How can we add Angular Momenta in Quantum Mechanics?

[M. next]

Hey!

While I was reading some book in Quantum Mechanics, I ran across the following, and couldn't
know how can this be true or actually how was it assumed.

How by adding equation (7.91)and (7.92), we get (7.110), see attachment.

 

[kevinferreira]

Well isn't
$$\vec{J}=\vec{J}_1+\vec{J}_2$$?
Then you can work component by component and obtain the result.

 

[M. next]

Yes, but this is not the 'real' addition, each of the operators you've listed belong to different spaces..

 

[kevinferreira]

Yes, but this is not the 'real' addition, each of the operators you've listed belong to different spaces..

But $\vec{J}$ may be defined in this way on the space defined as the direct sum of the spaces where 1 and 2 act, or not?

 

[M. next]

Please read carefully what's written in the attachment.

 

[Bill_K]

But J may be defined in this way on the space defined as the direct sum of the spaces where 1 and 2 act, or not?

They do act on different subspaces. But actually it's not the direct sum, it's the direct product. To be technical about it, J₁ is really J₁ ⊗ I, and J₂ is really I ⊗ J₂, and J = J₁ + J₂ = J₁ ⊗ J₂.

Now if you focus on two of the components, say x and y components, and look at their commutator,

[J_x, J_y] = [J_1x, J_1y] ⊗ [J_2x, J_2y] = i J_1z ⊗ J_2z = i J_z

 

[kevinferreira]

They do act on different subspaces. But actually it's not the direct sum, it's the direct product. To be technical about it, J₁ is really J₁ ⊗ I, and J₂ is really I ⊗ J₂, and J = J₁ + J₂ = J₁ ⊗ J₂.

Now if you focus on two of the components, say x and y components, and look at their commutator,

[J_x, J_y] = [J_1x, J_1y] ⊗ [J_2x, J_2y] = i J_1z ⊗ J_2z = i J_z

Yes, the direct product, I messed up my operations.

 

[M. next]

Thanks! This was helpful.