Lecture 10 Angular Momentum Source: D. Griffiths, Introduction to Quantum Mechanics (Prentice Hall, 2004) R. Scherrer, Quantum Mechanics An Accessible Introduction (Pearson Int’l Ed., 2006) R. Eisberg & R. Resnick, Quantum Physics of Atoms, Molecules, Solids, Nuclei and Particles (Wiley, 1974)

Topics Today Angular Momentum Angular Momentum Operators Raising and Lowering Operators Spherical Harmonics

Angular Momentum Operators Square of Total Angular Momentum Show that L2 commutes with Lx: Use:

Angular Momentum Operators where Therefore each component commutes with L2, so the square of the angular momentum and any component can be simultaneously known.

General Properties of Orbital Angular Momentum Note that the magnitude of L is while the maximum value of Lz is This means that the magnitude of L is greater than any component can be. The reason is this: If Lz were the same as the magnitude of L, then both Lx and Ly would have to be zero. But no two components can be simultaneously known, so they cannot both be zero. (The only exception is when all components are zero, i.e., L is zero.)

Angular Momentum Diagram For l=2 Arrows represent possible angular momenta in units of Length = Allowed values for m are the z-components. m= -2, -1, 0, 1, 2 Except for trivial case l = 0.

Problem 1

SUPERPOSITION OF EIGENFUNCTIONS OF ANGULAR MOMENTUM . is the probability of finding when these two variables are both measured. When only is measured the probability that l = 3 is the sum of all seven possible values of

To determine :

Question 12: Part (a)

Question 12: Part (b)

Question 12: Part (c)

PROBLEM 2

L+: Raising operator, increases eigenvalue of Lz by The commutator with Lz is So is an eigenfunction of Lz with new eigenvalues L+: Raising operator, increases eigenvalue of Lz by L-: Lowering operator, decreases eigenvalue of Lz by

Repeat the application of RAISING AND LOWERING OPERATORS Example: Use the raising/lowering operators to obtain the (Unnormalized) eigenfunctions For l=2 and m=2,1,0. SOLUTION: To get apply the operator once. Repeat the application of So that

Problem 3

Problem 4