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

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