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arXiv · 1307.5979

The Large-Volume Limit of a Quantum Tetrahedron is a Quantum Harmonic Oscillator

John Schliemann
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Abstract

It is shown that the volume operator of a quantum tetrahedron is, in the sector of large eigenvalues, accurately described by a quantum harmonic oscillator. This result relies on the fact that (i) the volume operator couples only neighboring states of its standard basis, and (ii) its matrix elements show a unique maximum as a function of internal angular momentum quantum numbers. These quantum numbers, considered as a continuous variable, are the coordinate of the oscillator describing its quadratic potential, while the corresponding derivative defines a momentum operator. We also analyze the scaling properties of the oscillator parameters as a function of the size of the tetrahedron, and the role of different angular momentum coupling schemes.

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John Schliemann. 2013-10-14. The Large-Volume Limit of a Quantum Tetrahedron is a Quantum Harmonic Oscillator. https://doi.org/10.1088/0264-9381%2F30%2F23%2F235018

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