Quantum Mechanics of Angular Momentum Classical Angular Momentum

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 p r

Quantum Mechanical Angular Momentum

Commutation Properties

Commutation Properties

Commutation Properties

Quantum Mechanical Angular Momentum Can measure one component and magnitude simultaneously

Spherical Polar Coordinates

Spherical Polar Coordinates

Spherical Polar Coordinates

Spherical Polar Coordinates

Ladder Operators

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

Ladder Operators commute

Eigenvalues/Eigenfunctions

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

Eigenvalues/Eigenfunctions ^ are symmetric about 0 Eigenvalues of L z n even n odd

Spherical Harmonics Find eigenfunctions same way as for Harmonic Oscillator

Legendre Polynomials /Associated Legendre functions

Legendre Polynomials /Associated Legendre functions

Legendre Polynomials /Associated Legendre functions

Spherical Harmonics

Rigid Rotator m 1 r 1 R m 2 r 2 Eigenfunctions are spherical harmonics

Rigid Rotator