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

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16-Dec-2015

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• Slide 1
• Applying the Correspondence Principle to the Three-Dimensional Rigid Rotor David Keeports Mills College dave@mills.edu
• Slide 2
• 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
• Slide 3
• The Instructional Challenge in Presenting the Correspondence Principle Consider obviously classical systems and show that they are really quantum systems
• Slide 4
• 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
• Slide 5
• Particle in 1-Dimensional Box uniform probability distribution from x = 0 to x = L
• Slide 6
• Particle in 3-Dimensional Box uniform probability distribution within 3-dimensional box
• Slide 7
• Harmonic Oscillator probability is enhanced at turning points
• Slide 8
• 2-Dimensional Rigid Rotor
• Slide 9
• In each case as a quantum number increases by 1, System energy appears to be a continuous function, i.e., quantization not evident
• Slide 10
• Consider a rigid rotor of binary star dimensions rotating in xy-plane A Classical Three-Dimensional Rigid Rotor
• Slide 11
• Assume both masses are solar masses M and separation is constant at r = 10 AU
• Slide 12
• Slide 13
• 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!
• Slide 14
• Eigen-Operators for 3-D Rigid Rotors Why Are Quantum Numbers Large?
• Slide 15
• Eigenvalues of Operators Spherical Harmonics Are Eigenfunctions
• Slide 16
• For assumed orbit in the xy-plane, angular momentum and its z-component are virtually indistinguishable, so
• Slide 17
• The Size of J = M J Large!
• Slide 18
• Energy and the Correspondence Principle Suppose that J increases by 1: Energy quantization unnoticed
• Slide 19
• Rotor Orientation From Spherical Harmonic Wavefunctions
• Slide 20
• Slide 21
• Because
• Slide 22
• probability is proportional to No is favored
• Slide 23
• 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
• Slide 24
• The Hydrogen Atom Problem in the Large Quantum Number Limit: Consider Earth-Sun System
• Slide 25
• Results for Quantum Earth
• Slide 26
• Assumed circular orbit implies consistent with correspondence principle
• Slide 27
• With, implies that Earths 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)
• Slide 28
• End