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Applying the Correspondence Principle to the Three-Dimensional Rigid Rotor David Keeports Mills College

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Presentation on theme: "Applying the Correspondence Principle to the Three-Dimensional Rigid Rotor David Keeports Mills College"— Presentation transcript:

1 Applying the Correspondence Principle to the Three-Dimensional Rigid Rotor David Keeports Mills College

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 …

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 star dimensions 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

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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!

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

15 Eigenvalues of Operators Spherical Harmonics Are Eigenfunctions

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

17 The Size of J = M J Large!

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

19 Rotor Orientation From Spherical Harmonic Wavefunctions

20

21 Because

22 probability is proportional to  No  is favored

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

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

25 Results for Quantum Earth

26 Assumed circular orbit implies consistent with correspondence principle

27 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 (2012)

28 End


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