1. Quantum Two

3. Angular Momentum and Rotations

4. Angular Momentum and Rotations Clebsch-Gordon Coefficients

5. In the last segment we stated and gave a constructive proof of the Angular Momentum Addition Theorem which shows how the subspace S(j₁, j₂) = S(j₁) ⊗ S(j₂) formed from the direct product of two irreducible invariant subspaces can be reduced to a direct sum
S(j₁) ⊗ S(j₂) = S(j₁+ j₂) ⊕ S(j₁+ j₂-1) ⊕ ⋯ ⊕ S(|j₁ - j₂|)
of its own irreducible invariant subspaces.
The proof thus shows how to construct the basis vectors
{|j, m〉}
of a standard representation of angular momentum eigenstates of J² and Jz for such a space, as linear combinations of the direct product basis states
{|j₁, j₂ ,m₁ , m₂〉}
which are eigenvectors of J₁², J₂², J1z , J2z and Jz but not, generally, of J².

6. In the last segment we stated and gave a constructive proof of the Angular Momentum Addition Theorem which shows how the subspace S(j₁, j₂) = S(j₁) ⊗ S(j₂) formed from the direct product of two irreducible invariant subspaces can be reduced to a direct sum
S(j₁) ⊗ S(j₂) = S(j₁+ j₂) ⊕ S(j₁+ j₂-1) ⊕ ⋯ ⊕ S(|j₁ - j₂|)
of its own irreducible invariant subspaces.
In the course of the proof we saw how to construct the basis vectors
{|j, m〉}
of a standard representation of angular momentum eigenstates of J² and Jz for such a space, as linear combinations of the direct product basis states
{|j₁, j₂ ,m₁ , m₂〉}
which are eigenvectors of J₁², J₂², J1z , J2z and Jz but not, generally, of J².

7. In the last segment we stated and gave a constructive proof of the Angular Momentum Addition Theorem which shows how the subspace S(j₁, j₂) = S(j₁) ⊗ S(j₂) formed from the direct product of two irreducible invariant subspaces can be reduced to a direct sum
S(j₁) ⊗ S(j₂) = S(j₁+ j₂) ⊕ S(j₁+ j₂-1) ⊕ ⋯ ⊕ S(|j₁ - j₂|)
of its own irreducible invariant subspaces.
In the course of the proof we saw how to construct the basis vectors
{|j, m〉}
of a standard representation of angular momentum eigenstates of J² and Jz for such a space, as linear combinations of the direct product basis states
{|j₁, j₂ ,m₁ , m₂〉}
which are eigenvectors of J₁², J₂², J1z , J2z and Jz but not, generally, of J².

8. With an appropriate choice of phase, the expansion coefficients that appear when expanding the states |j, m〉 in terms of the direct product states |j₁, j₂, m₁, m₂〉, take a standard form that can be computed once and for all.
Thus, as with the standard matrices that we have already discussed, they can be compiled and tabulated for later use in any problem that involves combining two or more angular momenta.
For example, using the completeness relation for the direct product states in the space S(j₁) ⊗ S(j₂) allows us to write the new basis states in the form

9. With an appropriate choice of phase, the expansion coefficients that appear when expanding the states |j, m〉 in terms of the direct product states |j₁, j₂, m₁, m₂〉, take a standard form that can be computed once and for all.
Thus, as with the standard matrices that we have already discussed, they can be compiled and tabulated for later use in any problem that involves combining two or more angular momenta.
For example, using the completeness relation for the direct product states in the space S(j₁) ⊗ S(j₂) allows us to write the new basis states in the form

10. With an appropriate choice of phase, the expansion coefficients that appear when expanding the states |j, m〉 in terms of the direct product states |j₁, j₂, m₁, m₂〉, take a standard form that can be computed once and for all.
Thus, as with the standard matrices that we have already discussed, they can be compiled and tabulated for later use in any problem that involves combining two or more angular momenta.
For example, using the completeness relation for the direct product states in the space S(j₁) ⊗ S(j₂) allows us to write the new basis states in the form

11. Similarly, the direct product states can be written as linear combinations of the new basis states |j, m〉, i.e.,
These expansions are completely determined once we know the corresponding expansion coefficients
〈j₁, j₂, m₁, m₂| j, m〉
which are referred to as Clebsch-Gordon (CG) coefficients.
Different authors denote these expansion coefficients in different ways, e.g.,

12. Similarly, the direct product states can be written as linear combinations of the new basis states |j, m〉, i.e.,
These expansions are completely determined once we know the corresponding expansion coefficients
〈j₁, j₂, m₁, m₂| j, m〉 or 〈 j, m | j₁, j₂, m₁, m₂〉
which are referred to as Clebsch-Gordon (CG) coefficients.
Different authors denote these expansion coefficients in different ways, e.g.,

13. Similarly, the direct product states can be written as linear combinations of the new basis states |j, m〉, i.e.,
These expansions are completely determined once we know the corresponding expansion coefficients
〈j₁, j₂, m₁, m₂| j, m〉 or 〈 j, m | j₁, j₂, m₁, m₂〉
which are referred to as Clebsch-Gordon (CG) coefficients.
Different authors denote these expansion coefficients in different ways, e.g.,

14. Similarly, the direct product states can be written as linear combinations of the new basis states |j, m〉, i.e.,
These expansions are completely determined once we know the corresponding expansion coefficients
〈j₁, j₂, m₁, m₂| j, m〉 or 〈 j, m | j₁, j₂, m₁, m₂〉
which are referred to as Clebsch-Gordon (CG) coefficients.
Different authors denote these expansion coefficients in different ways, e.g.,

15. They are also sometimes expressed in terms of what are referred to as
They are also sometimes expressed in terms of what are referred to as Wigner’s 3-j symbols through the relation
It is straightforward, using the procedure outlined in the proof of the angular momentum addition theorem, to generate the CG coefficients for given values of j₁, j₂, and j.
They obey certain properties that follow from their definition and from the way in which they are constructed.
In this segment we enumerate some of these properties below.

16. They are also sometimes expressed in terms of what are referred to as
They are also sometimes expressed in terms of what are referred to as Wigner’s 3-j symbols through the relation
It is straightforward, using the procedure outlined in the proof of the angular momentum addition theorem, to generate the CG coefficients for given values of j₁, j₂, and j.
They obey certain properties that follow from their definition and from the way in which they are constructed.
In this segment we enumerate some of these properties below.

17. They are also sometimes expressed in terms of what are referred to as
They are also sometimes expressed in terms of what are referred to as Wigner’s 3-j symbols through the relation
It is straightforward, using the procedure outlined in the proof of the angular momentum addition theorem, to generate the CG coefficients for given values of j₁, j₂, and j.
They obey certain properties that follow from their definition and from the way in which they are constructed.
In this segment we enumerate some of these properties below.

18. They are also sometimes expressed in terms of what are referred to as
They are also sometimes expressed in terms of what are referred to as Wigner’s 3-j symbols through the relation
It is straightforward, using the procedure outlined in the proof of the angular momentum addition theorem, to generate the CG coefficients for given values of j₁, j₂, and j.
They obey certain properties that follow from their definition and from the way in which they are constructed.
In this segment we enumerate some of these properties below.

19. They are also sometimes expressed in terms of what are referred to as
They are also sometimes expressed in terms of what are referred to as Wigner’s 3-j symbols through the relation
It is straightforward, using the procedure outlined in the proof of the angular momentum addition theorem, to generate the CG coefficients for given values of j₁, j₂, and j.
They obey certain properties that follow from their definition and from the way in which they are constructed.
In this segment we enumerate some of these properties below.

20. They are also sometimes expressed in terms of what are referred to as
They are also sometimes expressed in terms of what are referred to as Wigner’s 3-j symbols through the relation
It is straightforward, using the procedure outlined in the proof of the angular momentum addition theorem, to generate the CG coefficients for given values of j₁, j₂, and j.
They obey certain properties that follow from their definition and from the way in which they are constructed.
In this segment we enumerate some of their more important properties.

21. Restrictions on j and m - It is clear from the proof of the addition theorem detailed above that the CG coefficient 〈j₁, j₂, m₁, m₂| j, m〉 must vanish unless the two states in the inner product have the same z component of total angular momentum.
In addition, we must have the value of total j on the right lie within the range produced by the angular momenta j₁ and j₂. Thus we have the restriction
〈j₁, j₂, m₁, m₂| j, m〉 = 0 unless m = m₁ + m₂ ,
and j₁ + j₂ ≥ j ≥ | j₁ - j₂|.
The restriction on j is referred to as the triangle inequality, since it is equivalent to the condition that the positive numbers j, j₁, and j₂ be able to represent the lengths of the three sides of some triangle.
It is easily shown to apply to any permutation of these three numbers, i.e., its validity also implies that j + j₂ ≥ j₁ ≥ | j - j₂| and that j₁ + j ≥ j₂ ≥ | j - j₁ |. j₂ j j₁

22. Restrictions on j and m - It is clear from the proof of the addition theorem detailed above that the CG coefficient 〈j₁, j₂, m₁, m₂| j, m〉 must vanish unless the two states in the inner product have the same z component of total angular momentum.
In addition, we must have the value of total j on the right lie within the range produced by the angular momenta j₁ and j₂. Thus we have the restriction
〈j₁, j₂, m₁, m₂| j, m〉 = 0 unless m = m₁ + m₂ ,
and j₁ + j₂ ≥ j ≥ | j₁ - j₂|.
The restriction on j is referred to as the triangle inequality, since it is equivalent to the condition that the positive numbers j, j₁, and j₂ be able to represent the lengths of the three sides of some triangle.
It is easily shown to apply to any permutation of these three numbers, i.e., its validity also implies that j + j₂ ≥ j₁ ≥ | j - j₂| and that j₁ + j ≥ j₂ ≥ | j - j₁ |. j₂ j j₁

23. Restrictions on j and m - It is clear from the proof of the addition theorem detailed above that the CG coefficient 〈j₁, j₂, m₁, m₂| j, m〉 must vanish unless the two states in the inner product have the same z component of total angular momentum.
In addition, we must have the value of total j on the right lie within the range produced by the angular momenta j₁ and j₂. Thus we have the restriction
〈j₁, j₂, m₁, m₂| j, m〉 = 0 unless m = m₁ + m₂ ,
and j₁ + j₂ ≥ j ≥ | j₁ - j₂|.
The restriction on j is referred to as the triangle inequality, since it is equivalent to the condition that the positive numbers j, j₁, and j₂ be able to represent the lengths of the three sides of some triangle.
It is easily shown to apply to any permutation of these three numbers, i.e., its validity also implies that j + j₂ ≥ j₁ ≥ | j - j₂| and that j₁ + j ≥ j₂ ≥ | j - j₁ |. j₂ j j₁

24. Restrictions on j and m - It is clear from the proof of the addition theorem detailed above that the CG coefficient 〈j₁, j₂, m₁, m₂| j, m〉 must vanish unless the two states in the inner product have the same z component of total angular momentum.
In addition, we must have the value of total j on the right lie within the range produced by the angular momenta j₁ and j₂. Thus we have the restriction
〈j₁, j₂, m₁, m₂| j, m〉 = 0 unless m = m₁ + m₂ ,
and j₁ + j₂ ≥ j ≥ | j₁ - j₂|.
The restriction on j is referred to as the triangle inequality, since it is equivalent to the condition that the positive numbers j, j₁, and j₂ be able to represent the lengths of the three sides of some triangle.
It is easily shown to apply to any permutation of these three numbers, i.e., its validity also implies that j + j₂ ≥ j₁ ≥ | j - j₂| and that j₁ + j ≥ j₂ ≥ | j - j₁ |. j₂ j j₁

25. Restrictions on j and m - It is clear from the proof of the addition theorem detailed above that the CG coefficient 〈j₁, j₂, m₁, m₂| j, m〉 must vanish unless the two states in the inner product have the same z component of total angular momentum.
In addition, we must have the value of total j on the right lie within the range produced by the angular momenta j₁ and j₂. Thus we have the restriction
〈j₁, j₂, m₁, m₂| j, m〉 = 0 unless m = m₁ + m₂ ,
and j₁ + j₂ ≥ j ≥ | j₁ - j₂|.
The restriction on j is referred to as the triangle inequality, since it is equivalent to the condition that the positive numbers j, j₁, and j₂ be able to represent the lengths of the three sides of some triangle.
It is easily shown to apply to any permutation of these three numbers, i.e., its validity also implies that j + j₂ ≥ j₁ ≥ | j - j₂| and that j₁ + j ≥ j₂ ≥ | j - j₁ |. j₂ j j₁

26. Phase convention - The only ambiguity involved in constructing the states |j, m〉 from the direct product states |j₁, j₂, m₁, m₂〉 is at the point where we construct the maximally aligned vector | j, j〉 for each irreducible invariant subspace S(j).
This vector can always be constructed orthogonal to the states with the same value of m but higher values of j, but the phase of the state so constructed can, unless specified, take any value.
To specify this phase we define the CG coefficients so that the one coefficient
〈j₁, j₂, j₁, j - j₁ | j, j〉 = 〈 j, j| j₁, j₂, j₁, j - j₁〉 ≥ 0
is real and positive.
The remaining states |j, m〉, constructed from |j, j〉 using the lowering operator then have CG coefficients that are all real 〈j₁, j₂, m₁, m₂| j, m〉 = 〈 j, m | j₁, j₂, m₁, m₂〉 although not generally positive.

27. Phase convention - The only ambiguity involved in constructing the states |j, m〉 from the direct product states |j₁, j₂, m₁, m₂〉 is at the point where we construct the maximally aligned vector | j, j〉 for each irreducible invariant subspace S(j).
This vector can always be constructed orthogonal to the states with the same value of m (and higher values of j), but unless specified the phase of the state constructed at that point can take any value.
To specify this phase we define the CG coefficients so that the one coefficient
〈j₁, j₂, j₁, j - j₁ | j, j〉 = 〈 j, j| j₁, j₂, j₁, j - j₁〉 ≥ 0
is real and positive.
The remaining states |j, m〉, constructed from |j, j〉 using the lowering operator then have CG coefficients that are all real 〈j₁, j₂, m₁, m₂| j, m〉 = 〈 j, m | j₁, j₂, m₁, m₂〉 although not generally positive.

28. Phase convention - The only ambiguity involved in constructing the states |j, m〉 from the direct product states |j₁, j₂, m₁, m₂〉 is at the point where we construct the maximally aligned vector | j, j〉 for each irreducible invariant subspace S(j).
This vector can always be constructed orthogonal to the states with the same value of m (and higher values of j), but unless specified the phase of the state constructed at that point can take any value.
To specify this phase we define the CG coefficients so that the one coefficient
〈j₁, j₂, j₁, j - j₁ | j, j〉 = 〈 j, j| j₁, j₂, j₁, j - j₁〉 ≥ 0
is real and positive.
The remaining states |j, m〉, constructed from |j, j〉 using the lowering operator then have CG coefficients that are all real 〈j₁, j₂, m₁, m₂| j, m〉 = 〈 j, m | j₁, j₂, m₁, m₂〉 although not generally positive.

29. Phase convention - The only ambiguity involved in constructing the states |j, m〉 from the direct product states |j₁, j₂, m₁, m₂〉 is at the point where we construct the maximally aligned vector | j, j〉 for each irreducible invariant subspace S(j).
This vector can always be constructed orthogonal to the states with the same value of m (and higher values of j), but unless specified the phase of the state constructed at that point can take any value.
To specify this phase we define the CG coefficients so that the one coefficient
〈j₁, j₂, j₁, j - j₁ | j, j〉 = 〈 j, j| j₁, j₂, j₁, j - j₁〉 ≥ 0
is real and positive.
The remaining states |j, m〉, constructed from |j, j〉 using the lowering operator then have CG coefficients that are all real 〈j₁, j₂, m₁, m₂| j, m〉 = 〈 j, m | j₁, j₂, m₁, m₂〉 although not generally positive.

30. Orthogonality and completeness relations - Being eigenstates of Hermitian operators, the two sets of states {|j, m〉} and {| j₁, j₂, m₁, m₂〉} each form an ONB for the subspace S(j₁,j₂).
Orthonormality implies that
while completeness of each set within this subspace implies that
within S(j₁,j₂)

31. Orthogonality and completeness relations - Being eigenstates of Hermitian operators, the two sets of states {|j, m〉} and {| j₁, j₂, m₁, m₂〉} each form an ONB for the subspace S(j₁,j₂).
Orthonormality implies that
while completeness of each set within this subspace implies that
within S(j₁,j₂)

32. Orthogonality and completeness relations - Being eigenstates of Hermitian operators, the two sets of states {|j, m〉} and {| j₁, j₂, m₁, m₂〉} each form an ONB for the subspace S(j₁,j₂).
Orthonormality implies that
and
while completeness of each set within this subspace implies that
within S(j₁,j₂)

33. Orthogonality and completeness relations - Being eigenstates of Hermitian operators, the two sets of states {|j, m〉} and {| j₁, j₂, m₁, m₂〉} each form an ONB for the subspace S(j₁,j₂).
Orthonormality implies that
and
while completeness of each set within this subspace implies that
within S(j₁,j₂)

34. Orthogonality and completeness relations - Being eigenstates of Hermitian operators, the two sets of states {|j, m〉} and {| j₁, j₂, m₁, m₂〉} each form an ONB for the subspace S(j₁,j₂).
Orthonormality implies that
and
while completeness of each set within this subspace implies that
within S(j₁,j₂)

35. Inserting the completeness relations into the orthonormality relations gives corresponding orthonormality conditions for the CG coefficients, i.e.,
and

36. Inserting the completeness relations into the orthonormality relations gives corresponding orthonormality conditions for the CG coefficients, i.e.,
and

37. Inserting the completeness relations into the orthonormality relations gives corresponding orthonormality conditions for the CG coefficients, i.e.,
and

38. Inserting the completeness relations into the orthonormality relations gives corresponding orthonormality conditions for the CG coefficients, i.e.,
and

39. Recursion relation
The states |j, m〉 in each irreducible subspace S(j) are formed from the state |j, j〉 by application of the lowering operator.
It is possible, as a result, to use the lowering operator to obtain recursion relations for the Clebsch-Gordon coefficients associated with fixed values of j, j₁, and j₂ .
To develop these relations, consider the matrix element of J± between the states |j, m〉 and the states |j₁, j₂, m₁, m₂〉, i.e.,
On the left hand side of this expression we let J± act on the bra 〈j, m|.
But being the adjoint of J∓ |j, m〉, the role of the raising and lowering operators is reversed

40. Recursion relation
The states |j, m〉 in each irreducible subspace S(j) are formed from the state |j, j〉 by application of the lowering operator.
It is possible, as a result, to use the lowering operator to obtain recursion relations for the Clebsch-Gordon coefficients associated with fixed values of j, j₁, and j₂ .
To develop these relations, consider the matrix element of J± between the states |j, m〉 and the states |j₁, j₂, m₁, m₂〉, i.e.,
On the left hand side of this expression we let J± act on the bra 〈j, m|.
But being the adjoint of J∓ |j, m〉, the role of the raising and lowering operators is reversed

41. Recursion relation
The states |j, m〉 in each irreducible subspace S(j) are formed from the state |j, j〉 by application of the lowering operator.
It is possible, as a result, to use the lowering operator to obtain recursion relations for the Clebsch-Gordon coefficients associated with fixed values of j, j₁, and j₂ .
To develop these relations, consider the matrix element of J± between the states |j, m〉 and the states |j₁, j₂, m₁, m₂〉, i.e.,
On the left hand side of this expression we let J± act on the bra 〈j, m|.
But being the adjoint of J∓ |j, m〉, the role of the raising and lowering operators is reversed

42. Recursion relation
The states |j, m〉 in each irreducible subspace S(j) are formed from the state |j, j〉 by application of the lowering operator.
It is possible, as a result, to use the lowering operator to obtain recursion relations for the Clebsch-Gordon coefficients associated with fixed values of j, j₁, and j₂ .
To develop these relations, consider the matrix element of J± between the states |j, m〉 and the states |j₁, j₂, m₁, m₂〉, i.e.,
On the left hand side of this expression we let J± act on the bra 〈j, m|.
But being the adjoint of J∓ |j, m〉, the role of the raising and lowering operators is reversed

43. Recursion relation
The states |j, m〉 in each irreducible subspace S(j) are formed from the state |j, j〉 by application of the lowering operator.
It is possible, as a result, to use the lowering operator to obtain recursion relations for the Clebsch-Gordon coefficients associated with fixed values of j, j₁, and j₂ .
To develop these relations, consider the matrix element of J± between the states |j, m〉 and the states |j₁, j₂, m₁, m₂〉, i.e.,
On the left hand side of this expression we let J± act on the bra 〈j, m|.
But being the adjoint of J∓ |j, m〉, the role of the raising and lowering operators is reversed

44. Recursion relation
The states |j, m〉 in each irreducible subspace S(j) are formed from the state |j, j〉 by application of the lowering operator.
It is possible, as a result, to use the lowering operator to obtain recursion relations for the Clebsch-Gordon coefficients associated with fixed values of j, j₁, and j₂ .
To develop these relations, consider the matrix element of J± between the states |j, m〉 and the states |j₁, j₂, m₁, m₂〉, i.e.,
On the left hand side of this expression we let J± act on the bra 〈j, m|.
But being the adjoint of J∓ |j, m〉, the role of the raising and lowering operators is reversed

45. Recursion relation
The states |j, m〉 in each irreducible subspace S(j) are formed from the state |j, j〉 by application of the lowering operator.
It is possible, as a result, to use the lowering operator to obtain recursion relations for the Clebsch-Gordon coefficients associated with fixed values of j, j₁, and j₂ .
To develop these relations, consider the matrix element of J± between the states |j, m〉 and the states |j₁, j₂, m₁, m₂〉, i.e.,
On the left hand side of this expression we let J± act on the bra 〈j, m|.
But being the adjoint of J∓ |j, m〉, the role of the raising and lowering operators is reversed

46. Recursion relation
Substituting this in above and letting J1± and J₂± act to the right we obtain the relations
These relations allow all CG coefficients for fixed j, j₁, and j₂ to be obtained from, e.g., 〈 j, j| j₁, j₂, j₁, j - j₁〉 .

47. Recursion relation
Substituting this in above and letting J1± and J₂± act to the right we obtain the relations
These relations allow all CG coefficients for fixed j, j₁, and j₂ to be obtained from, e.g., 〈 j, j| j₁, j₂, j₁, j - j₁〉 .

48. Recursion relation
Substituting this in above and letting J1± and J₂± act to the right we obtain the relations
These relations allow all CG coefficients for fixed j, j₁, and j₂ to be obtained from, e.g., 〈 j, j| j₁, j₂, j₁, j - j₁〉 .

49. Clebsch-Gordon series - As a final property of the CG coefficients we derive a relation that follows from the fact that the space S(j₁, j₂) is formed from the direct product of irreducible invariant subspaces S₁(j₁) and S₂(j₂).
We know, e.g., that in S₁(j₁), the basis vectors |j₁, m₁〉 transform under rotations into linear combinations of the themselves according to the relation
where the are the matrix elements of the rotation matrix associated with an irreducible invariant subspace with angular momentum j₁.

50. Clebsch-Gordon series - As a final property of the CG coefficients we derive a relation that follows from the fact that the space S(j₁, j₂) is formed from the direct product of irreducible invariant subspaces S₁(j₁) and S₂(j₂).
We know, e.g., that in S₁(j₁), the basis vectors |j₁, m₁〉 transform under rotations into linear combinations of the themselves according to the relation
where the are the matrix elements of the rotation matrix associated with an irreducible invariant subspace with angular momentum j₁.

51. Clebsch-Gordon series - As a final property of the CG coefficients we derive a relation that follows from the fact that the space S(j₁, j₂) is formed from the direct product of irreducible invariant subspaces S₁(j₁) and S₂(j₂).
We know, e.g., that in S₁(j₁), the basis vectors |j₁, m₁〉 transform under rotations into linear combinations of the themselves according to the relation
where the are the matrix elements of the rotation matrix associated with an irreducible invariant subspace with angular momentum j₁.

52. Clebsch-Gordon series - As a final property of the CG coefficients we derive a relation that follows from the fact that the space S(j₁, j₂) is formed from the direct product of irreducible invariant subspaces S₁(j₁) and S₂(j₂).
We know, e.g., that in S₁(j₁), the basis vectors |j₁, m₁〉 transform under rotations into linear combinations of the themselves according to the relation
where the are the matrix elements of the rotation matrix associated with an irreducible invariant subspace with angular momentum j₁.

53. Clebsch-Gordon series Similarly, in S₂(j₂), the basis vectors |j₂, m₂ 〉 transform as
It follows that in the combined space the direct product states |j₁, j₂, m₁, m₂〉 transform under rotations as
On the other hand, we can also express the states |j₁, j₂, m₁, m₂〉 in terms of the states |j, m〉, i.e.,

54. Clebsch-Gordon series Similarly, in S₂(j₂), the basis vectors |j₂, m₂ 〉 transform as
It follows that in the combined space the direct product states |j₁, j₂, m₁, m₂〉 transform under rotations as
On the other hand, we can also express the states |j₁, j₂, m₁, m₂〉 in terms of the states |j, m〉, i.e.,

55. Clebsch-Gordon series Similarly, in S₂(j₂), the basis vectors |j₂, m₂ 〉 transform as
It follows that in the combined space the direct product states |j₁, j₂, m₁, m₂〉 transform under rotations as
On the other hand, we can also express the states |j₁, j₂, m₁, m₂〉 in terms of the states |j, m〉, i.e.,

56. Clebsch-Gordon series Similarly, in S₂(j₂), the basis vectors |j₂, m₂ 〉 transform as
It follows that in the combined space the direct product states |j₁, j₂, m₁, m₂〉 transform under rotations as
On the other hand, we can also express the states |j₁, j₂, m₁, m₂〉 in terms of the states |j, m〉.

57. Clebsch-Gordon series That is, we can write
But the states |j₂, m₂ 〉 are the basis vectors of an irreducible invariant subspace S(j) of the combined subspace, and so transform accordingly,
Thus, we deduce that

58. Clebsch-Gordon series That is, we can write
But the states |j₂, m₂ 〉 are the basis vectors of an irreducible invariant subspace S(j) of the combined subspace, and so transform accordingly,
Thus, we deduce that

59. Clebsch-Gordon series That is, we can write
But the states |j, m〉 are the basis vectors of an irreducible invariant subspace S(j) of the combined subspace, and so transform accordingly,
Thus, we deduce that

60. Clebsch-Gordon series That is, we can write
But the states |j, m〉 are the basis vectors of an irreducible invariant subspace S(j) of the combined subspace, and so transform accordingly,
Thus, we deduce that

61. Clebsch-Gordon series That is, we can write
But the states |j, m〉 are the basis vectors of an irreducible invariant subspace S(j) of the combined subspace, and so transform accordingly,
Thus, we deduce that

62. Clebsch-Gordon series
Comparing the two expressions for
and
We can clearly equate the coefficients of |j₁, j₂, m’₁, m’₂〉 . . .

63. Clebsch-Gordon series
Comparing the two expressions for
and
We can clearly equate the coefficients of |j₁, j₂, m’₁, m’₂〉 . . .

64. Clebsch-Gordon series
Comparing the two expressions for
and
We can clearly equate the coefficients of |j₁, j₂, m’₁, m’₂〉 . . .

65. to deduce a relation
between matrix elements of the rotation matrices for different values of j. This expression is referred to as the Clebsch-Gordon series.
The Clebsch-Gordon series, and the other properties of the CG coefficients derived in this segment can appear rather dry, formal, and abstract.
However they allow us to prove important things about certain collections of observables {Qk} called tensor operators, the components of which transform under rotations into linear combinations of themselves, providing a natural extension of the idea of scalar and vector observables, which do the same thing.
We explore the properties of tensor operators in the next segment.

66. to deduce a relation
between matrix elements of the rotation matrices for different values of j. This expression is referred to as the Clebsch-Gordon series.
The Clebsch-Gordon series, and the other properties of the CG coefficients derived in this segment can appear rather dry, formal, and abstract.
However they allow us to prove important things about certain collections of observables {Qk} called tensor operators, the components of which transform under rotations into linear combinations of themselves, providing a natural extension of the idea of scalar and vector observables, which do the same thing.
We explore the properties of tensor operators in the next segment.

67. to deduce a relation
between matrix elements of the rotation matrices for different values of j. This expression is referred to as the Clebsch-Gordon series.
The Clebsch-Gordon series, and the other properties of the CG coefficients derived in this segment can appear rather dry, formal, and abstract.
However they allow us to prove important things about certain collections of observables {Qk} called tensor operators, the components of which transform under rotations into linear combinations of themselves, providing a natural extension of the idea of scalar and vector observables, which do the same thing.
We explore the properties of tensor operators in the next segment.

68. to deduce a relation
between matrix elements of the rotation matrices for different values of j. This expression is referred to as the Clebsch-Gordon series.
The Clebsch-Gordon series, and the other properties of the CG coefficients derived in this segment can appear rather dry, formal, and abstract.
However they allow us to prove important things about certain collections of observables {Qk} called tensor operators, the components of which transform under rotations into linear combinations of themselves, providing a natural extension of the idea of scalar and vector observables, which do the same thing.
We explore the properties of tensor operators in the next segment.

69. to deduce a relation
between matrix elements of the rotation matrices for different values of j. This expression is referred to as the Clebsch-Gordon series.
The Clebsch-Gordon series, and the other properties of the CG coefficients derived in this segment can appear rather dry, formal, and abstract.
However they allow us to prove important things about certain collections of observables {Qk} called tensor operators, the components of which transform under rotations into linear combinations of themselves, providing a natural extension of the idea of scalar and vector observables, which do the same thing.
We explore the properties of tensor operators in the next segment.