1. Quantum Two 1

2. 2

3. Angular Momentum and Rotations 3

4. Tensor Commutation Relations 4

5. In the last segment, we found that the product of two tensors is itself a tensor, that in general is reducible into two or more tensors of lower rank. We then proved a tensor product reduction theorem, based entirely on similarity to the angular momentum addition theorem, that tells us explicitly how to reduce a product of spherical tensors into irreducible tensor components. In this segment we show that spherical tensors obey characteristic commutation relations with the components of the angular momentum operator, which reduce, when j = 0 to the scalar commutation relations that we have already derived, and when j = 1, to vector commutation relations. 5

6. In the last segment, we found that the product of two tensors is itself a tensor, that in general is reducible into two or more tensors of lower rank. We then proved a tensor product reduction theorem, based entirely on similarity to the angular momentum addition theorem, that tells us explicitly how to reduce a product of spherical tensors into irreducible tensor components. In this segment we show that spherical tensors obey characteristic commutation relations with the components of the angular momentum operator, which reduce, when j = 0 to the scalar commutation relations that we have already derived, and when j = 1, to vector commutation relations. 6

7. In the last segment, we found that the product of two tensors is itself a tensor, that in general is reducible into two or more tensors of lower rank. We then proved a tensor product reduction theorem, based entirely on similarity to the angular momentum addition theorem, that tells us explicitly how to reduce a product of spherical tensors into irreducible tensor components. In this segment we show that spherical tensors obey characteristic commutation relations with the components of the angular momentum operator, which reduce, when j = 0 to the scalar commutation relations that we have already derived, and when j = 1, to vector commutation relations. 7

8. Tensor Commutation Relations As with scalars and vectors, these commutation relations follow from the way that these operators transform under infinitesimal rotations. Recall that under an infinitesimal rotation an arbitrary operator Q is transformed into Thus, the m- th component of the spherical tensor is transformed by into 8

9. Tensor Commutation Relations As with scalars and vectors, these commutation relations follow from the way that these operators transform under infinitesimal rotations. Recall that under an infinitesimal rotation an arbitrary operator Q is transformed into Thus, the m- th component of the spherical tensor is transformed by into 9

10. Tensor Commutation Relations As with scalars and vectors, these commutation relations follow from the way that these operators transform under infinitesimal rotations. Recall that under an infinitesimal rotation an arbitrary operator Q is transformed into Thus, the m- th component of the spherical tensor is transformed by into 10

11. Tensor Commutation Relations As with scalars and vectors, these commutation relations follow from the way that these operators transform under infinitesimal rotations. Recall that under an infinitesimal rotation an arbitrary operator Q is transformed into Thus, the m- th component of the spherical tensor is transformed by into 11

12. Tensor Commutation Relations On the other hand, by definition, under any rotation is transformed into But for an infinitesimal rotation which implies that under this rotation... 12

13. Tensor Commutation Relations On the other hand, by definition, under any rotation is transformed into But for an infinitesimal rotation which implies that under this rotation... 13

14. Tensor Commutation Relations On the other hand, by definition, under any rotation is transformed into But for an infinitesimal rotation which implies that under this rotation... 14

15. Tensor Commutation Relations On the other hand, by definition, under any rotation is transformed into But for an infinitesimal rotation which implies that under this rotation... 15

16. Tensor Commutation Relations On the other hand, by definition, under any rotation is transformed into But for an infinitesimal rotation which implies that under this rotation... 16

17. Tensor Commutation Relations which gives Comparing this to our previous relation 17

18. Tensor Commutation Relations which gives Comparing this to our previous relation 18

19. Tensor Commutation Relations which gives Comparing this to our previous relation 19

20. Tensor Commutation Relations which gives Comparing this to our previous relation 20

21. Tensor Commutation Relations which gives Comparing this to our previous relation 21

22. Tensor Commutation Relations we deduce the general commutation relation: which relates the commutator of any component of angular momentum with the component of any spherical tensor of rank j, to the matrix elements of that component of angular momentum in any standard representation with total angular momentum quantum number j. Since this holds for any component of the angular momentum, it clearly holds for any Cartesian component, or any spherical components. 22

23. Tensor Commutation Relations we deduce the general commutation relation: which relates the commutator of any component of angular momentum with the component of any spherical tensor of rank j, to the matrix elements of that component of angular momentum in any standard representation with total angular momentum quantum number j. Since this holds for any component of the angular momentum, it clearly holds for any Cartesian component, or any spherical components. 23

24. Tensor Commutation Relations we deduce the general commutation relation: which relates the commutator of any component of angular momentum with the component of any spherical tensor of rank j, to the matrix elements of that component of angular momentum in any standard representation with total angular momentum quantum number j. Since this holds for any component of the angular momentum, it clearly holds for any Cartesian component, or any spherical components. 24

25. Thus for the z component of angular momentum we have : or which look like the eigenvalue equation for satisfied by a state of angular momentum. In a similar fashion we find that for the raising and lowering operators,... 25

26. Thus for the z component of angular momentum we have : or which look like the eigenvalue equation for satisfied by a state of angular momentum. In a similar fashion we find that for the raising and lowering operators,... 26

27. Thus for the z component of angular momentum we have : or which look like the eigenvalue equation for satisfied by a state of angular momentum. In a similar fashion we find that for the raising and lowering operators,... 27

28. Thus for the z component of angular momentum we have : or which look like the eigenvalue equation for satisfied by a state of angular momentum. In a similar fashion we find that for the raising and lowering operators,... 28

29. which reduces to which looks like the action of on a state a state of angular momentum. 29

30. which reduces to which looks like the action of on a state a state of angular momentum. 30

31. which reduces to which looks like the action of on a state a state of angular momentum. 31

32. which reduces to which looks like the action of on a state a state of angular momentum. 32

33. Thus, what we will refer to as tensor commutation relations are the expressions and which provide necessary and sufficient conditions for the operators to transform as the components of a spherical tensor of rank j. In the next segment we use these commutation relations to prove the Wigner- Eckart theorem. 33

34. Thus, what we will refer to as tensor commutation relations are the expressions and which provide necessary and sufficient conditions for the operators to transform as the components of a spherical tensor of rank j. In the next segment we use these commutation relations to prove the Wigner- Eckart theorem. 34

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