Addition of Angular Momenta

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.

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kevinferreira

Well isn't
[tex]
\vec{J}=\vec{J}_1+\vec{J}_2
[/tex]?
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

But [itex]\vec{J}[/itex] 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.

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Bill_K

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

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

M. next

Thanks! This was helpful.