Quantum Addition of Angular Momenta

1. Chapter 10 Addition of Angular Momenta

2. 10.A.1 10.A.2 Total angular momentum • Let us recall the definition of the classical angular momentum of a system of particles: • What is the quantum equivalent of the total angular momentum of the system in which there are parts with their own angular momenta? • The answer: • What are the eigenvalues and the eigenstates associated with this observable?

3. 10.B.1 Addition of two spin-½’s • Let us start with the simplest nontrivial case of two angular momenta being added, each of spin ½: • The state space is 4D: • The orthonormal basis: • These are the eigenstates of this set of operators: • It comprises a CSCO

4. 10.B.1 Addition of two spin-½’s • The eigneproblem: • S is also an angular momentum operator: • Similarly: • So:

5. 10.B.1 Addition of two spin-½’s • Another important operator is:

6. 10.B.1 Addition of two spin-½’s • Another important operator is:

7. 10.B.1 Addition of two spin-½’s • Important commutation relations: • What about this commutator?

8. 10.B.1 Addition of two spin-½’s • Let us calculate: • Since

9. 10.B.1 Basis change • We just showed that these observables commute: • Since • This set is not the same as the CSCO • Is this a CSCO?: • In other words, is there a new basis that will satisfy these equations?

10. 10.B.2 Basis change • Since • One can write: • Thus: • I.e.,

11. 10.B.2 Basis change • So:

12. 10.B.3 Basis change • Let us recall: • Then:

13. 10.B.3 Basis change • Let us recall: • Then:

14. 10.B.3 Basis change • Let us recall: • Then:

15. 10.B.3 Basis change • Let us recall: • Then:

16. 10.B.3 Basis change • Synopsizing: • Thus:

17. 10.B.3 Basis change • Let us diagonalize this matrix: • Secular equation: • Roots: + -

18. 10.B.3 Basis change • Normalizing:

19. 10.B.3 Basis change • Thus, we found all four eigenstates: • The corresponding values of M: • Thus the corresponding values of S:

20. 10.B.4 Basis change • So, we successfully obtained the new orthonormal basis: • The first three states with S = 1 are called a triplet, and the fourth state with S = 0 is called a singlet

21. 10.B.4 Basis change • So, we successfully obtained the new orthonormal basis: • As a result of the basis change the subspace allocation became different:

22. 10.C.4 Basis change • The same results can be obtained using the ladder operators • Recall:

23. 10.C.4 Basis change • The same results can be obtained using the ladder operators • Recall:

24. 10.C.4 Basis change • On the other hand • Similarly:

25. 10.C.2 Case of two arbitrary angular momenta • Let us consider addition of two angular momenta:

26. 10.C.2 Case of two arbitrary angular momenta • The state space is: • Each subspace in the tensor sum has a dimension of (2j1 + 1)·(2j2 + 1)

27. 10.C.2 Case of two arbitrary angular momenta • J is also an angular momentum operator: • Similarly: • So: • Moreover:

28. 10.C.2 Case of two arbitrary angular momenta • Important commutation relations:

29. 10.C.2 Basis change • The eigneproblem: • We just showed that these observables commute: • Since • This set is not the same as: • Is there a new state set satisfying these equations?

30. 10.C.2 Basis change • If this is the case, the following tensor sum can be considered: • The following questions have to be answered: • 1) What are the values of J for a given pair of j1 and j2? • 2) How can the eigenvectors of J2 and Jz be expanded via the basis of eigenvectors of Ji2 and Jiz?

31. 10.C.3 Eigenvalues of J2 and Jz • Let us assume that • Thus • Since • One can write • What is the degeneracy of these values?

32. 10.C.3 Eigenvalues of J2 and Jz • Consider a case

33. 10.C.3 Eigenvalues of J2 and Jz • Consider a case

34. 10.C.3 Eigenvalues of J2 and Jz • Consider a case

35. 10.C.3 Eigenvalues of J2 and Jz • Consider a case

36. 10.C.3 Eigenvalues of J2 and Jz • Consider a case

37. 10.C.3 Eigenvalues of J2 and Jz • The following dependence is obvious:

38. 10.C.3 Eigenvalues of J2 and Jz • The following dependence is obvious:

39. 10.C.3 Eigenvalues of J2 and Jz • The following dependence is obvious:

40. 10.C.3 Eigenvalues of J2 and Jz • Thus, for given j1 and j2, the eigenvalues of J2 are: • One can associate a single invariant subspace E (J) with each of these values • For each value of J and each value of M, compatible with it, there is only one state vector in E (j1, j2) • Therefore, J2 and Jz comprise a CSCO in E (j1, j2)

41. 10.C.4 Eigenstates of J2 and Jz • As a result: • Let us start with the first subspace E (j1 + j2) • Setting: • Application of the lowering operator should yield:

42. 10.C.4 Eigenstates of J2 and Jz

43. 10.C.4 Eigenstates of J2 and Jz

44. 10.C.4 Eigenstates of J2 and Jz • This procedure can be repeated by further applying the lowering ladder operator until reaching • The same approach can be used for • Here, the new maximum value of M is: • The corresponding eigenvector should be proportional to

45. 10.C.4 Eigenstates of J2 and Jz • Assuming • With • It should be orthogonal to: • Thus: • Solution:

46. 10.C.4 Eigenstates of J2 and Jz • Thereby: • The rest of the states can be obtained by a repeated application of the lowering ladder operator until reaching • Then, the tensor sum is considered: • Etc.

47. 10.C.4 Clebsch-Gordan coefficients • Recall that inside a given space E (j1, j2) this set is a basis: • It is complete, thus the closure relation applies: • Using it we can write down the following expansion:

48. 10.C.4 Clebsch-Gordan coefficients • These coefficients are called Clebsch-Gordan coefficients Paul Albert Gordan (1837 – 1912) Rudolf Friedrich Alfred Clebsch (1833 – 1872)

49. 10.C.4 Clebsch-Gordan coefficients • Conventionally, the Clebsch-Gordan coefficients are chosen to be real • On the other hand, since this set is a complete basis: • The closure relation applies: • Thus:

50. 10.C.4 Clebsch-Gordan coefficients • The Clebsch-Gordan coefficients can be calculated by iteration using the ladder operators • Usually they are compiled in a tabular form: