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I have a question regarding the invariance of a 'mixed' Casimir operator under rotation,

By 'mixed' Casimir operator I refer to:

[itex]\vec{J}_1\cdot \vec{J}_2[/itex]

Where J_{1}and J_{2}are two independent angular momenta.

I want to show that this 'mixed' Casimir operator is invariant under rotations,

The rotation matrix for J_{1}+ J_{2}will be of the form,

[itex]D(R) = D_1 (R)\otimes D_2 (R)[/itex]

Where D_{1}and D_{2}are the rotation matrices for J_{1}and J_{2}.

So what I am really trying to show is that:

[itex]D^{\dagger}(R) \vec{J}_1\cdot \vec{J}_2 D(R) = \vec{J}_1\cdot \vec{J}_2 [/itex]

But I am having trouble seeing how to do this.

I have read in Schwinger's paper that the rotation matrix for J_{1}+ J_{2}is of the form:

D_{1}D_{2}

But does that mean that [itex]D_2 (R) D_1 (R) = D_1 (R)\otimes D_2 (R)[/itex] ?

If that is the case then I now have:

[itex]D_{2}^{\dagger }D_{1}^{\dagger }{{J}_{1}}{{J}_{2}}{{D}_{1}}{{D}_{2}} [/itex]

And expanding the product gives:

[itex]D_{2}^{\dagger }D_{1}^{\dagger }{{J}_{1,x}}{{J}_{2,x}}{{D}_{1}}{{D}_{2}}+\,\,D_{2}^{\dagger }D_{1}^{\dagger }{{J}_{1,y}}{{J}_{2,y}}{{D}_{1}}{{D}_{2}}+D_{2}^{\dagger} D_{1}^{\dagger }{{J}_{1,z}}{{J}_{2,z}}{{D}_{1}}{{D}_{2}} [/itex]

And some operators can be switched around since the the D_{1}commutes with J_{2}and so on,

But this gets me to here:

[itex]D_{1}^{\dagger }{{J}_{1,x}}{{D}_{1}}D_{2}^{\dagger }{{J}_{2,x}}{{D}_{2}}+\,\,D_{1}^{\dagger }{{J}_{1,y}}{{D}_{1}}D_{2}^{\dagger }{{J}_{2,y}}{{D}_{2}}+D_{1}^{\dagger }{{J}_{1,z}}{{D}_{1}}D_{2}^{\dagger }{{J}_{2,z}}{{D}_{2}} [/itex]

And the D_{1}and J_{1}operators don't necessarily commute so I'm not sure how to progress from here,

Does anyone have any ideas?

Thanks for your time,

Jess

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# Homework Help: Addition of angular momenta, rotation operators

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