23. Angular Momentum Operators

Question 1

What are the eigenfunctions of the wave function?

Answer 1

The spherical harmonics

Question 2

How is the z component of the angular momentum quantised?

Answer 2

In integer multiples of h_bar with integers of m_l running from -l to +l

Question 3

Which component is L_z a function of in the spherical coordinate system

Answer 3

Φ

Question 4

What is the consequence if m_l is equal to 0?

Answer 4

The spherical harmonic does not depend on Φ, and vanishes upon differentiating

Question 5

What is the consequence if m_l is NOT equal to 0?

Answer 5

Get a factor of i m_l coming down when differentiating.

• End up with + h_bar in front of the original function
• These are eigenfunctions of L_z operator, with eigenvalues of h_bar m_l

Question 6

See page 15 for the relationship between the angular momentum operators

Answer 6

Do it

Question 7

Describe how the L_x, L_y, and L_z differ

Answer 7

There is a cyclic permutation of the indicies

Question 8

Are two individul angular momentum operators compatible, and what does this mean?

Answer 8

No - Cannot know both of them simultaneously and precisely

Question 9

Are L^2 and L_z compatible (or L_y or L_z)?

Answer 9

Yes - Can know them precisely and simultaneously

Question 10

See book for the derivation involving the angular momentum ladder operators

Answer 10

If u want

Question 11

See page 16 for an overview of the angular momentum ladder operator

Answer 11

do it

Question 12

Does the ladder operator and L^2 commute?

Answer 12

Yes

Question 13

Describe how the ladder operator works

Answer 13

It raises or lowers the z component of the orbital angular momentum by h bar