Quantum Two 1

2

Angular Momentum and Rotations 3

Rotationally Invariant Subspaces 4

In our discussion of commuting or compatible observables we encountered the idea of global invariance and saw, e.g., that the eigenspaces S(a) of an observable A are globally invariant with respect to any operator B that commutes with A. This means, that the action of such an operator B on any state lying in the eigenspace S(a) is to map it onto another state within that subspace. This means also that the matrix representing any operator B that commutes with A is block-diagonal in any representation { |a, τ 〉 } of eigenstates of A. That is, since B|a, τ 〉 is in the same eigenspace S(a) of A as the state | a, τ 〉, 〈 a′,τ′|B |a, τ 〉 =0 if a′ ≠ a the matrix element of B between states in different eigenspaces vanishes. 5 S(a ‘’’) S(a ’) S(a ’’) S(a)S(a)

In our discussion of commuting or compatible observables we encountered the idea of global invariance and saw, e.g., that the eigenspaces S(a) of an observable A are globally invariant with respect to any operator B that commutes with A. This means, that the action of such an operator B on any state lying in the eigenspace S(a) is to map it onto another state within that same subspace. It respects the boundaries of S(a). This means also that the matrix representing any operator B that commutes with A is block-diagonal in any representation { |a, τ 〉 } of eigenstates of A. That is, since B|a, τ 〉 is in the same eigenspace S(a) of A as the state | a, τ 〉, 〈 a′,τ′|B |a, τ 〉 =0 if a′ ≠ a the matrix element of B between states in different eigenspaces vanishes. 6 S(a ‘’’) S(a ’) S(a ’’) S(a)S(a)    

In our discussion of commuting or compatible observables we encountered the idea of global invariance and saw, e.g., that the eigenspaces S(a) of an observable A are globally invariant with respect to any operator B that commutes with A. This means, that the action of such an operator B on any state lying in the eigenspace S(a) is to map it onto another state within that same subspace. It respects the boundaries of S(a). This means also that the matrix representing any operator B that commutes with A is block-diagonal in any representation { |a, τ 〉 } of eigenstates of A. That is, since B|a, τ 〉 is in the same eigenspace S(a) of A as the state | a, τ 〉, 〈 a′,τ′|B |a, τ 〉 =0 if a′ ≠ a the matrix element of B between states in different eigenspaces vanishes. 7 S(a ‘’’) S(a ’) S(a ’’) S(a)S(a)    

In our discussion of commuting or compatible observables we encountered the idea of global invariance and saw, e.g., that the eigenspaces S(a) of an observable A are globally invariant with respect to any operator B that commutes with A. This means, that the action of such an operator B on any state lying in the eigenspace S(a) is to map it onto another state within that same subspace. It respects the boundaries of S(a). This means also that the matrix representing any operator B that commutes with A is block-diagonal in any representation { |a, τ 〉 } of eigenstates of A. That is, since B|a, τ 〉 stays in the same eigenspace S(a) of A as the state | a, τ 〉, 〈 a ′, τ ′ | B |a, τ 〉  0 if a′ ≠ a. The matrix element of B between states in different eigenspaces vanishes. 8 S(a ‘’’) S(a ’) S(a ’’) S(a)S(a)    

In our discussion of commuting or compatible observables we encountered the idea of global invariance and saw, e.g., that the eigenspaces S(a) of an observable A are globally invariant with respect to any operator B that commutes with A. This means, that the action of such an operator B on any state lying in the eigenspace S(a) is to map it onto another state within that same subspace. It respects the boundaries of S(a). This means also that the matrix representing any operator B that commutes with A is block-diagonal in any representation { |a, τ 〉 } of eigenstates of A. That is, since B|a, τ 〉 stays in the same eigenspace S(a) of A as the state | a, τ 〉, 〈 a ′, τ ′ | B |a, τ 〉  0 if a′ ≠ a. it is orthogonal to states in different eigenspaces. 9 S(a ‘’’) S(a ’) S(a ’’) S(a)S(a)    

It is also not hard to see that if [A, B]  0, then A commutes with any power of B, i.e., that [A, B n ]  0, since, e.g., A BBB  B A BB  BBB A B  BBB A That is, we can just push A through each factor of B individually, until we have commuted all the factors. It also then follows that A commutes with any operator function of B, i.e., if F ( B )  ∑ n f n B n then 1. [A, F(B)] = The eigenspaces of A will be globally invariant with respect to F ( B ). 3. The operator F ( B ) will be block diagonal with respect to the eigenspaces of A. 10

It is also not hard to see that if [A, B]  0, then A commutes with any power of B, i.e., that [A, B n ]  0, since, e.g., A BBB  B A BB  BBB A B  BBB A That is, we can just push A through each factor of B individually, until we have commuted all the factors. It also then follows that A commutes with any operator function of B, i.e., if F ( B )  ∑ n f n B n then 1. [A, F(B)] = The eigenspaces of A will be globally invariant with respect to F ( B ). 3. The operator F ( B ) will be block diagonal with respect to the eigenspaces of A. 11

It is also not hard to see that if [A, B]  0, then A commutes with any power of B, i.e., that [A, B n ]  0, since, e.g., A BBB  B A BB  BBB A B  BBB A That is, we can just push A through each factor of B individually, until we have commuted all the factors. It also then follows that A commutes with any operator function of B, i.e., if F ( B )  ∑ n f n B n then 1. [A, F(B)] = The eigenspaces of A will be globally invariant with respect to F ( B ). 3. The operator F ( B ) will be block diagonal with respect to the eigenspaces of A. 12

It is also not hard to see that if [A, B]  0, then A commutes with any power of B, i.e., that [A, B n ]  0, since, e.g., A BBB  B A BB  BB A B  BBB A That is, we can just push A through each factor of B individually, until we have commuted all the factors. It also then follows that A commutes with any operator function of B, i.e., if F ( B )  ∑ n f n B n then 1. [A, F(B)] = The eigenspaces of A will be globally invariant with respect to F ( B ). 3. The operator F ( B ) will be block diagonal with respect to the eigenspaces of A. 13

It is also not hard to see that if [A, B]  0, then A commutes with any power of B, i.e., that [A, B n ]  0, since, e.g., A BBB  B A BB  BB A B  BBB A Thus, we can just push A through each factor of B individually, until we have commuted all the factors. It also then follows that A commutes with any operator function of B, i.e., if F ( B )  ∑ n f n B n then 1. [A, F(B)] = The eigenspaces of A will be globally invariant with respect to F ( B ). 3. The operator F ( B ) will be block diagonal with respect to the eigenspaces of A. 14

It is also not hard to see that if [A, B]  0, then A commutes with any power of B, i.e., that [A, B n ]  0, since, e.g., A BBB  B A BB  BBB A B  BBB A That is, we can just push A through each factor of B individually, until we have commuted all the factors. It also then follows that A commutes with any operator function of B, i.e., if F ( B )  ∑ n f n B n then 1. [A, F(B)] = The eigenspaces of A will be globally invariant with respect to F ( B ). 3. The operator F ( B ) will be block diagonal with respect to the eigenspaces of A. 15

It is also not hard to see that if [A, B]  0, then A commutes with any power of B, i.e., that [A, B n ]  0, since, e.g., A BBB  B A BB  BBB A B  BBB A That is, we can just push A through each factor of B individually, until we have commuted all the factors. It also then follows that A commutes with any operator function of B, i.e., if F ( B )  ∑ n f n B n then 1. [A, F(B)] = The eigenspaces of A will be globally invariant with respect to F ( B ). 3. The operator F ( B ) will be block diagonal with respect to the eigenspaces of A. 16

It is also not hard to see that if [A, B]  0, then A commutes with any power of B, i.e., that [A, B n ]  0, since, e.g., A BBB  B A BB  BBB A B  BBB A That is, we can just push A through each factor of B individually, until we have commuted all the factors. It also then follows that A commutes with any operator function of B, i.e., if F ( B )  ∑ n f n B n then 1. [A, F(B)] = The eigenspaces of A will be globally invariant with respect to F ( B ). 3. The operator F ( B ) will be block diagonal with respect to the eigenspaces of A. 17

It is also not hard to see that if [A, B]  0, then A commutes with any power of B, i.e., that [A, B n ]  0, since, e.g., A BBB  B A BB  BBB A B  BBB A That is, we can just push A through each factor of B individually, until we have commuted all the factors. It also then follows that A commutes with any operator function of B, i.e., if F ( B )  ∑ n f n B n then 1. [A, F(B)] = The eigenspaces of A will be globally invariant with respect to F ( B ). 3. The operator F ( B ) will be block diagonal with respect to the eigenspaces of A. 18 S(a ‘’’) S(a ’) S(a ’’) S(a)S(a)    

It is also not hard to see that if [A, B]  0, then A commutes with any power of B, i.e., that [A, B n ]  0, since, e.g., A BBB  B A BB  BBB A B  BBB A That is, we can just push A through each factor of B individually, until we have commuted all the factors. It also then follows that A commutes with any operator function of B, i.e., if F ( B )  ∑ n f n B n then 1. [A, F(B)] = The eigenspaces of A will be globally invariant with respect to F ( B ). 3. The operator F ( B ) will be block diagonal with respect to the eigenspaces of A. 19 S(a ‘’’) S(a ’) S(a ’’) S(a)S(a)    

It is useful to extend this idea to apply to more than one operator at a time, and, in the process, to introduce the idea of invariant subspaces. A subspace S ′ of the state space S is globally invariant (or simply) invariant with respect to the action of a set or group G  {R ₁, R ₂, ⋯ } of operators if, for every |ψ 〉 in S ′, each of the vectors {R i |ψ 〉 } = {R ₁ |ψ 〉,R ₂ |ψ 〉, ⋯ } are in S ′ as well. That is, the operators R i all respect the boundaries of the subspace S ′. Any state that starts in S ′ stays in S ′. The subspace S′ is than said to be an invariant subspace of the specified set or group of operators. 20

It is useful to extend this idea to apply to more than one operator at a time, and, in the process, to introduce the idea of invariant subspaces. A subspace S ′ of the state space S is globally invariant (or simply) invariant with respect to the action of a set or group of operators G  {R ₁, R ₂, ⋯ } if for every |ψ 〉 in S ′, each of the vectors {R i |ψ 〉 } = {R ₁ |ψ 〉,R ₂ |ψ 〉, ⋯ } are in S ′ as well. That is, the operators R i all respect the boundaries of the subspace S ′. Any state that starts in S ′ stays in S ′. The subspace S′ is than said to be an invariant subspace of the specified set or group of operators. 21 S ‘’’ S’ S ’’ S

It is useful to extend this idea to apply to more than one operator at a time, and, in the process, to introduce the idea of invariant subspaces. A subspace S ′ of the state space S is globally invariant (or simply) invariant with respect to the action of a set or group of operators G  {R ₁, R ₂, ⋯ } if for every |ψ 〉 in S ′, each of the vectors {R i |ψ 〉 } = {R ₁ |ψ 〉, R ₂ |ψ 〉, ⋯ } are in S ′ as well. That is, the operators R i all respect the boundaries of the subspace S ′. Any state that starts in S ′ stays in S ′. The subspace S′ is than said to be an invariant subspace of the specified set or group of operators. 22 S ‘’’ S’ S ’’ S

It is useful to extend this idea to apply to more than one operator at a time, and, in the process, to introduce the idea of invariant subspaces. A subspace S ′ of the state space S is globally invariant (or simply) invariant with respect to the action of a set or group of operators G  {R ₁, R ₂, ⋯ } if for every |ψ 〉 in S ′, each of the vectors {R i |ψ 〉 } = {R ₁ |ψ 〉, R ₂ |ψ 〉, ⋯ } are in S ′ as well. That is, the operators R i all respect the boundaries of the subspace S ′. Any state that starts in S ′ stays in S ′. The subspace S′ is than said to be an invariant subspace of the specified set or group of operators. 23 S ‘’’ S’ S ’’ S

It is useful to extend this idea to apply to more than one operator at a time, and, in the process, to introduce the idea of invariant subspaces. A subspace S ′ of the state space S is globally invariant (or simply) invariant with respect to the action of a set or group of operators G  {R ₁, R ₂, ⋯ } if for every |ψ 〉 in S ′, each of the vectors {R i |ψ 〉 } = {R ₁ |ψ 〉, R ₂ |ψ 〉, ⋯ } are in S ′ as well. That is, the operators R i all respect the boundaries of the subspace S ′. Any state that starts in S ′, stays in S ′. The subspace S′ is than said to be an invariant subspace of the specified set or group of operators. 24 S ‘’’ S’S’ S ’’ S    

It is useful to extend this idea to apply to more than one operator at a time, and, in the process, to introduce the idea of invariant subspaces. A subspace S ′ of the state space S is globally invariant (or simply) invariant with respect to the action of a set or group of operators G  {R ₁, R ₂, ⋯ } if for every |ψ 〉 in S ′, each of the vectors {R i |ψ 〉 } = {R ₁ |ψ 〉, R ₂ |ψ 〉, ⋯ } are in S ′ as well. That is, the operators R i all respect the boundaries of the subspace S ′. Any state that starts in S ′, stays in S ′. The subspace S ′ is than said to be an invariant subspace of the specified set or group of operators. 25 S ‘’’ S’S’ S ’’ S    

With this definition, we consider a quantum mechanical system with state space S, characterized by total angular momentum operator with components J i. As we have seen, the state space S can be expressed as a direct sum of orthogonal eigenspaces associated with any observable, and in particular can be written as (or reduced to) a direct sum S  S(j) ⊕ S(j′) ⊕ S(j′′)  ⊕  ⋯ of eigenspaces S(j) associated with the operator J 2. Now, since [J i, J 2 ] = 0 = [J u, J 2 ], the eigenspaces S(j) of J 2 are globally invariant with respect to the action of the component J u of the angular momentum along any direction. 26

With this definition, we consider a quantum mechanical system with state space S, characterized by total angular momentum operator with components J i. As we have seen, the state space S can be expressed as a direct sum of orthogonal eigenspaces associated with any observable, and in particular can be written as (or reduced to) a direct sum S  S(j) ⊕ S(j′) ⊕ S(j′′)  ⊕  ⋯ of eigenspaces S(j) associated with the operator J 2. Now, since [J i, J 2 ] = 0 = [J u, J 2 ], the eigenspaces S(j) of J 2 are globally invariant with respect to the action of the component J u of the angular momentum along any direction. 27 S(j ‘’’) S(j ’) S(j ’’) S(j)S(j)

With this definition, we consider a quantum mechanical system with state space S, characterized by total angular momentum operator with components J i. As we have seen, the state space S can be expressed as a direct sum of orthogonal eigenspaces associated with any observable, and in particular can be written as (or reduced to) a direct sum S  S(j) ⊕ S(j′) ⊕ S(j′′)  ⊕  ⋯ of eigenspaces S(j) associated with the operator J 2. Now, since [J i, J 2 ] = 0 = [J u, J 2 ], the eigenspaces S(j) of J 2 are globally invariant with respect to the action of the component J u of the angular momentum along any direction. 28 S(j ‘’’) S(j ’) S(j ’’) S(j)S(j)    

But from our earlier arguments, these eigenspaces must also be invariant with respect to the action of any operator function of J u. In particular, the S(j) must be invariant subspaces of the group of unitary operators U u (α)  exp (  iαJ u ) that carry out the effects of rotation on the entire state space. Thus under an arbitrary rotation U u (α), any state that starts in S(j) stays in S(j). Since these unitary operators form a representation of the 3DRG, each of the eigenspaces S(j) of J 2 are said to form an invariant subspace of the 3DRG. 29 S(j ‘’’) S(j ’) S(j ’’) S(j)S(j)

But from our earlier arguments, these eigenspaces must also be invariant with respect to the action of any operator function of J u. In particular, the S(j) must be invariant subspaces of the group of unitary operators U u (α)  exp (  iαJ u ) that carry out the effects of rotation on the entire state space. Thus under an arbitrary rotation U u (α), any state that starts in S(j) stays in S(j). Since these unitary operators form a representation of the 3DRG, each of the eigenspaces S(j) of J 2 are said to form an invariant subspace of the 3DRG. 30 S(j ‘’’) S(j ’) S(j ’’) S(j)S(j)    

But from our earlier arguments, these eigenspaces must also be invariant with respect to the action of any operator function of J u. In particular, the S(j) must be invariant subspaces of the group of unitary operators U u (α)  exp (  iαJ u ) that carry out the effects of rotation on the entire state space. Thus under an arbitrary rotation U u (α), any state that starts in S(j) stays in S(j). Since these unitary operators form a representation of the 3DRG, each of the eigenspaces S(j) of J 2 are said to form an invariant subspace of the 3DRG. 31 S(j ‘’’) S(j ’) S(j ’’) S(j)S(j)    

But from our earlier arguments, these eigenspaces must also be invariant with respect to the action of any operator function of J u. In particular, the S(j) must be invariant subspaces of the group of unitary operators U u (α)  exp (  iαJ u ) that carry out the effects of rotation on the entire state space. Thus under an arbitrary rotation U u (α), any state that starts in S(j) stays in S(j). Since these unitary operators form a representation of the 3DRG, each of the eigenspaces S(j) of J 2 is said to form an invariant subspace of the 3DRG. 32 S(j ‘’’) S(j ’) S(j ’’) S(j)S(j)    

But, typically, each one of the spaces S(j) associated with a particular eigenvalue j(j+1) of J² can, itself, be decomposed or reduced into a direct sum S(j)  S(j, τ) ⊕ S(j, τ ′ ) ⊕ S(j, τ ′′ )  ⊕  ⋯ of essentially identical 2j +1 dimensional subspaces S(j, τ) spanned by the basis vectors {|τ, j, m 〉 | m  j, ⋯, j} of any standard representation of angular momentum eigenstates for the space S with fixed values of j and τ. We have already noted that the matrices that represent J² and any of the components J i or J u of angular momentum are block diagonal with respect to these subspaces, i.e., that the matrix element 〈 τ′, j′, m′| J i |τ, j, m 〉  0 if (j ′, τ ′ ) ≠ (j, τ) of any component of angular momentum between states in different subspaces S(j, τ) and S(j ′, τ ′ ) vanishes. 33 SjSj Sj’Sj’ S j ’’ S j ’’’   ’    ’’   ’

But, typically, each one of the spaces S(j) associated with a particular eigenvalue j(j+1) of J² can, itself, be decomposed or reduced into a direct sum S(j)  S(j, τ) ⊕ S(j, τ ′ ) ⊕ S(j, τ ′′ )  ⊕  ⋯ of essentially identical 2j +1 dimensional subspaces S(j, τ) spanned by the basis vectors {|τ, j, m 〉 | m  j, ⋯, j} of any standard representation of angular momentum eigenstates for the space S with fixed values of j and τ. We have already noted that the matrices that represent J² and any of the components J i or J u of angular momentum are block diagonal with respect to these subspaces, i.e., that the matrix element 〈 τ′, j′, m′| J i |τ, j, m 〉  0 if (j ′, τ ′ ) ≠ (j, τ) of any component of angular momentum between states in different subspaces S(j, τ) and S(j ′, τ ′ ) vanishes. 34 SjSj Sj’Sj’ S j ’’ S j ’’’   ’    ’’   ’

But, typically, each one of the spaces S(j) associated with a particular eigenvalue j(j+1) of J² can, itself, be decomposed or reduced into a direct sum S(j)  S(j, τ) ⊕ S(j, τ ′ ) ⊕ S(j, τ ′′ )  ⊕  ⋯ of essentially identical 2j +1 dimensional subspaces S(j, τ) spanned by the basis vectors {|τ, j, m 〉 | m  j, ⋯, j} of any standard representation of angular momentum eigenstates for the space S with fixed values of j and τ. We have already noted that the matrices that represent J² and any of the components J i or J u of angular momentum are block diagonal with respect to these subspaces, i.e., that the matrix element 〈 τ′, j′, m′| J i |τ, j, m 〉  0 if (j ′, τ ′ ) ≠ (j, τ) of any component of angular momentum between states in different subspaces S(j, τ) and S(j ′, τ ′ ) vanishes. 35 SjSj Sj’Sj’ S j ’’ S j ’’’   ’    ’’   ’

Thus, when J i acts on a linear combination of states |τ, j, m 〉 in S ( j,τ ) it takes it onto a different linear combination of states in the same space. In other words, the 2j +1 dimensional subspaces S ( j,τ ) are also globally invariant with respect to the action of the components J u of angular momentum along any direction, and thus to any operator valued function of any of those components. We deduce therefore that, like the eigenspaces S ( j ) of J² themselves, each of the 2j +1 dimensional subspaces S ( j,τ ) are invariant subspaces of the three dimensional rotation group. 36 SjSj Sj’Sj’ S j ’’ S j ’’’   ’    ’’   ’ 

Thus, when J i acts on a linear combination of states |τ, j, m 〉 in S ( j,τ ) it takes it onto a different linear combination of states in the same space. In other words, the 2j +1 dimensional subspaces S ( j,τ ) are also globally invariant with respect to the action of the components J u of angular momentum along any direction, and thus to any operator valued function of any of those components. We deduce therefore that, like the eigenspaces S ( j ) of J² themselves, each of the 2j +1 dimensional subspaces S ( j,τ ) are invariant subspaces of the three dimensional rotation group. 37 SjSj Sj’Sj’ S j ’’ S j ’’’   ’    ’’   ’ 

Thus, when J i acts on a linear combination of states |τ, j, m 〉 in S ( j,τ ) it takes it onto a different linear combination of states in the same space. In other words, the 2j +1 dimensional subspaces S ( j,τ ) are also globally invariant with respect to the action of the components J u of angular momentum along any direction, and thus to any operator valued function of any of those components. We deduce therefore that, like the eigenspaces S ( j ) of J² themselves, each of the 2j +1 dimensional subspaces S ( j,τ ) are invariant subspaces of the three dimensional rotation group. 38 SjSj Sj’Sj’ S j ’’ S j ’’’   ’    ’’   ’         

We now point out the following suggestive hierarchy. 1.The state space S itself is clearly invariant under arbitrary rotations, since under such a rotation, any state in S remains in S. 39

We now point out the following suggestive hierarchy. 1.The state space S itself is clearly invariant under arbitrary rotations, since under such a rotation, any state in S remains in S. But we know that in general S is reducible S  S(j) ⊕ S(j′) ⊕ S(j′′)  ⊕  ⋯ into rotationally invariant subspaces S(j) of dimension smaller than S itself. 40 SjSj Sj’Sj’ S j ’’ S j ’’’

We now point out the following suggestive hierarchy. 1.The state space S itself is clearly invariant under arbitrary rotations, since under such a rotation, any state in S remains in S. But we know that in general S is reducible S  S(j) ⊕ S(j′) ⊕ S(j′′)  ⊕  ⋯ into rotationally invariant subspaces S(j) of dimension smaller than S itself. 2.But the rotationally invariant subspaces S(j) are themselves, typically, reducible S(j)  S(j, τ) ⊕ S(j, τ ′ ) ⊕ S(j, τ ′′ )  ⊕  ⋯ into a certain number of rotationally invariant subspaces S(j, τ) having a dimension smaller than that of S(j). 41 SjSj Sj’Sj’ S j ’’ S j ’’’   ’    ’’   ’

The obvious question arises: are each of the 2j +1 dimensional rotationally invariant subspaces S(j, τ) further reducible into rotationally invariant subspaces having a dimension smaller than 2j +1 ? The answer, it turns out, is that they are not. The 2j +1 dimensional subspaces S(j, τ) spanned by the basis vectors {|τ, j, m 〉 | m  j, ⋯, j} of a standard representation with fixed j and τ each form an irreducible invariant subspace of the 3DRG. This idea of reducibility or irreducibility seems fairly clear, I hope, from the pattern that emerged in the preceding discussion. To make it a bit more precise, however, let us introduce a definition. 42 SjSj Sj’Sj’ S j ’’ S j ’’’   ’    ’’   ’

The obvious question arises: are each of the 2j +1 dimensional rotationally invariant subspaces S(j, τ) further reducible into rotationally invariant subspaces having a dimension smaller than 2j +1 ? The answer, it turns out, is that they are not. The 2j +1 dimensional subspaces S(j, τ) spanned by the basis vectors {|τ, j, m 〉 | m  j, ⋯, j} of a standard representation with fixed j and τ each form an irreducible invariant subspace of the 3DRG. This idea of reducibility or irreducibility seems fairly clear, I hope, from the pattern that emerged in the preceding discussion. To make it a bit more precise, however, let us introduce a definition. 43 SjSj Sj’Sj’ S j ’’ S j ’’’   ’    ’’   ’

The obvious question arises: are each of the 2j +1 dimensional rotationally invariant subspaces S(j, τ) further reducible into rotationally invariant subspaces having a dimension smaller than 2j +1 ? The answer, it turns out, is that they are not. The 2j +1 dimensional subspaces S(j, τ) spanned by the basis vectors {|τ, j, m 〉 | m  j, ⋯, j} of a standard representation with fixed j and τ each form an irreducible invariant subspace of the 3DRG. This idea of reducibility or irreducibility seems fairly clear, I hope, from the pattern that emerged in the preceding discussion. To make it a bit more precise, however, let us introduce a definition. 44 SjSj Sj’Sj’ S j ’’ S j ’’’   ’    ’’   ’

The obvious question arises: are each of the 2j +1 dimensional rotationally invariant subspaces S(j, τ) further reducible into rotationally invariant subspaces having a dimension smaller than 2j +1 ? The answer, it turns out, is that they are not. The 2j +1 dimensional subspaces S(j, τ) spanned by the basis vectors {|τ, j, m 〉 | m  j, ⋯, j} of a standard representation with fixed j and τ each form an irreducible invariant subspace of the 3DRG. This idea of reducibility or irreducibility seems fairly clear, I hope, from the pattern that emerged in the preceding discussion. To make it a bit more precise, however, let us introduce a definition. 45 SjSj Sj’Sj’ S j ’’ S j ’’’   ’    ’’   ’

The obvious question arises: are each of the 2j +1 dimensional rotationally invariant subspaces S(j, τ) further reducible into rotationally invariant subspaces having a dimension smaller than 2j +1 ? The answer, it turns out, is that they are not. The 2j +1 dimensional subspaces S(j, τ) spanned by the basis vectors {|τ, j, m 〉 | m  j, ⋯, j} of a standard representation with fixed j and τ each form an irreducible invariant subspace of the 3DRG. This idea of reducibility or irreducibility seems fairly clear, I hope, from the pattern that emerged in the preceding discussion. To make it a bit more precise, however, let us introduce a definition. 46 SjSj Sj’Sj’ S j ’’ S j ’’’   ’    ’’   ’

47 Definition: An invariant subspace S ′ of a group of operators G  {R ₁, R ₂, ⋯ } is said to be irreducible with respect to G (or is an irreducible invariant subspace of G ) if, for every non-zero vector |ψ 〉 in S ′, the vectors {R i |ψ 〉 } span S ′. Conversely, S ′ is reducible if there exists a nonzero vector |ψ 〉 in S ′ for which the vectors {R i |ψ 〉 } fail to span the space. Clearly, in the latter case the vectors spanned by the set {R i |ψ 〉 } form a subspace of S ′ that is itself invariant with respect to G.

48 Definition: An invariant subspace S ′ of a group of operators G  {R ₁, R ₂, ⋯ } is said to be irreducible with respect to G (or is an irreducible invariant subspace of G ) if, for every non-zero vector |ψ 〉 in S ′, the vectors {R i |ψ 〉 } span S ′. Conversely, S ′ is reducible if there exists a nonzero vector |ψ 〉 in S ′ for which the vectors {R i |ψ 〉 } fail to span the space. Clearly, in the latter case the vectors spanned by the set {R i |ψ 〉 } form a subspace of S ′ that is itself invariant with respect to G.

49 Definition: An invariant subspace S ′ of a group of operators G  {R ₁, R ₂, ⋯ } is said to be irreducible with respect to G (or is an irreducible invariant subspace of G ) if, for every non-zero vector |ψ 〉 in S ′, the vectors {R i |ψ 〉 } span S ′. Conversely, S ′ is reducible if there exists a nonzero vector |ψ 〉 in S ′ for which the vectors {R i |ψ 〉 } fail to span the space. Clearly, in the latter case the vectors spanned by the set {R i |ψ 〉 } form a subspace of S ′ that is itself invariant with respect to G.

50 Thus to show explicitly that the subspace S(j, τ) associated with a standard representation for the state space S is, in fact, an irreducible invariant subspace of the 3D rotation group we have to show that from any nonzero state in this subspace, it is possible to construct a basis for this space as a linear combination of states {U R |ψ 〉 }. This is carried out in the class text in some detail. It is a little wonkish and technical, and does not provide a great deal of physical insight. In short, by considering infinitesimal rotations it is possible to show that …

51 Thus to show explicitly that the subspace S(j, τ) associated with a standard representation for the state space S is, in fact, an irreducible invariant subspace of the 3D rotation group we have to show that from any nonzero state in this subspace, it is possible to construct a basis for this space as a linear combination of states {U R |ψ 〉 }. This is carried out in the class text in some detail. It is a little wonkish and technical, and does not provide a great deal of physical insight. In short, by considering infinitesimal rotations it is possible to show that …

52 Thus to show explicitly that the subspace S(j, τ) associated with a standard representation for the state space S is, in fact, an irreducible invariant subspace of the 3D rotation group we have to show that from any nonzero state in this subspace, it is possible to construct a basis for this space as a linear combination of states {U R |ψ 〉 }. This is carried out in the class text in some detail. It is a little wonkish and technical, and does not provide a great deal of physical insight. In short, by considering infinitesimal rotations it is possible to show that …

53 1. The vectors J u |ψ 〉 can all be written as a linear combination of states of the form {U u (δα)|ψ 〉 } 2. The states (J + )ⁿ|ψ 〉 = (J x + iJ y )ⁿ|ψ 〉 thus can also be written as such a linear combination. 3. For some n < 2j + 1 this repeated raising must produce a state proportional to |τ, j, j 〉. 4. That repeated lowering on that state produces the remaining basis states, and thus that 5. It is possible to write the basis states for S(j, τ) as linear combinations of the states {U u (δα)|ψ 〉 }. 6. Thus the states {U u (δα)|ψ 〉 } span the entire space S(j, τ), making it an irreducible invariant subspace of the 3DRG.

54 1. The vectors J u |ψ 〉 can all be written as a linear combination of states of the form {U u (δα)|ψ 〉 } 2. The states (J + )ⁿ|ψ 〉 = (J x + iJ y )ⁿ|ψ 〉 thus can also be written as such a linear combination. 3. For some n < 2j + 1 this repeated raising must produce a state proportional to |τ, j, j 〉. 4. That repeated lowering on that state produces the remaining basis states, and thus that 5. It is possible to write the basis states for S(j, τ) as linear combinations of the states {U u (δα)|ψ 〉 }. 6. Thus the states {U u (δα)|ψ 〉 } span the entire space S(j, τ), making it an irreducible invariant subspace of the 3DRG.

55 1. The vectors J u |ψ 〉 can all be written as a linear combination of states of the form {U u (δα)|ψ 〉 } 2. The states (J + )ⁿ|ψ 〉 = (J x + iJ y )ⁿ|ψ 〉 thus can also be written as such a linear combination. 3. For some n < 2j + 1 this repeated raising must produce a state proportional to |τ, j, j 〉. 4. That repeated lowering on that state produces the remaining basis states, and thus that 5. It is possible to write the basis states for S(j, τ) as linear combinations of the states {U u (δα)|ψ 〉 }. 6. Thus the states {U u (δα)|ψ 〉 } span the entire space S(j, τ), making it an irreducible invariant subspace of the 3DRG.

56 1. The vectors J u |ψ 〉 can all be written as a linear combination of states of the form {U u (δα)|ψ 〉 } 2. The states (J + )ⁿ|ψ 〉 = (J x + iJ y )ⁿ|ψ 〉 thus can also be written as such a linear combination. 3. For some n < 2j + 1 this repeated raising must produce a state proportional to |τ, j, j 〉. 4. That repeated lowering on that state produces the remaining basis states, and thus that 5. It is possible to write the basis states for S(j, τ) as linear combinations of the states {U u (δα)|ψ 〉 }. 6. Thus the states {U u (δα)|ψ 〉 } span the entire space S(j, τ), making it an irreducible invariant subspace of the 3DRG.

57 1. The vectors J u |ψ 〉 can all be written as a linear combination of states of the form {U u (δα)|ψ 〉 } 2. The states (J + )ⁿ|ψ 〉 = (J x + iJ y )ⁿ|ψ 〉 thus can also be written as such a linear combination. 3. For some n < 2j + 1 this repeated raising must produce a state proportional to |τ, j, j 〉. 4. That repeated lowering on that state produces the remaining basis states, and thus that 5. It is possible to write the basis states for S(j, τ) as linear combinations of the states {U u (δα)|ψ 〉 }. 6. Thus the states {U u (δα)|ψ 〉 } span the entire space S(j, τ), making it an irreducible invariant subspace of the 3DRG.

58 1. The vectors J u |ψ 〉 can all be written as a linear combination of states of the form {U u (δα)|ψ 〉 } 2. The states (J + )ⁿ|ψ 〉 = (J x + iJ y )ⁿ|ψ 〉 thus can also be written as such a linear combination. 3. For some n < 2j + 1 this repeated raising must produce a state proportional to |τ, j, j 〉. 4. That repeated lowering on that state produces the remaining basis states, and thus that 5. It is possible to write the basis states for S(j, τ) as linear combinations of the states {U u (δα)|ψ 〉 }. 6. Thus the states {U u (δα)|ψ 〉 } span the entire space S(j, τ), making it an irreducible invariant subspace of the 3DRG.

59 It follows that the under the unitary transformation associated with a general rotation, the basis vectors {|τ, j, m 〉 } of S(j, τ) are each transformed into new basis vectors for the same invariant subspace. Indeed, it is not hard to see, based upon our earlier description of the rotation process, that a rotation R that takes the unit vector z onto a new direction z ′ will take the basis states |τ, j, m z 〉 of J² and J z onto a new set of basis kets |τ, j, m z ′ 〉 for the same space that are now eigenstates of J² and the component J z ′ of angular momentum along the new rotated direction. These new vectors can, of course, be expressed as linear combinations of the original ones. The picture that emerges is that, under rotations, the vectors |τ, j, m 〉 transform (irreducibly) into linear combinations of themselves.

60 It follows that the under the unitary transformation associated with a general rotation, the basis vectors {|τ, j, m 〉 } of S(j, τ) are each transformed into new basis vectors for the same invariant subspace. Indeed, it is not hard to see, based upon our earlier description of the rotation process, that a rotation R that takes the unit vector z onto a new direction z ′ will take the basis states |τ, j, m z 〉 of J² and J z onto a new set of basis kets |τ, j, m z ′ 〉 for the same space that are now eigenstates of J² and the component J z ′ of angular momentum along the new rotated direction. These new vectors can, of course, be expressed as linear combinations of the original ones. The picture that emerges is that, under rotations, the vectors |τ, j, m 〉 transform (irreducibly) into linear combinations of themselves.

61 It follows that the under the unitary transformation associated with a general rotation, the basis vectors {|τ, j, m 〉 } of S(j, τ) are each transformed into new basis vectors for the same invariant subspace. Indeed, it is not hard to see, based upon our earlier description of the rotation process, that a rotation R that takes the unit vector z onto a new direction z ′ will take the basis states |τ, j, m z 〉 of J² and J z onto a new set of basis kets |τ, j, m z ′ 〉 for the same space that are now eigenstates of J² and the component J z ′ of angular momentum along the new rotated direction. These new vectors can, of course, be expressed as linear combinations of the original ones. The picture that emerges is that, under rotations, the vectors |τ, j, m 〉 transform (irreducibly) into linear combinations of themselves.

62 It follows that the under the unitary transformation associated with a general rotation, the basis vectors {|τ, j, m 〉 } of S(j, τ) are each transformed into new basis vectors for the same invariant subspace. Indeed, it is not hard to see, based upon our earlier description of the rotation process, that a rotation R that takes the unit vector z onto a new direction z ′ will take the basis states |τ, j, m z 〉 of J² and J z onto a new set of basis kets |τ, j, m z ′ 〉 for the same space that are now eigenstates of J² and the component J z ′ of angular momentum along the new rotated direction. These new vectors can, of course, be expressed as linear combinations of the original ones. The picture that emerges is that, under rotations, the vectors |τ, j, m 〉 transform (irreducibly) into linear combinations of themselves.

63 This transformation is essentially geometric in nature and is analogous to the way that normal basis vectors in R³ transform into linear combinations of one another. Indeed, by analogy, the coefficients of this linear transformation are identical in any 2j +1 dimensional subspace S(j, τ) of a standard representation having the same value of j, since the basis vectors of such a representation have been constructed using the angular momentum operators in precisely the same fashion. This leads to the concept of rotation matrices, [R (j) ], i.e., a set of standard matrices representing the rotation operators U u (α) in terms of their effect on the vectors within any irreducible invariant subspace S(j, τ). Just as with the matrices representing the components of angular momentum within any irreducible subspace S(j, τ), the elements of the rotation matrices [R (j) ] will depend upon j and m but are independent of τ.

64 This transformation is essentially geometric in nature and is analogous to the way that normal basis vectors in R³ transform into linear combinations of one another. Indeed, by analogy, the coefficients of this linear transformation are identical in any 2j +1 dimensional subspace S(j, τ) of a standard representation having the same value of j, since the basis vectors of such a representation have been constructed using the angular momentum operators in precisely the same fashion. This leads to the concept of rotation matrices, [R (j) ], i.e., a set of standard matrices representing the rotation operators U u (α) in terms of their effect on the vectors within any irreducible invariant subspace S(j, τ). Just as with the matrices representing the components of angular momentum within any irreducible subspace S(j, τ), the elements of the rotation matrices [R (j) ] will depend upon j and m but are independent of τ.

65 This transformation is essentially geometric in nature and is analogous to the way that normal basis vectors in R³ transform into linear combinations of one another. Indeed, by analogy, the coefficients of this linear transformation are identical in any 2j +1 dimensional subspace S(j, τ) of a standard representation having the same value of j, since the basis vectors of such a representation have been constructed using the angular momentum operators in precisely the same fashion. This leads to the concept of rotation matrices, [R (j) ], i.e., a set of standard matrices representing the rotation operators U u (α) in terms of their effect on the vectors within any irreducible invariant subspace S(j, τ). Just as with the matrices representing the components of angular momentum within any irreducible subspace S(j, τ), the elements of the rotation matrices [R (j) ] will depend upon j and m but are independent of τ.

66 This transformation is essentially geometric in nature and is analogous to the way that normal basis vectors in R³ transform into linear combinations of one another. Indeed, by analogy, the coefficients of this linear transformation are identical in any 2j +1 dimensional subspace S(j, τ) of a standard representation having the same value of j, since the basis vectors of such a representation have been constructed using the angular momentum operators in precisely the same fashion. This leads to the concept of rotation matrices, [R (j) ], i.e., a set of standard matrices representing the rotation operators U u (α) in terms of their effect on the vectors within any irreducible invariant subspace S(j, τ). Just as with the matrices representing the components of angular momentum within any irreducible subspace S(j, τ), the elements of the rotation matrices [R (j) ] will depend upon j and m but are independent of τ.

67 In terms of the elements of the rotation matrices, the invariance of the subspaces S(j, τ) under rotations leads to the general relation These matrices are straightforward to compute for low dimensional subspaces, and general formulas have been developed for calculating the matrices for rotations associated with the Euler angles (α,β,γ). For rotations about the z axis the matrices take a particulalrly simple form, since the rotation operator in this case is a simple function of the operator J z of which the states |j,m 〉 are eigenstates. Thus, e.g.,

68 In terms of the elements of the rotation matrices, the invariance of the subspaces S(j, τ) under rotations leads to the general relation These matrices are straightforward to compute for low dimensional subspaces, and general formulas have been developed for calculating the matrices for rotations associated with the Euler angles (α,β,γ). For rotations about the z axis the matrices take a particulalrly simple form, since the rotation operator in this case is a simple function of the operator J z of which the states |j,m 〉 are eigenstates. Thus, e.g.,

69 In terms of the elements of the rotation matrices, the invariance of the subspaces S(j, τ) under rotations leads to the general relation These matrices are straightforward to compute for low dimensional subspaces, and general formulas have been developed for calculating the matrices for rotations associated with the Euler angles (α,β,γ). For rotations about the z axis the matrices take a particulalrly simple form, since the rotation operator in this case is a simple function of the operator J z of which the states |j,m 〉 are eigenstates. Thus, e.g.,

70 In terms of the elements of the rotation matrices, the invariance of the subspaces S(j, τ) under rotations leads to the general relation These matrices are straightforward to compute for low dimensional subspaces, and general formulas have been developed for calculating the matrices for rotations associated with the Euler angles (α,β,γ). For rotations about the z axis the matrices take a particulalrly simple form, since the rotation operator in this case is a simple function of the operator J z of which the states |j,m 〉 are eigenstates. Thus, e.g.,

71 In a subspace with j , for example, this takes the form while in a space with j  we have The point is that, once the rotation matrices have been worked out for a given value of j, they can be used for a standard representation of any quantum mechanical system in which that value of j appears.

72 In a subspace with j , for example, this takes the form while in a space with j  we have The point is that, once the rotation matrices have been worked out for a given value of j, they can be used for a standard representation of any quantum mechanical system in which that value of j appears.

73 In a subspace with j , for example, this takes the form while in a space with j  we have The point is that, once the rotation matrices have been worked out for a given value of j, they can be used for a standard representation of any quantum mechanical system in which that value of j appears.

74