The tables summarize the important elements for treating and understanding angular momentum.

  1. Start with as a vector operator.
  2. Find its commutation rules with all other observables
  3. Choose to define the relevant eigenstates
    1. Normalize
  4. Introduce raising an lowering operators
  5. Find there commutation relationships and some reformulations of the other operators in terms of these new operators (e.g. )
    1. show that    explicitly commutes
  6. Show that are integers and that  
  7. Consider spherical coordinates as a more natural parameterization of the position states for cases with spherical symmetry. Review definitions.
  8. Express the angular momentum operators wrt states
  9. Find the eigenstates in this representation  spherical harmonics
    1. Separation of variables (assume: sol=)
    2. Legendre’s equation in
    3. Simple  differential equation

 

1, 2, 3

7

8

4,5

 

9

6

 

9c

9a

9

Angular  solution: Spherical harmonics

l,m are integers

9a

General Legendre Equation

Solution: Associated Legendre Ploynomials

 

 

  1. Allow for two carriers of angular momentun
    1. two separate particles
    2. external and internal
  2. States must be labeled by both operators but these operators operate on different subspaces and therefore commute.
  3. Introduce the total angular momentum
  4. Introduce associated raising and lowering operators
  5. Show that J has the same algebra as L
    1. commutation relationships among Ji are same as Li.
    2.  have the same relationship as
  6.  should have eigenstates
  7. Recognize that there are other operators required to label the total angular momentum states because there are many ways that one can combine two vectors to reach a particular value of J. [Vector addition]
    1. Find commutation relationships between

                                                    i.     L components do not commute with J2

                                                  ii.     L2,S2,J2 commute

  1. Choose
    1.   or

10

11

12

 

13

14a,b

15

 

16

16

 

17a

17b

 

6  Are there restrictions on m values? YES

Therefore m has a maximum and a minimum value based on the total angular momentum. This is simply the generally accepted rule that a component can not be longer than the length of vector. A component is equal to the length when the vector and the component are aligned (parallel).

This leads to the constraint

 

In order for the extreme values to be reached using the raising and lowering operators which increment m by ± 1.  Either the state m=0 or m=± 1/2 must be included.

3-a Normalization

Commutation relationships for L,S components and J2

5a - Sum up both and get zero

 

Commutation relationships for L2, S2 components and J2

 

 

Commutation relationships for Li and L2