps are decreased. The starting positions of the particles are linearly distributed around the peaks of the wave density at

Reference

[1] N. W. Evans and P. E. Verrier, "Superintegrability of the Caged Anisotropic Oscillator," *Journal of Mathematical Physics*, 49(9), 2008 pp. 092902–092902-10. dx.doi.org/10.1063/1.2988133 or arxiv.org/abs/0808.2146.

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