1. Chapter 6
Angular Momentum

2. 6.B.1
Definition
In classical mechanics the angular momentum of a point mass relative to some axis is defined as:
In quantum mechanics the orbital angular momentum is defined by applying a quantization operation to the classical expression – replacing classical physical quantities with the corresponding observables (operators):
This definition does not require symmetrization with respect to non-commuting operators, e.g.:

3. 6.B.1
Definition
For a system of (spinless) particles the total orbital angular momentum is defined as:

4. Commutation relations
6.B.1
Commutation relations
Let us consider:
Similarly, one can obtain:
Thereby:

5. Total angular momentum
6.B.2
Total angular momentum
Previously we introduced the spin:
Let us consider:
Similarly, one can obtain:

6. Total angular momentum
6.B.2
Total angular momentum
Thereby, the spin can be treated as an intrinsic (non-orbital) angular momentum
Combination of the orbital and intrinsic angular momenta is the total angular momentum of the system J, defined via the commutation relations:
Let us introduce an operator:
This operator is Hermitian, and we will assume it is an observable

7. Total angular momentum
6.B.2
Total angular momentum
Let us consider:
Similarly, one can obtain:
Thereby:

8. Total angular momentum
6.B.2
Total angular momentum
What is the physical meaning of the commutation relations?
It is impossible to measure simultaneously the three components of the angular momentum, however, J2 and any component of J are compatible and could be measured simultaneously
Therefore, there is a possibility to find simultaneous eigenstates of J2 and any component of J (e.g., Jz)

9. Total angular momentum
6.C.1
Total angular momentum
We introduce (non-Hermitian) operators:
Let us consider:
Also

10. Total angular momentum
6.C.1
Total angular momentum
We introduce (non-Hermitian) operators:
Let us consider:
Synopsizing:

11. Total angular momentum
6.C.1
Total angular momentum
Let us also calculate:
Similarly:
Therefore:

12. Eigenvalues Let us consider an eigenvalue problem:
Recalling the expression
Moreover

13. Eigenvalues Let us consider an eigenvalue problem:
This also can be written as:
And this also can be written as:

14. Eigenvalues Let us consider an eigenvalue problem:
This also can be written as:
And this also can be written as:

15. 6.C.1
6.C.2
Eigenvalues
Since:
On the other hand, the square of the eigenvalue of the z-component of the angular momentum cannot exceed the eigenvalue of its magnitude squared:
Therefore, there should be top and bottom “rungs” for integers m and n

16. Eigenvalues Let us assume that for the top “rung” the eigenstate is:
6.C.2
Eigenvalues
Let us assume that for the top “rung” the eigenstate is:
Then:
And:
Recall:

17. 6.C.2
Eigenvalues
Let us assume that for the bottom “rung” the eigenstate is:
Then:
And:
Recall:

18. Eigenvalues Combining: There are two solutions: Thereby:
j must be integer or half-integer

19. Eigenvalues Using: So:
6.C.2
Eigenvalues
Using:
So:
We will therefore use indices j and m to label the eigenstates common to J2 and Jz
However, J2 and Jz do not necessarily constitute a CSCO, thus we introduce a third index k to distinguish between different eigenstates corresponding to the same j and m

20. Eigenvalues Synopsizing:
6.C.2
Eigenvalues
Synopsizing:
We thus have found the eigenvalues of the angular momentum
What are the eigenstates?

21. Eigenstates Let us recall: And Thus Therefore
We need to calculate the a± constants

22. Eigenstates Since We have Calculating a matrix element
On the other hand

23. Eigenstates Since We have Calculating a matrix element
On the other hand

24. Eigenstates Since We have Calculating a matrix element
On the other hand

25. Eigenstates Since We have Calculating a matrix element
On the other hand

26. Eigenstates Therefore A possible solution:
6.C.3
Eigenstates
Therefore
A possible solution:
Calculating a matrix element
On the other hand

27. Eigenstates Therefore A possible solution:
6.C.3
Eigenstates
Therefore
A possible solution:
Assuming a normalized set of eigenstates
Similarly

28. Eigenstates We assumed that the considered eigenstates are normalized
In fact, it is practical to work with a complete orthonormal basis
Thus, we should require:
And the closure relation:

29. 6.C.3
Eigenstates
A matrix representation of such basis can be built using subspaces E (k,j) as follows:
It easy to show that:
E (k,j)
E (k’,j)
E (k’,j’) …
matrix
(2j + 1) × (2j + 1)
(2j’ + 1) × (2j’+ 1)

30. 6.C.3
Eigenstates
A matrix representation of such basis can be built using subspaces E (k,j) as follows:
It easy to show that:
E (k,j)
E (k’,j)
E (k’,j’) …
matrix
(2j + 1) × (2j + 1)
(2j’ + 1) × (2j’+ 1)

31. 6.C.3
Eigenstates
Using:

32. 6.C.3
Eigenstates
Using:

33. 6.C.3
Eigenstates
Matrices corresponding to subspaces E (k,j) depend on the value of j, which is determined by the specificity of the studied system
For example, when j = 1, the dimensionality of the matrices is (2j + 1) × (2j + 1) = 3 × 3
In this case:

34. 6.C.3
Eigenstates
Using the expression for matrix elements below:

35. Eigenstates Using the expression for matrix elements below: Using
6.C.3
Eigenstates
Using the expression for matrix elements below:
Using

36. Eigenstates Using the expression for matrix elements below: Using
6.C.3
Eigenstates
Using the expression for matrix elements below:
Using

37. Eigenproblem for orbital momentum
6.D.1
Eigenproblem for orbital momentum
We return to the orbital angular momentum of a spinless particle
Let us find relevant eigenstates in the r-representation
The Cartesian components of the orbital angular momentum operator:

38. Eigenproblem for orbital momentum
6.D.1
Eigenproblem for orbital momentum
It is convenient to work in spherical coordinates

39. Eigenproblem for orbital momentum
6.D.1
Eigenproblem for orbital momentum
Then

40. Eigenproblem for orbital momentum
6.D.1
Eigenproblem for orbital momentum
Then
And

41. Eigenproblem for orbital momentum
6.D.1
Eigenproblem for orbital momentum
Recall:
For the orbital angular momentum:
In the r-representation (and spherical coordinates):

42. Eigenproblem for orbital momentum
6.D.1
Eigenproblem for orbital momentum
In these equations r does not appear in the differential operators, so we will consider it as a parameter
Thus, the wavefunction can be written as:
We try separating the variables:

43. Eigenproblem for orbital momentum
6.D.1
Eigenproblem for orbital momentum
In these equations r does not appear in the differential operators, so we will consider it as a parameter
Thus, the wavefunction can be written as:
We try separating the variables:

44. Eigenproblem for orbital momentum
6.D.1
Eigenproblem for orbital momentum
In these equations r does not appear in the differential operators, so we will consider it as a parameter
Thus, the wavefunction can be written as:
We try separating the variables:

45. Eigenproblem for orbital momentum
6.D.1
Eigenproblem for orbital momentum
Then:

46. Eigenproblem for orbital momentum
6.D.1
Eigenproblem for orbital momentum
Then:

47. Eigenproblem for orbital momentum
6.D.1
Eigenproblem for orbital momentum
Then:

48. Eigenproblem for orbital momentum
6.D.1
Eigenproblem for orbital momentum
Then:
Since m is integer, l shoud be also an integer (not a half-integer)

49. Eigenproblem for orbital momentum
6.D.1
Eigenproblem for orbital momentum
We successfully separated variables but still need to solve this equation:
Solutions:
Here Pl are the Legendre polynomials:
Adrien-Marie
Legendre
(1752 –1833)

50. Eigenproblem for orbital momentum
6.D.1
Eigenproblem for orbital momentum
Legendre polynomials:
Adrien-Marie
Legendre
(1752 –1833)

51. Eigenproblem for orbital momentum
6.D.1
Eigenproblem for orbital momentum
The resulting solutions:
They have to be normalized
This yields:
Functions Y are called spherical harmonics

52. Eigenproblem for orbital momentum
6.D.1
Eigenproblem for orbital momentum
Spherical harmonics:

53. Eigenproblem for orbital momentum
6.D.1
Eigenproblem for orbital momentum
Spherical harmonics:

54. Eigenproblem for orbital momentum
6.D.1
Eigenproblem for orbital momentum
These harmonics constitute an orthonormal basis:
Any function of θ and ϕ can be expanded in terms of the spherical harmonics:

55. Eigenproblem for orbital momentum
6.D.1
Eigenproblem for orbital momentum
Spherical harmonics are functions with a definite parity:

56. Eigenproblem for orbital momentum
6.D.1
Eigenproblem for orbital momentum
The original eigenproblem:
In the r-representation it looks like:
Where

57. Eigenproblem for orbital momentum
6.D.1
Eigenproblem for orbital momentum
This equation:
In the r-representation becomes:
Since
One can write:

58. Eigenproblem for orbital momentum
6.D.1
Eigenproblem for orbital momentum
Therefore:
Orthonormalization relation:
In the r-prerestnation it yields:
Since

59. Properties of eigenstates
6.D.2
Properties of eigenstates
Let us calculate this average value
Similarly:

60. Properties of eigenstates
6.D.2
Properties of eigenstates
Since
Then:
Therefore

61. Properties of eigenstates
6.D.2
Properties of eigenstates

62. Properties of eigenstates
6.D.2
Properties of eigenstates
Similarly:
Therefore

63. Measurements Consider a particle with a wave function:
In can be expanded as:
The probability of finding in a simultaneous measurement of L2 and Lz values ћ2l(l+1) and ћm is

64. 6.D.2
Measurements
The probability of measurement of L2 with the value ћ2l(l+1) is
The probability of measurement of Lz with the value ћm is
It’s more convenient to use the following expansion:
Where

65. 6.D.2
Measurements
Then
And