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Quantum Mechanics of 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. Classical Angular Momentum.
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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
Quantum Mechanical Angular Momentum Can measure one component and magnitude simultaneously
Spherical Polar Coordinates Chain rule
Spherical Polar Coordinates Do not depend on r
Ladder Operators Produces new eigenfunction with eigenvalue Step-up operator Produces new eigenfunction with eigenvalue Step-down operator
Ladder Operators commute
Eigenvalues/Eigenfunctions For a given a there is a max and min b
^ Eigenvalues of Lz are symmetric about 0 Eigenvalues/Eigenfunctions n odd n even Not physically meaningful
Spherical Harmonics Find eigenfunctions same way as for Harmonic Oscillator
r1 m1 R r2 m2 Rigid Rotator Eigenfunctions are spherical harmonics