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Applying the Correspondence Principle to the Three-Dimensional Rigid Rotor David Keeports Mills College dave@mills.edu

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Quantum Mechanical Correspondence Principle “Quantum systems appear to be classical when their quantum numbers are very large.” No system strictly obeys classical mechanics Instead, all systems are quantum systems, but …

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The Instructional Challenge in Presenting the Correspondence Principle Consider “obviously classical” systems and show that they are really quantum systems

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Correspondence Principle Applied to Fundamental Quantum Systems Particle in 1-Dimensional Box Particle in 3-Dimensional Box Harmonic Oscillator 2-Dimensional Rigid Rotor 3-Dimensional Rigid Rotor Hydrogen Atom

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Particle in 1-Dimensional Box uniform probability distribution from x = 0 to x = L

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Particle in 3-Dimensional Box uniform probability distribution within 3-dimensional box

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Harmonic Oscillator probability is enhanced at turning points

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2-Dimensional Rigid Rotor

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In each case as a quantum number increases by 1, System energy appears to be a continuous function, i.e., quantization not evident

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Consider a rigid rotor of binary star dimensions rotating in xy-plane A Classical Three-Dimensional Rigid Rotor

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Assume both masses are solar masses M and separation is constant at r = 10 AU

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But does this 3-D rotor really obey classical mechanics? No, it is a quantum system that only appears to obey classical mechanics because its quantum numbers are very large!

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Eigen-Operators for 3-D Rigid Rotors Why Are Quantum Numbers Large?

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Eigenvalues of Operators Spherical Harmonics Are Eigenfunctions

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For assumed orbit in the xy-plane, angular momentum and its z-component are virtually indistinguishable, so …

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The Size of J = M J Large!

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Energy and the Correspondence Principle Suppose that J increases by 1: Energy quantization unnoticed

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Rotor Orientation From Spherical Harmonic Wavefunctions

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Because

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probability is proportional to No is favored

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Localization of axis at a particular requires superposition of wavefunctions with a range of angular momentum values Uncertainty principle: Angular certainty comes at the expense of angular momentum certainty

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The Hydrogen Atom Problem in the Large Quantum Number Limit: Consider Earth-Sun System

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Results for Quantum Earth

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Assumed circular orbit implies consistent with correspondence principle

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With, implies that Earth’s spatial probability distribution is y 0x Earth is in a hydrogen-like orbital characterized by huge quantum numbers Quantum Mechanical Earth: Where Orbitals Become Orbits. European Journal of Physics, Vol. 33, pp. 1587-98 (2012)

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