1. 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)

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

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

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

6. 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.)

7. 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.

8. Problem 1

9. 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

10. To determine :

12. Question 12:
Part (a)

13. Question 12:
Part (b)

14. Question 12:
Part (c)

15. PROBLEM 2

16. 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

18. 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

20. Problem 3

21. Problem 4