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When we combine two Angular Momentum operators $J_1$ and $J_2$, which each exist in the Hilbert Space $H_1$ and $H_2$, $J = J_1 + J_2$, the total Hilbert space for $J$ is $H_1 \otimes H_2$.

In my lecture notes it says $J = J_1 \otimes H_2 + H_1 \otimes J_2$. Shouldn't this be $J = J_1 \otimes H_1 + J_2 \otimes H_2$.

Furthermore, does $H_2 \otimes J_2$ mean the same thing as $J_2 \otimes H_2$.

When you find $J^2$, the expression reads $J^2 = J_1^2 \otimes H_2 + H_1 \otimes J_2^2 + 2(J_1 \otimes J_2)$. But if I expand out the definition of $J^2$:

$J^2 = (J_1 + J_2)(J_1 + J_2) = J_1^2 + J_2^2 + J_1 . J_2 + J_2 . J_1$. However, $J_2 . J_1$ surely can't be equal to $J_1 . J_2$, since the commutator of this isn't zero.

Tobias Fünke
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Icy_Boi
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1 Answers1

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Your lecturer is making a small mistake. Quite a lot of those need to be identity operators, rather than H. But, continuing on,

Furthermore, does $H_2\otimes J_2$ mean the same thing as $J_2\otimes H_2$ [?]

No. This would be the same mistake as ``is (x,y) the same as (y,x) ? "

And it should be identity operators rather than H in the above expression.

In my lecture notes it says $J=J_1\otimes H_2+H_1\otimes J_2$. Shouldn't this be $J=J_1\otimes H_1+J_2\otimes H_2$ [?]

No. This should be $J=J_1\otimes\mathbb I_2+\mathbb I_1\otimes J_2$, which means that the full operator $J$ is gotten by taking the angular momentum operator for part 1 of system and expanding it by the identity operator on part 2, added to the identity operator on part 1 expanded by the angular momentum operator for part 2. What you had, was making angular momentum of part 1 expanded by identity operator of part 1, added to part 2 versions thereof, so that then it will impossible to add. An easy example is the addition of spin half to spin 1, your version will be having matrix sizes mismatched.

However, $J_2\cdot J_1$ surely can't be equal to $J_1\cdot J_2$

You are getting confused. They commute because they are acting on different particles, and the thing we were doing before was $\otimes$ not dot product (even though dot product is clearly going to be in the result of that expression).

Note that you cannot have $J_2\otimes J_1$ because you cannot use the angular momentum operator of the 2nd part to act on part 1, and vice versa, even though this matrix would have the correct matrix size and shape. So, really, it is not about commutation at all, just carefully following the new mathematical system that you are now learning to work with.

naturallyInconsistent
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