1. Jan 14, 2013

### 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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2. Jan 14, 2013

### kevinferreira

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

3. Jan 14, 2013

### M. next

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

4. Jan 14, 2013

### kevinferreira

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?

5. Jan 14, 2013

### M. next

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6. Jan 14, 2013

### 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,

[Jx, Jy] = [J1x, J1y] ⊗ [J2x, J2y] = i J1z ⊗ J2z = i Jz

7. Jan 15, 2013

### kevinferreira

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

8. Jan 15, 2013