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Applying the Correspondence Principle to the Three-Dimensional Rigid Rotor

Applying the Correspondence Principle to the Three-Dimensional Rigid Rotor. David Keeports Mills College dave@mills.edu. Quantum Mechanical Correspondence Principle. “Quantum systems appear to be classical when their quantum numbers are very large.” .

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Applying the Correspondence Principle to the Three-Dimensional Rigid Rotor

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

  2. Quantum MechanicalCorrespondence 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 …

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

  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

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

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

  7. Harmonic Oscillator probability is enhanced at turning points

  8. 2-Dimensional Rigid Rotor

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

  10. Consider a rigid rotor of binary stardimensions rotating in xy-plane A Classical Three-Dimensional Rigid Rotor

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

  12. 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!

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

  14. Eigenvalues of Operators Spherical Harmonics Are Eigenfunctions

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

  16. The Size of J = MJ Large!

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

  18. Rotor Orientation From Spherical Harmonic Wavefunctions

  19. Because

  20. probability is proportional to Df No f is favored

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

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

  23. Results for Quantum Earth

  24. Assumed circular orbit implies consistent with correspondence principle

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

  26. End

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