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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<p>A Binary Star as a Quantum System</p> <p>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</p> <p>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</p> <p>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</p> <p>3Correspondence Principle Applied to Fundamental Quantum SystemsParticle in 1-Dimensional BoxParticle in 3-Dimensional BoxHarmonic Oscillator2-Dimensional Rigid Rotor</p> <p>3-Dimensional Rigid Rotor</p> <p>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</p> <p>uniform probability distribution from x = 0 to x = L----- Meeting Notes (1/6/14 09:54) -----3-Dimensional Rigid RotorCorrespondence Principle</p> <p>5Particle in 3-Dimensional Boxuniform probability distributionwithin 3-dimensional box</p> <p>----- Meeting Notes (1/6/14 09:54) -----3-Dimensional Rigid RotorCorrespondence Principle</p> <p>6Harmonic Oscillator</p> <p>probability is enhanced at turning points----- Meeting Notes (1/6/14 09:54) -----3-Dimensional Rigid RotorCorrespondence Principle</p> <p>72-Dimensional Rigid Rotor</p> <p>----- Meeting Notes (1/6/14 09:54) -----3-Dimensional Rigid RotorCorrespondence Principle</p> <p>8In each case as a quantum number increases by 1,</p> <p>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</p> <p>9Consider a rigid rotor of binary stardimensions rotating in xy-plane A Classical Three-Dimensional Rigid Rotor</p> <p>Assume both masses are solar masses M and separation is constant at r = 10 AU</p> <p>----- Meeting Notes (1/4/14 12:17) -----Can calculate many classical properties.11</p> <p>----- 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</p> <p>13Eigen-Operators for 3-D Rigid Rotors</p> <p>Why Are Quantum Numbers Large?----- Meeting Notes (1/6/14 09:54) -----3-Dimensional Rigid RotorCorrespondence Principle</p> <p>14Eigenvalues of Operators</p> <p>Spherical Harmonics Are Eigenfunctions----- Meeting Notes (1/6/14 09:54) -----3-Dimensional Rigid RotorCorrespondence Principle</p> <p>15For assumed orbit in the xy-plane, angular momentum and its z-component are virtually indistinguishable, so </p> <p>----- Meeting Notes (1/4/14 12:17) -----Hand gesture for angular momentum vectors.16The Size of J = MJ</p> <p>Large!----- Meeting Notes (1/6/14 09:54) -----3-Dimensional Rigid RotorCorrespondence Principle</p> <p>17Energy and the Correspondence PrincipleSuppose that J increases by 1:</p> <p>Energy quantization unnoticed----- Meeting Notes (1/6/14 09:54) -----3-Dimensional Rigid RotorCorrespondence Principle</p> <p>18</p> <p>Rotor Orientation From Spherical Harmonic Wavefunctions----- Meeting Notes (1/4/14 12:21) -----P's are associated Legendre functions.19</p> <p>----- Meeting Notes (1/6/14 09:54) -----3-Dimensional Rigid RotorCorrespondence Principle</p> <p>20</p> <p>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</p> <p>probability is proportional to Df No f is favored</p> <p>----- Meeting Notes (1/6/14 09:54) -----3-Dimensional Rigid RotorCorrespondence Principle</p> <p>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</p> <p>----- Meeting Notes (1/6/14 09:54) -----3-Dimensional Rigid RotorCorrespondence Principle</p> <p>24Results for Quantum Earth</p> <p>----- Meeting Notes (1/6/14 09:54) -----3-Dimensional Rigid RotorCorrespondence Principle</p> <p>25</p> <p>Assumed circular orbit impliesconsistent with correspondence principle----- Meeting Notes (1/6/14 09:54) -----3-Dimensional Rigid RotorCorrespondence Principle</p> <p>26With </p> <p>, </p> <p>implies that Earths spatial probability distribution is</p> <p>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) </p> <p>----- Meeting Notes (1/6/14 09:54) -----3-Dimensional Rigid RotorCorrespondence Principle</p> <p>27End----- Meeting Notes (1/6/14 09:54) -----3-Dimensional Rigid RotorCorrespondence Principle</p> <p>28</p>

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