I am confused regarding equations for the square of total angular momentum in quantum mechanics. I understand that summing two independent angular momenta can be represented as $$J_1 \otimes I + I \otimes J_2$$ To evaluate the square of this angular momentum I can then write $$J^2 = (J_1 \otimes 1 + 1 \otimes J_2)(J_1 \otimes 1 + 1 \otimes J_2) = (J_1 \otimes 1 )^2+(1 \otimes J_2)^2+2(J_1 \otimes J_2)$$ But I see in texts that $$J_1 \otimes J_2 = J_{1x} \otimes J_{2x} + J_{1y} \otimes J_{2y}+ J_{1z } \otimes J_{2z}$$ However what about terms such as $$J_{1x} \otimes J_{2y}$$ etc. ? I note that texts may also use the inner product to show that $$J^2=J_1^2+J_2^2+2J_1.J_2$$ which leads to $$J_{1,x}J_{2x}+J_{1y}J_{2y}+J_{1z}J_{2z}$$ as required. But I am also confused by the use of inner products versus tensor products and how the two relate. I would really appreciate some answers to these two questions.
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3 Answers
The total angular momentum is defined as $$\vec J=\vec J_1\otimes 1+1\otimes\vec J_2$$ which, by laziness (answer to your second question), is often written as $$\vec J=\vec J_1+\vec J_2$$ As a consequence, in terms of components $$\eqalign{ J_x&=J_{1x}\otimes 1+1\otimes J_{2x} \cr J_y&=J_{1y}\otimes 1+1\otimes J_{2y} \cr J_z&=J_{1z}\otimes 1+1\otimes J_{2z} \cr }$$ The square of the angular momentum is therefore $$J^2=J_x^2+J_y^2+J_z^2$$ After expanding the squares, you will see that there is no mixing terms like $J_{1x}\otimes J_{2y}$.
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$\begingroup$ Thank you. I am still a bit confused. Perhaps it is best if I work through my logic which will come up with the wrong result so that it will be clear where I am in error. $\endgroup$ Commented 11 hours ago
I am confused regarding equations for the square of total angular momentum in quantum mechanics. I understand that summing two independent angular momenta can be represented as $$J_1 \otimes I + I \otimes J_2$$ To evaluate the square of this angular momentum I can then write $$J^2 = (J_1 \otimes 1 + 1 \otimes J_2)(J_1 \otimes 1 + 1 \otimes J_2) = (J_1 \otimes 1 )^2+(1 \otimes J_2)^2+2(J_1 \otimes J_2)$$
Angular momentum is a vector. The notation used here does not seem to indicate that OP is treating angular momentum as a vector.
However what about terms such as $$J_{1x} \otimes J_{2y}$$ etc. ?
Those terms do not appear.
In component form, we have $$ J_i^2 = {(J^{(1)}_i)}^2\otimes 1 + 1\otimes {(J^{(2)}_i)}^2 + 2J_i^{(1)}\otimes J_i^{(2)} $$ and so $$ J^2 = (J^{(1)})^2\otimes 1 + 1\otimes{( J^{(2)})}^2 + 2\sum_i J_i^{(1)}\otimes J_i^{(2)} $$ $$ =(J^{(1)})^2\otimes 1 + 1\otimes{( J^{(2)})}^2 + 2{\vec J_i^{(1)}}\cdot {\vec J_i^{(2)}}\;, $$ where I didn't explicitly write the $\otimes$ in the last term on the last line.
I would really appreciate some answers to these two questions.
I only see one question.
The correct expressions are $J_x=J^{(1)}_x\otimes I+I\otimes J^{(2)}_x$ etc so summing $J_x^2+J_y^2+J_z^2$ does not produce cross terms.
The mathematical statement $J_x=J^{(1)}_x\otimes I+I\otimes J^{(2)}_x$ is just the statement that the total component of $J$ along $\hat x$ is the sum of the individual components along $\hat x$, and that each part of the sum in $J_x$ acts only on the appropriate subspace of ${\cal H}_1\otimes {\cal H}_2$.
