Applying the Correspondence Principle to the Three-Dimensional Rigid Rotor

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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. . - PowerPoint PPT Presentation

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A Binary Star as a Quantum System

Applying the Correspondence Principle to the Three-Dimensional Rigid RotorDavid KeeportsMills Collegedave@mills.edu----- Meeting Notes (1/6/14 09:49) -----3-Dimensional Rigid RotorCorrespondence Principle

1Quantum MechanicalCorrespondence PrincipleQuantum systems appear to be classical when their quantum numbers are very large. No system strictly obeys classical mechanicsInstead, all systems are quantum systems, but ----- Meeting Notes (1/6/14 09:49) -----3-Dimensional Rigid RotorCorrespondence Principle

2The Instructional Challenge in Presenting the Correspondence PrincipleConsider obviously classical systems and show that they are really quantum systems----- Meeting Notes (1/6/14 09:49) -----3-Dimensional Rigid RotorCorrespondence Principle

3Correspondence Principle Applied to Fundamental Quantum SystemsParticle in 1-Dimensional BoxParticle in 3-Dimensional BoxHarmonic Oscillator2-Dimensional Rigid Rotor

3-Dimensional Rigid Rotor

Hydrogen Atom ----- Meeting Notes (1/4/14 12:17) -----I teach QM to chemistry majors and place considerable emphasis upon correspondence principle. These systems illustrate methods of QM, with hydrogen atom most important problem in chemistry.4Particle in 1-Dimensional Box

uniform probability distribution from x = 0 to x = L----- Meeting Notes (1/6/14 09:54) -----3-Dimensional Rigid RotorCorrespondence Principle

5Particle in 3-Dimensional Boxuniform probability distributionwithin 3-dimensional box

----- Meeting Notes (1/6/14 09:54) -----3-Dimensional Rigid RotorCorrespondence Principle

6Harmonic Oscillator

probability is enhanced at turning points----- Meeting Notes (1/6/14 09:54) -----3-Dimensional Rigid RotorCorrespondence Principle

72-Dimensional Rigid Rotor

----- Meeting Notes (1/6/14 09:54) -----3-Dimensional Rigid RotorCorrespondence Principle

8In each case as a quantum number increases by 1,

System energy appears to be a continuous function, i.e.,quantization not evident----- Meeting Notes (1/6/14 09:54) -----3-Dimensional Rigid RotorCorrespondence Principle

9Consider a rigid rotor of binary stardimensions rotating in xy-plane A Classical Three-Dimensional Rigid Rotor

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

----- Meeting Notes (1/4/14 12:17) -----Can calculate many classical properties.11

----- Meeting Notes (1/4/14 12:17) -----For emphasis, I am discussing a rigid rotor and not true binary star, which has potential energy.12But does this 3-D rotor really obey classical mechanics?No, it is a quantum system that only appears to obey classical mechanicsbecause its quantum numbers are very large!----- Meeting Notes (1/6/14 09:54) -----3-Dimensional Rigid RotorCorrespondence Principle

13Eigen-Operators for 3-D Rigid Rotors

Why Are Quantum Numbers Large?----- Meeting Notes (1/6/14 09:54) -----3-Dimensional Rigid RotorCorrespondence Principle

14Eigenvalues of Operators

Spherical Harmonics Are Eigenfunctions----- Meeting Notes (1/6/14 09:54) -----3-Dimensional Rigid RotorCorrespondence Principle

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

----- Meeting Notes (1/4/14 12:17) -----Hand gesture for angular momentum vectors.16The Size of J = MJ

Large!----- Meeting Notes (1/6/14 09:54) -----3-Dimensional Rigid RotorCorrespondence Principle

17Energy and the Correspondence PrincipleSuppose that J increases by 1:

Energy quantization unnoticed----- Meeting Notes (1/6/14 09:54) -----3-Dimensional Rigid RotorCorrespondence Principle

18

Rotor Orientation From Spherical Harmonic Wavefunctions----- Meeting Notes (1/4/14 12:21) -----P's are associated Legendre functions.19

----- Meeting Notes (1/6/14 09:54) -----3-Dimensional Rigid RotorCorrespondence Principle

20

BecauseTheta for the orientation of the rotor and theta for the direction of the angular momentum vector differ by pi/2. If the rotor rotates in the xy-plane, the first angle is pi/2 and the second angle is 0. 21

probability is proportional to Df No f is favored

----- Meeting Notes (1/6/14 09:54) -----3-Dimensional Rigid RotorCorrespondence Principle

22Localization of axis at a particular f requires superposition of wavefunctions with a range of angular momentum valuesUncertainty principle: Angular certainty comes at the expense of angular momentum certainty23The Hydrogen Atom Problem in the Large Quantum Number Limit:Consider Earth-Sun System

----- Meeting Notes (1/6/14 09:54) -----3-Dimensional Rigid RotorCorrespondence Principle

24Results for Quantum Earth

----- Meeting Notes (1/6/14 09:54) -----3-Dimensional Rigid RotorCorrespondence Principle

25

Assumed circular orbit impliesconsistent with correspondence principle----- Meeting Notes (1/6/14 09:54) -----3-Dimensional Rigid RotorCorrespondence Principle

26With

,

implies that Earths spatial probability distribution is

y0xEarth is in a hydrogen-like orbital characterized by huge quantum numbersQuantum Mechanical Earth: Where Orbitals Become Orbits. European Journal of Physics, Vol. 33, pp. 1587-98 (2012)

----- Meeting Notes (1/6/14 09:54) -----3-Dimensional Rigid RotorCorrespondence Principle

27End----- Meeting Notes (1/6/14 09:54) -----3-Dimensional Rigid RotorCorrespondence Principle

28

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