Addition of angular momenta, rotation operators

[Jesssa]

Hey,

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

By 'mixed' Casimir operator I refer to:

$\vec{J}_1\cdot \vec{J}_2$

Where J₁ and J₂ are two independent angular momenta.

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

The rotation matrix for J₁ + J₂ will be of the form,

$D(R) = D_1 (R)\otimes D_2 (R)$

Where D₁ and D₂ are the rotation matrices for J₁ and J₂.

So what I am really trying to show is that:

$D^{\dagger}(R) \vec{J}_1\cdot \vec{J}_2 D(R) = \vec{J}_1\cdot \vec{J}_2$

But I am having trouble seeing how to do this.

I have read in Schwinger's paper that the rotation matrix for J₁ + J₂ is of the form:

D₁D₂

But does that mean that $D_2 (R) D_1 (R) = D_1 (R)\otimes D_2 (R)$ ?

If that is the case then I now have:

$D_{2}^{\dagger }D_{1}^{\dagger }{{J}_{1}}{{J}_{2}}{{D}_{1}}{{D}_{2}}$

And expanding the product gives:

$D_{2}^{\dagger }D_{1}^{\dagger }{{J}_{1,x}}{{J}_{2,x}}{{D}_{1}}{{D}_{2}}+\,\,D_{2}^{\dagger }D_{1}^{\dagger }{{J}_{1,y}}{{J}_{2,y}}{{D}_{1}}{{D}_{2}}+D_{2}^{\dagger} D_{1}^{\dagger }{{J}_{1,z}}{{J}_{2,z}}{{D}_{1}}{{D}_{2}}$

And some operators can be switched around since the the D₁ commutes with J₂ and so on,

But this gets me to here:

$D_{1}^{\dagger }{{J}_{1,x}}{{D}_{1}}D_{2}^{\dagger }{{J}_{2,x}}{{D}_{2}}+\,\,D_{1}^{\dagger }{{J}_{1,y}}{{D}_{1}}D_{2}^{\dagger }{{J}_{2,y}}{{D}_{2}}+D_{1}^{\dagger }{{J}_{1,z}}{{D}_{1}}D_{2}^{\dagger }{{J}_{2,z}}{{D}_{2}}$

And the D₁ and J₁ 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

 

[Oxvillian]

But does that mean that $D_2 (R) D_1 (R) = D_1 (R)\otimes D_2 (R)$ ?

That looks right to me because the rotations in the two different spaces must commute.

I think you can get your result immediately by using the fact that any J is a "vector operator",
$$D^{\dagger} J_i D = R_{ik}J_k.$$

 

[Jesssa]

Hey Oxvillan,

Thanks a lot for your reply,

Sorry what I was trying to ask was if the tensor product between D_1 and D_2 is equal to the multiplication of the two rotation matrices, D_1 D_2. I have worked out why this is true now.

Sorry I just realized I was inconsistent with my indexes and wrote the 1 and 2 back to front.

Would you happen to know where I could find more information regarding

$$D^{\dagger} J_i D = R_{ik}J_k.$$

The i = x, y or z right? But what does the k represent?

Thanks again for your reply!

 

[Oxvillian]

Hey Oxvillan,

Thanks a lot for your reply,

Sorry what I was trying to ask was if the tensor product between D_1 and D_2 is equal to the multiplication of the two rotation matrices, D_1 D_2. I have worked out why this is true now.

Ah I see

I would write something like

$$D_1 D_2 = (D \otimes I)(I \otimes D) = D \otimes D.$$

Would you happen to know where I could find more information regarding

$$D^{\dagger} J_i D = R_{ik}J_k.$$

The i = x, y or z right? But what does the k represent?

Thanks again for your reply!

I would recommend Shankar chapter 12.

The index k also runs from 1 to 3 (for x,y and z). R_ik is just a rotation matrix.

The idea here is that we have a "unitary representation of the rotation group". Ordinary 3-vectors are rotated using rotation matrices, but state vectors in a Hilbert space are rotated using unitary operators. For things to be consistent, a vector operator (eg. the set of operators {J_x, J_y, J_z}) has to transform according to the relation I gave.

 

[Jesssa]

Cool thanks a lot!

So this reduces

$D_{1}^{\dagger }{{J}_{1,x}}{{D}_{1}}D_{2}^{\dagger }{{J}_{2,x}}{{D}_{2}}+\,\,D_{1}^{\dagger }{{J}_{1,y}}{{D}_{1}}D_{2}^{\dagger }{{J}_{2,y}}{{D}_{2}}+D_{1}^{\dagger }{{J}_{1,z}}{{D}_{1}}D_{2}^{\dagger }{{J}_{2,z}}{{D}_{2}}$

To

${{R}_{1,xk}}{{J}_{1,x}}{{R}_{2,xk}}{{J}_{2,x}}+\,\,{{R}_{1,yk}}{{J}_{1,y}}{{R}_{2,yk}}{{J}_{2,y}}+{{R}_{1,zk}}{{J}_{1,z}}{{R}_{2,zk}}{{J}_{2,z}}$

Then as the R_2 matrices commute with the J_1 operators

${{R}_{1,xk}}{{R}_{2,xk}}{{J}_{1,x}}{{J}_{2,x}}+\,\,{{R}_{1,yk}}{{R}_{2,yk}}{{J}_{1,y}}{{J}_{2,y}}+{{R}_{1,zk}}{{R}_{2,zk}}{{J}_{1,z}}{{J}_{2,z}}$

Does this look right?

So this implies that the two rotation matrices in front of each J_1 J_2 component must equal to the identity matrix, but how is this possible?

 

[Oxvillian]

You're almost there!

I'd recommend just using the Einstein summation convention - then you only have to write out a third as many terms...

There should not be 1 or 2 subscripts on the R's because R is just your rotation matrix:

$${\bf J}_1 \cdot {\bf J}_2 = J_{1i} J_{2i} → R_{ip} J_{1p} R_{iq} J_{2q}$$
One more little trick and you're home and dry...

 

[Jesssa]

I am having a bit of trouble seeing what the trick is,

Does

$$R_{ip} J_{1p} R_{iq} J_{2q}=J_{1p} J_{2q}$$

Because the whole system is being rotated by the same rotation?

Then p and q both run from 1-3 right, (x y and z)?

But I'm really not sure,

Or is it something like $$R_{iq}=R^T_{ip} ?$$

Thanks again for helping Oxvillian

 

[Oxvillian]

I am having a bit of trouble seeing what the trick is,

Does

$$R_{ip} J_{1p} R_{iq} J_{2q}=J_{1p} J_{2q}$$

No, that wouldn't make sense because p and q are dummy indices that we are summing over,
$$\sum_i \sum_p \sum_q R_{ip} J_{1p} R_{iq} J_{2q}.$$
What you want to do first is sum over i. Take a look at
$$\sum_i R_{ip} R_{iq} = \sum_i R^T_{pi} R_{iq}.$$
How can you simplify that, bearing in mind that R is a rotation matrix?

 

[Jesssa]

Do you mean that the R's are orthogonal (since they're rotation matrices) so

$R^T_{pi}R_{iq} = \hat{1}$

Is that correct? It seems like the indices would make it incorrect.

Haha, I don't know why but the indices are giving me the most confusing,

How come it is okay to write,

$R^T_{pi} = R_{ip}$

Sorry if I am being a nuisance.

 

[Oxvillian]

Do you mean that the R's are orthogonal (since they're rotation matrices) so

$R^T_{pi}R_{iq} = \hat{1}$

Is that correct? It seems like the indices would make it incorrect.

Haha, I don't know why but the indices are giving me the most confusing,

You're mixing up the matrix notation and the index notation here. The matrix notation is
$$R^T R = I$$
while the index notation would be
$$\sum_i R^T_{pi} R_{iq} = \delta_{pq}$$

How come it is okay to write,

$R^T_{pi} = R_{ip}$

That's just the definition of the transpose of a matrix

 

[Jesssa]

Ah okay cool,

Thanks!

So

$$\sum_i R_{ip} R_{iq} = \sum_i R^T_{pi} R_{iq} = \delta_{pq}$$

Then

$$\sum_i \sum_p \sum_q R_{ip} J_{1p} R_{iq} J_{2q}= \sum_p \sum_q \delta_{pq} J_{1p} J_{2q} = \sum_p J_{1p} J_{2p} = J_{1x}J_{2x} + J_{1y}J_{2y} + J_{1z}J_{2z}$$

Which is what I needed!

Thanks heaps!