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Quantum Mechanics of Angular Momentum

Dive into the realm of quantum angular momentum, covering topics such as classical vs. quantum properties, spherical polar coordinates, ladder operators, eigenvalues/eigenfunctions, spherical harmonics, and Legendre polynomials. Discover the commutation properties, measurement methods, and detailed calculations for understanding rigid rotator systems in three dimensions.

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Quantum Mechanics of Angular Momentum

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  1. Quantum Mechanicsof Angular Momentum • Classical Angular Momentum • Quantum Mechanical Angular Momentum • Spherical Polar Coordinates • Ladder Operators • Eigenvalues / Eigenfunctions • Spherical Harmonics • Legendre Polynomials /Associated Legendre functions • Rigid Rotator

  2. Classical Angular Momentum p r

  3. Quantum Mechanical Angular Momentum

  4. Commutation Properties

  5. Commutation Properties

  6. Commutation Properties

  7. Quantum Mechanical Angular Momentum Can measure one component and magnitude simultaneously

  8. Spherical Polar Coordinates

  9. Spherical Polar Coordinates Chain rule

  10. Spherical Polar Coordinates

  11. Spherical Polar Coordinates Do not depend on r

  12. Ladder Operators

  13. Ladder Operators Produces new eigenfunction with eigenvalue Step-up operator Produces new eigenfunction with eigenvalue Step-down operator

  14. Ladder Operators commute

  15. Eigenvalues/Eigenfunctions

  16. Eigenvalues/Eigenfunctions For a given a there is a max and min b

  17. ^ Eigenvalues of Lz are symmetric about 0 Eigenvalues/Eigenfunctions n odd n even Not physically meaningful

  18. Spherical Harmonics Find eigenfunctions same way as for Harmonic Oscillator

  19. Legendre Polynomials /Associated Legendre functions

  20. Legendre Polynomials /Associated Legendre functions

  21. Legendre Polynomials /Associated Legendre functions

  22. Spherical Harmonics

  23. r1 m1 R r2 m2 Rigid Rotator Eigenfunctions are spherical harmonics

  24. Rigid Rotator

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